# The Chow ring of the moduli space of degree $2$ quasi-polarized K3 surfaces

Samir Canning  Department of Mathematics, ETH Zürich
<EMAIL_ADDRESS>,  Dragos Oprea  Department of Mathematics,
University of California, San Diego<EMAIL_ADDRESS>and  Rahul
Pandharipande  Department of Mathematics, ETH Zürich<EMAIL_ADDRESS>
(Date: June 2024)

###### Abstract.

We study the Chow ring with rational coefficients of the moduli space
$\mathcal{F}_{2}$ of quasi-polarized $K3$ surfaces of degree $2$. We find
generators, relations, and calculate the Chow Betti numbers. The highest
nonvanishing Chow group is
$\mathsf{A}^{17}(\mathcal{F}_{2})\cong{\mathbb{Q}}$. We prove that the Chow
ring consists of tautological classes and is isomorphic to the even
cohomology. The Chow ring is not generated by divisors and does not satisfy
duality with respect to the pairing into $\mathsf{A}^{17}(\mathcal{F}_{2})$.
The kernel of the pairing is a 1-dimensional subspace of
$\mathsf{A}^{9}(\mathcal{F}_{2})$ which we calculate explicitly. In the
appendix, we revisit Kirwan-Lee’s calculation of the Poincaré polynomial of
$\mathcal{F}_{2}$.

###### Contents

  1. 1 Introduction
  2. 2 Tautological classes
  3. 3 Shah’s construction
  4. 4 Relations from the locus of multiple lines
  5. 5 Relations from quadruple points
  6. 6 The weighted blowup
  7. 7 The cycle map
  8. A The Poincaré polynomial of the moduli space

##  1\. Introduction

###  1.1. Main results

Let $\mathcal{F}_{2\ell}$ denote the moduli space of quasi-polarized $K3$
surfaces of degree $2\ell$. The space $\mathcal{F}_{2\ell}$ is a nonsingular
Deligne-Mumford stack of dimension $19$. We consider the Chow ring
$\mathsf{A}^{*}(\mathcal{F}_{2\ell})$, which will be taken with
$\mathbb{Q}$-coefficients throughout the paper. The Chow of ring admits a
tautological subring

$\mathsf{R}^{*}(\mathcal{F}_{2\ell})\subset\mathsf{A}^{*}(\mathcal{F}_{2\ell})\,,$

which was defined in [MOP] and will be reviewed in Section 2.

We focus here on the moduli space $\mathcal{F}_{2}$ of $K3$ surfaces of degree
$2\ell=2$. The generic $K3$ surface $(S,L)\in\mathcal{F}_{2}$ is a double
cover

$\epsilon:S\rightarrow{\mathbb{P}^{2}}$

branched along a nonsingular sextic curve with quasi-polarization
$L=\epsilon^{*}{\mathcal{O}}_{\mathbb{P}^{2}}(1)$. The geometry of
$\mathcal{F}_{2}$ is therefore closely related to the classical geometry of
sextic plane curves.

######  Theorem 1.

The following results hold for the Chow ring
$\mathsf{A}^{*}(\mathcal{F}_{2})$:

  * (i)

The Chow ring is tautological,

$\mathsf{R}^{*}(\mathcal{F}_{2})=\mathsf{A}^{*}(\mathcal{F}_{2})\,,$

and is generated by 4 elements of degrees $1,1,2,3$ respectively.

  * (ii)

$\mathsf{R}^{17}(\mathcal{F}_{2})=\mathbb{Q}$ and
$\mathsf{R}^{18}(\mathcal{F}_{2})=\mathsf{R}^{19}(\mathcal{F}_{2})=0$.

  * (iii)

The dimensions are given by

$\displaystyle\sum_{k=0}^{19}t^{k}\dim\mathsf{R}^{k}(\mathcal{F}_{2})=1+$
$\displaystyle 2t+3t^{2}+5t^{3}+6t^{4}+8t^{5}+10t^{6}+12t^{7}+13t^{8}+$
$\displaystyle+14t^{9}+12t^{10}+10t^{11}+8t^{12}+6t^{13}+5t^{14}+3t^{15}+2t^{16}+t^{17}.$

  * (iv)

The cycle class map is an isomorphism onto the even cohomology:

$\forall k\geq 0\,,\ \ \ \mathsf{R}^{k}(\mathcal{F}_{2})\cong
H^{2k}(\mathcal{F}_{2})\,.$

Theorem 1 is the first complete Chow calculation for the moduli spaces
${\mathcal{F}}_{2\ell}$ of quasi-polarized $K3$ surfaces. There are several
immediate connections and consequences:

$\bullet$ The generators (i) of $\mathsf{R}^{*}(\mathcal{F}_{2})$ are not all
divisor classes. Indeed, the Chow Betti numbers given in (iii) grow too
quickly to be generated by the two divisor classes.

$\bullet$ In the proof of Theorem 1, we determine the structure of the Chow
ring of $\mathcal{F}_{2}$: there are $14$ relations111The relations are
recorded in Remark 7 (3 relations), Remark 9 (3 relations), Remark 13 (2
relations), Remark 17 and Remark 18 (together 6 relations). In these 14
relations, the $4$ generators are denoted by $H$ (degree 1), $\mathsf{Ell}$
(degree 1), $c_{2}$ (degree $2$), $c_{3}$ (degree 3). between the 4 generators
(i).

$\bullet$ The socle and vanishing results of (ii) were proposed earlier as
analogues of Faber’s conjectures [Fa] for the tautological ring of the moduli
space of curves $\mathcal{M}_{g}$:

$\mathsf{R}^{>g-2}(\mathcal{M}_{g})=0\,,\quad\mathsf{R}^{g-2}(\mathcal{M}_{g})=\mathbb{Q}\,.$

######  Conjecture 2 (Oprea–Pandharipande (2015)).

Let $\Gamma$ be the Picard lattice for a K3 surface such that $d=20-\text{\em
rank}(\Gamma)>3$. For the moduli space $\mathcal{F}_{\Gamma}$ of
$\Gamma$-polarized $K3$ surfaces, we have

$\mathsf{R}^{d-2}(\mathcal{F}_{\Gamma})\cong\mathbb{Q}\ \ \ \text{and}\ \ \
\mathsf{R}^{d-1}(\mathcal{F}_{\Gamma})=\mathsf{R}^{d}(\mathcal{F}_{\Gamma})=0\,.$

In cohomology with ${\mathbb{Q}}$-coefficients, the vanishing part of
Conjecture 2 is established in [Pe1]. For the hyperbolic lattice $U$, the
moduli space $\mathcal{F}_{U}$ corresponds to elliptic $K3$ surfaces with
section. The socle and Chow vanishing properties of Conjecture 2 for
${\mathcal{F}}_{U}$ are established in [CK]. The moduli space
$\mathcal{F}_{2}$ is the first rank 1 case where Conjecture 2 is proven.

$\bullet$ The Chow Betti number calculation (iii) shows that the pairing into
$\mathsf{R}^{17}(\mathcal{F}_{2})$ is not perfect because the middle
dimensions are not equal:

$\dim\mathsf{R}^{8}(\mathcal{F}_{2})\neq\dim\mathsf{R}^{9}(\mathcal{F}_{2})\,.$

In fact, the kernel of the pairing

$\mathsf{R}^{8}(\mathcal{F}_{2})\times\mathsf{R}^{9}(\mathcal{F}_{2})\to\mathsf{R}^{17}(\mathcal{F}_{2})$

has rank $1$. A formula for the class in $\mathsf{R}^{9}(\mathcal{F}_{2})$
which spans the Gorenstein kernel is given in Remark 21. Finding a geometric
interpretation of the kernel class is a very interesting open direction. For
$k$ not equal to 8 or 9, the pairing

$\mathsf{R}^{k}(\mathcal{F}_{2})\times\mathsf{R}^{17-k}(\mathcal{F}_{2})\to\mathsf{R}^{17}(\mathcal{F}_{2})$

is perfect. The results about the pairing are proven using our explicit
presentation for $\mathsf{R}^{*}(\mathcal{F}_{2})$.

$\bullet$ A construction of the moduli space $\mathcal{F}_{2}$ as an open
subset of a weighted blowup of the moduli space of sextic curves in
$\mathbb{P}^{2}$ is given in [S], see [La, Lo] for a summary. A partial
desingularization of the full GIT compactification of the space of sextics is
obtained in [KL1] and requires three further blowups. Using all four blowups,
the cohomology of $\mathcal{F}_{2}$ was studied in [KL1, KL2]. A main result
of [KL2] is the calculation of the Poincaré polynomial222The value of the
Poincaré polynomial given in [KL2] is
$1+2q^{2}+3q^{4}+5q^{6}+6q^{8}+8q^{10}+10q^{12}+12q^{14}+13q^{16}+14q^{18}+12q^{20}\\\
+10q^{22}+8q^{24}+6q^{26}+q^{27}+5q^{28}+3q^{30}+q^{31}+2q^{32}+q^{33}+3q^{35}\,,$
which differs from the statement above by $q^{33}+q^{34}$. We will explain the
necessary correction in Appendix A.

$\sum_{k=0}^{38}q^{k}\dim
H^{k}(\mathcal{F}_{2})=1+2q^{2}+3q^{4}+5q^{6}+6q^{8}+8q^{10}+10q^{12}+12q^{14}+13q^{16}+14q^{18}+12q^{20}\\\
+10q^{22}+8q^{24}+6q^{26}+q^{27}+5q^{28}+3q^{30}+q^{31}+2q^{32}+2q^{33}+q^{34}+3q^{35}.$

While there are odd cohomology classes, the dimensions of the even cohomology
of $\mathcal{F}_{2}$ match the Chow Betti numbers (iii) as required by (iv).

$\bullet$ By a result of [BLMM], the even cohomology
$H^{2k}(\mathcal{F}_{2\ell})$ for $k\leq 4$ is tautological for all $\ell\geq
1$. Isomorphism (iv) is a much stronger property which holds for the moduli
space $\mathcal{F}_{2}$.

###  1.2. Plan of the paper

Definitions and basic results related to tautological classes on moduli spaces
of $K3$ surfaces are reviewed in Section 2. Our approach to the geometry of
$\mathcal{F}_{2}$ relies upon Shah’s blowup construction [S] which is
discussed in Section 3. Part (i) of Theorem 1 is proven in Section 3.2.

Shah describes $\mathcal{F}_{2}$ as an open subset of a weighted blowup of the
space of sextic plane curves. The heart of our Chow calculation for
$\mathcal{F}_{2}$ is presented in Sections 4–6, where relations obtained from
the removal of various loci are determined. Parts (ii) and (iii) of Theorem 1
are proven in Section 6.

Part (iv) of Theorem 1, the isomorphism of the cycle class map onto the even
cohomology, is proven in Section 7.

The Kirwan-Lee calculation of the Poincaré polynomial of $\mathcal{F}_{2}$ is
discussed carefully in Appendix A. In particular, we explain how to correct
the calculations in [KL1, KL2].

###  1.3. Future directions

While complete Chow calculations for the moduli spaces $\mathcal{F}_{2\ell}$
will likely become intractable for large $\ell$, the study of
$\mathcal{F}_{4}$ should be possible as there is a parallel (though more
complicated) construction starting from the moduli of quartic surfaces, see
[LOG].

Another direction of study is to find structure in the tautological ring
$\mathsf{R}^{*}(\mathcal{F}_{2\ell})$ beyond Conjecture 2. The parallel
direction in the study of the moduli space of curves has led to the surprising
discovery of uniform sets of tautological relations, see [P] for a survey.
Whether any analogues of the Faber-Zagier and Pixton relations hold for the
moduli of $K3$ surfaces is an interesting question.

Finding algebraic cycle classes on $\mathcal{F}_{2\ell}$ which are non-
tautological in cohomology is another open direction. Since such classes for
the moduli of curves and abelian varieties can be constructed using the
geometry of Hurwitz covers of higher genus curves [COP2, GP, Z], a simple idea
for $K3$ surfaces is the following. Let
$\mathcal{B}_{g}\subset{\mathcal{F}}_{2\ell}$ be the closure of the locus of
$K3$ surfaces for which there exists a nonsingular linear section (of genus
$\ell+1$) admitting a degree 2 map to a genus $g\geq 1$ curve. A reasonable
expectation is that the image in cohomology of the algebraic cycle class

$[\mathcal{B}_{g}]\in\mathsf{A}^{*}(\mathcal{F}_{2\ell})$

is non-tautological for appropriate choices of $g$ and $\ell$.

### Acknowledgments

We thank Dan Abramovich, Veronica Arena, Adrian Clingher, Carel Faber, Frances
Kirwan, Bochao Kong, Hannah Larson, Zhiyuan Li, Alina Marian, Davesh Maulik,
Stephen Obinna, Dan Petersen, and Burt Totaro for many related conversations
over the years. We are very grateful to the referee for several comments and
corrections.

An earlier attempt to calculate the Chow ring of ${\mathcal{F}}_{2}$ with
Qizheng Yin and Fei Si did not succeed (due to the geometric and computational
complexity of the approach). A different path is taken here, leading to
several simplifications. We thank Qizheng Yin and Fei Si for the previous
collaboration.

S.C. was supported by a Hermann-Weyl-Instructorship from the
Forschungsinstitut für Mathematik at ETH Zürich. D.O. was supported by NSF-DMS
1802228. R.P. was supported by SNF-200020-182181, SNF-200020-219369,
ERC-2017-AdG-786580-MACI, and SwissMAP.

This project has received funding from the European Research Council (ERC)
under the Euro- pean Union Horizon 2020 research and innovation program (grant
agreement No. 786580).

##  2\. Tautological classes

We review here the definition of the tautological rings of the moduli spaces
$\mathcal{F}_{2\ell}$. Consider the universal $K3$ surface and quasi-
polarization

$\pi:\mathcal{S}\to\mathcal{F}_{2\ell}\ ,\ \ \ \mathcal{L}\to\mathcal{S}\,.$

The most basic tautological classes are:

$\bullet$ Hodge classes. The dual Hodge bundle is defined as the pushforward

$\mathbb{E}^{\vee}={\mathbf{R}}^{2}\pi_{*}{\mathcal{O}}_{\mathcal{S}}\to\mathcal{F}_{2\ell}\,.$

Let $\lambda\in\mathsf{A}^{1}({\mathcal{F}}_{2\ell})$ denote the first Chern
class

$\lambda=c_{1}(\mathbb{E})=-c_{1}\left({\mathbf{R}}^{2}\pi_{*}{\mathcal{O}}_{\mathcal{S}}\right)\,.$

The ring
$\Lambda^{*}(\mathcal{F}_{2\ell})\subset\mathsf{A}^{*}({\mathcal{F}}_{2\ell})$
generated by powers of $\lambda$ was studied in [GK]. It is shown there that

$\Lambda^{*}(\mathcal{F}_{2\ell})=\mathbb{Q}[\lambda]/(\lambda^{18})\,.$

The Chern classes of the tangent bundle of $\mathcal{F}_{2\ell}$ belong to the
ring $\Lambda^{*}.$ In fact, by [GK] (see the paragraph following Proposition
3.2) we have

$\mathrm{ch}(T\mathcal{F}_{2\ell})=-1+21e^{-\lambda}-e^{-2\lambda}\,.$

$\bullet$ Noether-Lefschetz classes. Let $\Gamma$ be an even lattice of
signature $(1,r-1)$ for an integer $r\leq 19$. Consider the moduli space
$\mathcal{F}_{\Gamma}$ parametrizing $\Gamma$-polarized $K3$ surfaces:

$\iota:\Gamma\to\text{Pic}(S)\,,$

with the image of $\iota$ containing a quasi-polarization of the surface $S$.
Upon fixing a primitive $v\in\Gamma$ with $v^{2}=2\ell$ mapping to the
quasipolarization, there is a forgetful map

$\mathcal{F}_{\Gamma}\to\mathcal{F}_{2\ell}\,,$

whose image determines a Noether-Lefschetz cycle in $\mathcal{F}_{2\ell}$.
Define

$\mathsf{NL}^{*}(\mathcal{F}_{2\ell})\subset\mathsf{A}^{*}(\mathcal{F}_{2\ell})$

to be the $\mathbb{Q}$-subalgebra generated by all Noether-Lefschetz cycle
classes. A more extensive discussion of the Noether-Lefschetz cycles can be
found for instance in [MP].

$\bullet$ Kappa classes. Let $T_{\pi}^{\text{rel}}$ be the relative tangent
bundle of the universal surface $\pi:\mathcal{S}\to\mathcal{F}_{2\ell}$. The
first Chern class is related to the Hodge class,

$c_{1}\left(T^{\text{rel}}_{\pi}\right)=-\pi^{*}\lambda\,.$

Define $t=c_{2}\left(T_{\pi}^{\text{rel}}\right)$. By [GK, Proposition 3.1],
the pushforwards $\kappa_{0,n}=\pi_{*}(t^{n})$ are contained in $\Lambda^{*}$.
More general kappa classes can be defined by including the class of the quasi-
polarization $c_{1}(\mathcal{L})$ and considering the pushforwards

$\kappa_{m,n}=\pi_{*}\left(c_{1}(\mathcal{L})^{m}\cdot t^{n}\right)\,.$

These classes depend upon the normalization of $\mathcal{L}$ by line bundles
pulled back from $\mathcal{F}_{2\ell}$. By defining canonical normalizations
for admissible $\mathcal{L}$, the ambiguity can be removed, see [PY].

More generally, we define enriched kappa classes over $\mathcal{F}_{\Gamma}$
and consider their pushforwards to $\mathcal{F}_{2\ell}$. After picking a
basis $\mathsf{B}$ for $\Gamma$, we obtain line bundles
$\mathcal{L}_{1},\ldots,\mathcal{L}_{r}\to\mathcal{F}_{\Gamma}$, and we set

(1)
$\kappa^{\Gamma,\mathsf{B}}_{a_{1},\ldots,a_{r},b}=\pi_{*}\left(c_{1}(\mathcal{L}_{1})^{a_{1}}\cdot\ldots\cdot
c_{1}(\mathcal{L}_{r})^{a_{r}}\cdot
c_{2}(T^{\textrm{rel}}_{\pi})^{b}\right)\,.$

The tautological ring of the moduli space ${\mathcal{F}_{2\ell}}$,

$\mathsf{R}^{*}(\mathcal{F}_{2\ell})\subset\mathsf{A}^{*}(\mathcal{F}_{2\ell})\,,$

is defined in [MOP] to be the $\mathbb{Q}$-subalgebra generated by the
pushforwards of all enriched kappa classes (1) for all possible $\Gamma$.

By definition,
$\mathsf{R}^{*}(\mathcal{F}_{2\ell})\supset\mathsf{NL}^{*}(\mathcal{F}_{2\ell})$.
The following isomorphism was proven in [PY].

######  Theorem 3 (Pandharipande–Yin (2020)).

For all $\ell\geq 1$, we have
$\mathsf{R}^{*}(\mathcal{F}_{2\ell})=\mathsf{NL}^{*}(\mathcal{F}_{2\ell})$.

Stronger results hold for divisor classes. In codimension $1$, the isomorphism

$\text{Pic}(\mathcal{F}_{2\ell})_{\mathbb{Q}}=\mathsf{NL}^{1}(\mathcal{F}_{2\ell})$

was conjectured in [MP] and proven in [BLMM]. Combined with [B], this
isomorphism determines the Picard rank. The integral Picard group has recently
been considered in [LFV]. Furthermore, bases for the rational Picard group for
small $\ell$ are given in [OG] and [GLT].

For ${\mathcal{F}_{2}}$, consider the divisor $\mathsf{Ell}$ of elliptic $K3$
surfaces with a section and the divisor $\mathsf{Sing}$ of $K3$ surfaces for
which the quasi-polarization fails to be ample. These are Noether-Lefschetz
divisors corresponding to the lattices

$\begin{bmatrix}-2&1\\\ 1&0\end{bmatrix}\ \ \ \text{and}\ \ \
\begin{bmatrix}2&0\\\ 0&-2\end{bmatrix}\,,$

respectively. By [OG], we have

$\text{Pic}(\mathcal{F}_{2})_{\mathbb{Q}}=\langle\left[\mathsf{Ell}\right],\left[\mathsf{Sing}\right]\rangle$

which is consistent with part (iii) of Theorem 1. In fact, the arguments of
[OG] or alternatively [MOP, Proposition 1] imply that $\lambda$ is a
combination of $\left[\mathsf{Ell}\right]$ and $\left[\mathsf{Sing}\right]$
with nonzero coefficients. Thus, we also have

(2)
$\text{Pic}(\mathcal{F}_{2})_{\mathbb{Q}}=\langle\lambda,\left[\mathsf{Ell}\right]\rangle\,.$

##  3\. Shah’s construction

###  3.1. Overview

We review here the construction of $\mathcal{F}_{2}$ described in [S]. We
start with the moduli space of sextics with suitably restricted singularities.
The moduli space $\mathcal{F}_{2}$ is then obtained as an open subset of a
weighted blowup of the locus of triple conics. Using the construction, we will
prove part (i) of Theorem 1.

###  3.2. ADE sextics

By a result of [Ma], a quasi-polarized $K3$ surface $(S,L)$ of degree $2$ must
take one of the following two geometric forms:

  * (a)

$S$ is the resolution of a double cover
$\epsilon:\widehat{S}\to\mathbb{P}^{2}$ branched along a sextic curve $C$ with
$L=\epsilon^{*}{\mathcal{O}}_{\mathbb{P}^{2}}(1)$. For $S$ to be a nonsingular
$K3$ surface, $C$ must have ADE singularities [BHPV].

  * (b)

$S$ is an elliptic fibration $S\to\mathbb{P}^{1}$ with fiber class $f$,
section $\sigma$, and $L=\sigma+2f$.

Surfaces $(S,L)$ of form (b) constitute the Noether-Lefschetz divisor
$\mathsf{Ell}$ in $\mathcal{F}_{2}$. The complement
$\mathcal{F}_{2}\smallsetminus\mathsf{Ell}$ is a gerbe banded by a finite
group over the moduli space of plane sextics with ADE singularities.

We will construct the moduli space of plane sextics with ADE singularities as
a quotient stack. Let $V=\mathbb{C}^{3}$, and let

$X=\mathbb{P}(\operatorname{Sym}^{6}V^{*})=\mathbb{P}^{27}$

be the space of all sextic curves in $\mathbb{P}(V)$. A sextic
$C\subset\mathbb{P}^{2}$ has at worst ADE singularities if and only if the
following three conditions are satisfied simultaneously

  * (i)

$C$ is reduced,

  * (ii)

$C$ does not contain a consecutive triple point,

  * (iii)

$C$ does not contain a quadruple point.

By definition, a consecutive triple point $p\in C$ is a triple point such that
the strict transform of $C$ in the blowup of $\mathbb{P}^{2}$ at $p$ continues
to admit a triple point. In local coordinates $(x,y)$ around $p$, the equation
for $C$ lies in the ideal $(x,y^{2})^{3}$. We write $\mathsf{NR}$,
$\mathsf{CTP}$, $\mathsf{QP}$ for the three loci failing conditions (i)-(iii)
respectively. We have a morphism

(3)
$\mathcal{F}_{2}\smallsetminus\mathsf{Ell}\rightarrow\left[\left(X\smallsetminus(\mathsf{NR}\cup\mathsf{CTP}\cup\mathsf{QP})\right)/\operatorname{PGL}(V)\right]\,,$

of degree $1/2$ (because of the extra $\mathbb{Z}_{2}$ automorphisms of the
$K3$ surfaces). We also have a degree $1/3$ morphism

(4)
$\left[\left(X\smallsetminus(\mathsf{NR}\cup\mathsf{CTP}\cup\mathsf{QP})\right)/\operatorname{SL}(V)\right]\rightarrow\left[\left(X\smallsetminus(\mathsf{NR}\cup\mathsf{CTP}\cup\mathsf{QP})\right)/\operatorname{PGL}(V)\right]\,.$

induced by the map $\operatorname{SL}(V)\rightarrow\operatorname{PGL}(V)$.
Both (3) and (4) yield isomorphisms on Chow groups with
$\mathbb{Q}$-coefficients via pullback. The group $G=\operatorname{SL}(V)$
will be geometrically simpler for us to work with because
$\operatorname{BSL}(V)$ carries a universal vector bundle.

The locus $\mathsf{NR}$ is reducible with components corresponding to the
images of the maps

$m_{r}:\mathbb{P}(\operatorname{Sym}^{6-2r}V^{*})\times\mathbb{P}(\operatorname{Sym}^{r}V^{*})\rightarrow
X,\quad m_{r}(f,g)=fg^{2}$

for $r=1,2,3$. Let $\mathsf{ML}\subset\mathsf{NR}$ denote the irreducible
component of $\mathsf{NR}$ corresponding to sextics containing a multiple line
(the image of $m_{1}$). The locus $\mathsf{ML}$ has codimension $11$ in $X$.
The other components of $\mathsf{NR}$ are the double conic locus (the image of
$m_{2}$) and the double cubic locus (the image of $m_{3}$). These have
codimensions $17$ and $18$ in $X$. We will consider these two components
together, thus writing $\mathsf{DC}$ for their union. The quasi-projective
locus of cubes of nonsingular conics will be denoted $\mathsf{TC}$. It is easy
to see that the loci

$\mathsf{QP},\,\mathsf{CTP},\,\mathsf{TC}\subset X$

have codimensions $8$, $9$ and $22$ respectively.

###### Proof of Theorem 1 (i)..

Throughout the paper, we identify Chow groups of quotient stacks with the
corresponding equivariant Chow groups [EG, Kr].

We first consider the Chow classes in

$\mathsf{A}^{*}(\mathcal{F}_{2}\smallsetminus\mathsf{Ell})=\mathsf{A}^{*}_{G}(X\smallsetminus(\mathsf{NR}\cup\mathsf{CTP}\cup\mathsf{QP}))\,.$

Note that $\mathsf{A}_{G}^{*}=\mathbb{Z}[c_{2},c_{3}]$, where $c_{2},c_{3}$
are the second and third Chern classes of the universal bundle over
$\mathrm{B}G$. The $\mathsf{A}_{G}^{*}$-algebra $\mathsf{A}_{G}^{*}(X)$ is
generated by $H$, the equivariant hyperplane class. Since we have a surjection

$\mathsf{A}_{G}^{*}(X)\to\mathsf{A}^{*}_{G}(X\smallsetminus(\mathsf{NR}\cup\mathsf{CTP}\cup\mathsf{QP}))\,,$

we see that $\mathsf{A}^{*}(\mathcal{F}_{2}\smallsetminus\mathsf{Ell})$ is
generated by the images of $H,$ $c_{2},$ $c_{3}$. By (2),
$\mathsf{Pic}(\mathcal{F}_{2})$ is generated by $\lambda$ and
$\left[\mathsf{Ell}\right]$. The class $H$ is the restriction of a linear
combination $a\lambda+b[\mathsf{Ell}]$ to
$\mathcal{F}_{2}\smallsetminus\mathsf{Ell}$, and therefore must be a multiple
of $\lambda$. Thus, the ring
$\mathsf{A}^{*}(\mathcal{F}_{2}\smallsetminus\mathsf{Ell})$ is generated by
$\lambda,$ $c_{2},$ $c_{3}.$

Over the locus $\mathcal{F}_{2}\smallsetminus\mathsf{Ell}$, the classes
$c_{2},$ $c_{3}$ are the Chern classes of a rank $3$ vector bundle
$\mathcal{V}$ which can be explicitly described. Let

$\pi:(\mathcal{S},\mathcal{L})\to\mathcal{F}_{2}$

be the universal surface and the universal polarization. For surfaces in the
locus $\mathcal{F}_{2}\smallsetminus\mathsf{Ell}$, the quasi-polarization
$L\to S$ is globally generated inducing a morphism
$S\to\mathbb{P}(H^{0}(S,L))$. Then $c_{2},c_{3}$ are the Chern classes of the
bundle333 The third root $\det(\pi_{*}(\mathcal{L}))^{-\frac{1}{3}}$ can
either be viewed formally for the Chern class calculation or can be viewed as
an actual line bundle on the fiber product of (3) and (4).

$\mathcal{V}=\pi_{*}\left(\mathcal{L}\right)\otimes\det(\pi_{*}\left(\mathcal{L})\right)^{-\frac{1}{3}}\,.$

A Grothendieck-Riemann-Roch calculation explained in shows that $c_{2},c_{3}$
are the restrictions of tautological classes on
$\mathcal{F}_{2}\smallsetminus\mathsf{Ell}.$ The Grothendieck-Riemann-Roch
calculation will be made more precise in Remark 5 below.

As shown in [CK, Theorem 4.1], the Chow classes on $\mathsf{Ell}$ are
polynomials in (the restrictions of) $\lambda$ and a certain codimension $2$
class. In fact, there is a quotient presentation

$\mathsf{Ell}=[Z/\operatorname{PSL}_{2}]\to\textrm{BPSL}_{2}$

and the codimension $2$ class is obtained by pullback from the base. As we
will see below, the stack $\mathcal{F}_{2}$ also admits a quotient
presentation

$\mathcal{F}_{2}=[W/G]\to\textrm{B}G$

hence the class $c_{2}$ also makes sense on $\mathcal{F}_{2}$ by pullback from
the base. Furthermore, there is a natural morphism

$\operatorname{PSL}_{2}\to G\,,\ \ \ g\mapsto\operatorname{Sym}^{2}g$

compatible with the above maps. Geometrically, this corresponds to the fact
that the linear series $|L|$ induces a map from $S$ to a nonsingular conic in
$\mathbb{P}^{2}$ in the elliptic case. Consequently, the restriction of
$c_{2}$ from $\mathcal{F}_{2}$ to $\mathsf{Ell}$ can be chosen to be the
degree $2$ generator on $\mathsf{Ell}.$ Thus $\mathsf{A}^{*}(\mathsf{Ell})$ is
generated by $\lambda$ and $c_{2}$. Using the push-pull formula, we conclude
that the image of $\mathsf{A}^{*}(\mathsf{Ell})$ in
$\mathsf{A}^{*}(\mathcal{F}_{2})$ lies in the subring generated by
$[\mathsf{Ell}],$ $\lambda,$ $c_{2}.$

By excision, there is an exact sequence

$\mathsf{A}^{*-1}(\mathsf{Ell})\rightarrow\mathsf{A}^{*}(\mathcal{F}_{2})\rightarrow\mathsf{A}^{*}(\mathcal{F}_{2}\smallsetminus\mathsf{Ell})\rightarrow
0\,.$

As a result, $\lambda$, $[\mathsf{Ell}],$ $c_{2}$ and $c_{3}$ suffice to
generate the ring $\mathsf{A}^{*}(\mathcal{F}_{2}).$ We also conclude that the
entire Chow ring is tautological. ∎

######  Remark 4.

The proof of Theorem 1 (i) shows that we can choose the degree $1$ generators
to be $\lambda$ and $\left[\mathsf{Ell}\right].$ However, the degree 1
generators

$\alpha_{1}=H,\quad\alpha_{2}=\left[\mathsf{Ell}\right]$

will be more convenient for us. A simple calculation using test curves in
$\mathcal{F}_{2}$ yields the change of basis equation

$\lambda=\frac{1}{2}H-[\mathsf{Ell}]\,.$

Indeed, the identity is correct on any complete curve contained in
$\mathsf{Ell}$ since

$[\mathsf{Ell}]^{2}=-\lambda\cdot[\mathsf{Ell}]\,,\
\,H\cdot[\mathsf{Ell}]=0\,,$

see [OG, Lemma 1.2] for the first intersection. For a second test curve, we
consider a double cover of $\mathbb{P}^{1}\times\mathbb{P}^{2}$ branched along
a smooth $(2,6)$ hypersurface. This construction yields a family of sextic
double planes, for which $\lambda$, $H$ and $[\mathsf{Ell}]$ evaluate to $1$
(using Riemann-Hurwitz), $2$ and $0$ respectively.

######  Remark 5.

We give the details of the Grothendieck-Riemann-Roch calculation alluded to in
the proof above. Recall that $c_{2},c_{3}$ are the Chern classes of the bundle

$\mathcal{V}=\pi_{*}\left(\mathcal{L}\right)\otimes\det(\pi_{*}\left(\mathcal{L})\right)^{-\frac{1}{3}}\,.$

By Grothendieck-Riemann-Roch, we have

$\displaystyle\text{ch}(\pi_{*}\mathcal{L})$
$\displaystyle=\pi_{*}\left(\exp(c_{1}(\mathcal{L}))\cdot\text{Todd}(T_{\pi}^{\text{rel}})\right)\,.$

We expand

$\exp(c_{1}(\mathcal{L}))=1+c_{1}(\mathcal{L})+\frac{c_{1}(\mathcal{L})^{2}}{2}+\frac{c_{1}(\mathcal{L})^{3}}{6}+\frac{c_{1}(\mathcal{L})^{4}}{24}+\ldots\,,$
$\text{Todd}(T_{\pi}^{\text{rel}})=1-\frac{\lambda}{2}+\frac{\lambda^{2}+t}{12}-\frac{\lambda
t}{24}+\frac{-\lambda^{4}+4\lambda^{2}t+3t^{2}}{720}+\ldots\,,$

where $t=c_{2}(T_{\pi}^{\text{rel}}).$ From here, using the definition of the
$\kappa$-classes and the fact that $\pi_{*}(t)=24$,
$\pi_{*}(t^{2})=88\lambda^{2}$ by [GK, Section 3], we immediately obtain

$\text{ch}_{1}(\pi_{*}\mathcal{L})=\frac{2\kappa_{3,0}+\kappa_{1,1}}{12}-\frac{3\lambda}{2},\quad\text{ch}_{2}(\pi_{*}\mathcal{L})=\frac{\kappa_{4,0}+\kappa_{2,1}}{24}-\frac{\lambda}{24}\cdot(2\kappa_{3,0}+\kappa_{1,1})+\frac{7\lambda^{2}}{12}\,.$

Using $c_{1}(\mathcal{V})=0$, we find

$c_{2}(\mathcal{V})=-\text{ch}_{2}(\mathcal{V})=-\text{ch}_{2}(\pi_{*}(\mathcal{L}))+\frac{1}{6}\left(\text{ch}_{1}(\pi_{*}(\mathcal{L}))\right)^{2}\,.$

Substituting, we obtain

$c_{2}=-\frac{1}{24}\left(\kappa_{4,0}+\kappa_{2,1}-\frac{1}{36}(2\kappa_{3,0}+\kappa_{1,1})^{2}+{5\lambda^{2}}\right)\,.$

The calculation for $c_{3}$ is entirely similar, but yields a more complicated
expression:

$\displaystyle
c_{3}=\frac{1}{360}(3\kappa_{1,2}+10\kappa_{3,1}+6\kappa_{5,0})$
$\displaystyle+\frac{1}{23328}(2\kappa_{3,0}+\kappa_{1,1})^{3}-\frac{1}{432}(2\kappa_{3,0}+\kappa_{1,1})(\kappa_{2,1}+\kappa_{4,0})$
$\displaystyle-\frac{1}{1080}(23\kappa_{1,1}+40\kappa_{3,0})\lambda^{2}.$

Both Chow classes $c_{2}$, $c_{3}$ are well-defined on the moduli space
$\mathcal{F}_{2}$ (and not only on the related quotients (3), (4) which appear
in the proof of Theorem 1(i) above). Furthermore, $c_{2}$ and $c_{3}$ belong
to the tautological ring.

###  3.3. Blowing up the triple conic locus

We would like to study the whole moduli space $\mathcal{F}_{2}$, not just
$\mathcal{F}_{2}\smallsetminus\mathsf{Ell}$. Shah showed how to construct
$\mathcal{F}_{2}$ as a GIT quotient [S]. The discussion below is just an
adaptation of his work in the language of stacks.

The sextic given by the triple nonsingular conic,
$\delta=(x_{1}x_{3}-x_{2}^{2})^{3}$, plays a special role. In fact, in the
quotient $[X/G]$, all surfaces in the divisor $\mathsf{Ell}$ correspond to the
orbit of $\delta$. Following [S], in order to construct the moduli space
$\mathcal{F}_{2},$ we blow up the orbit of $\delta$ and remove further loci
from the blowup. We make this more precise.

Elliptic surfaces $(S,L)$ in $\mathsf{Ell}$ can also be exhibited as
resolutions of certain branched double covers. In this case, the linear series
$|L|=|\sigma+2f|$ has fixed part $\sigma$, and $|L-\sigma|$ induces the map
$S\to Q\subset\mathbb{P}^{2}$, where
$Q=\\{x_{1}x_{3}-x_{2}^{2}=0\\}\simeq\mathbb{P}^{1}$ is the nonsingular conic.
However, the linear series $|2L|$ has no fixed components and induces a
morphism

$S\to\widehat{S}\to\mathbb{P}^{5}\,.$

Here $S\to\widehat{S}$ contracts all nonsingular rational curves on which $L$
restricts trivially. Under the Veronese embedding

$\mathsf{v}:\mathbb{P}^{2}\to\mathbb{P}^{5},$

the conic $Q$ corresponds to a rational normal curve $\mathsf{v}(Q)$ contained
in a hyperplane section $\mathbb{P}^{4}$ of $\mathbb{P}^{5}.$ The cone over
${\mathsf{v}}(Q)$ is denoted $\Sigma_{4}^{0}\subset\mathbb{P}^{5}.$ This cone
is a flat deformation of the Veronese surface $\mathsf{v}(\mathbb{P}^{2})$.
Blowing up the vertex, we obtain the Hirzebruch surface $\Sigma_{4}$. The
surface $S$ is the resolution of the double cover

$\widehat{S}\to\Sigma_{4}^{0}$

ramified over the vertex of the cone and over a curve $D$. For $S$ to be a
$K3$ surface, $D$ must be a cubic section of
$\Sigma_{4}^{0}\subset\mathbb{P}^{5}$ not passing through the vertex, and must
have ADE singularities.

Following [La, T], we note the following uniform description of all surfaces
in $\mathcal{F}_{2}$ as resolutions of branched double covers $\widehat{S}$.
Indeed, in all cases, $\widehat{S}$ arises as a complete intersection

$\\{z^{2}-f_{6}(x_{1},x_{2},x_{3},y)=f_{2}(x_{1},x_{2},x_{3},y)=0\\}\subset\mathbb{P}(1,1,1,2,3)\,,$

where $f_{2},f_{6}$ have the indicated weighted degrees. When
$f_{2}(0,0,0,1)\neq 0,$ we can change coordinates so that
$f_{2}(x_{1},x_{2},x_{3},y)=y$ and we recover the sextic double planes. When
$f_{2}(0,0,0,1)=0$, and $f_{2}(x_{1},x_{2},x_{3},0)$ is non degenerate, we may
assume $f_{2}(x_{1},x_{2},x_{3},y)=x_{1}x_{3}-x_{2}^{2}$ and we can write

$f_{6}(x,y)=y^{3}+yg_{4}(x)+g_{6}(x)\,.$

This corresponds to the branched double covers of $\Sigma_{4}^{0}$. Indeed, we
have $\Sigma_{4}^{0}=\mathbb{P}(1,1,4)$, see [D], and the latter is cut out by
$x_{1}x_{3}-x_{2}^{2}=0$ in $\mathbb{P}(1,1,1,2).$

We return to the moduli space $\mathcal{F}_{2}$. There is a $G$-equivariant
cubing map

$m_{3}:\mathbb{P}\operatorname{Sym}^{2}V^{*}\rightarrow X\,.$

Let $\Delta_{2}\subset\mathbb{P}\operatorname{Sym}^{2}V^{*}$ denote the
divisor parametrizing singular conics. The induced map

$\mathbb{P}\operatorname{Sym}^{2}V^{*}\smallsetminus\Delta_{2}\rightarrow
X\smallsetminus(\mathsf{QP}\cup\mathsf{ML})$

is a $G$-equivariant closed embedding whose image is the locus $\mathsf{TC}$
parametrizing cubes of nonsingular conics. A local calculation shows that

(5) $\mathsf{TC}\subset\mathsf{CTP}.$

All nonsingular conics are equivalent under the action of $G$. Let
$W=\mathbb{C}^{2}$ as an $\operatorname{SL}_{2}$ representation. We can
identify $V=\operatorname{Sym}^{2}W$, as the conic is a $\mathbb{P}^{1}$
embedded in $\mathbb{P}^{2}$ via the Veronese map. By Luna’s étale slice
theorem, the quotient $[X/G]$ is locally identified around $[\mathsf{TC}/G]$
with a quotient by $\operatorname{PSL}(W)$ of the normal slice to the orbit of
the triple conic. The normal slice is

$\operatorname{Sym}^{8}W^{*}\oplus\operatorname{Sym}^{12}W^{*}\subset\operatorname{Sym}^{6}(\operatorname{Sym}^{2}W^{*})\,.$

This identification is more thoroughly explained in [S, Section 5], [La, Lemma
4.9], and in Section 6.6 below. We assign weights $2$ and $3$ to the two
summands of the normal slice. 444The moduli space of elliptic $K3$ surfaces
requires weights $(4,6)$ on the two summands $\operatorname{Sym}^{8}W^{*}$ and
$\operatorname{Sym}^{12}W^{*}$, see [M]. However, our calculation is carried
out on the moduli of plane sextics (while ensuring that the relevant classes
get identified correctly on ${\mathcal{F}}_{2}$). By doing so, we do not
account for the additional $\mathbb{Z}/2\mathbb{Z}$-symmetry of the double
cover. In order to match the automorphisms on the exceptional divisor, the
correct weights are $(2,3)$. The geometry here is consistent with the
constructions in [S]. Next, we carry out the $(2,3)$ weighted blowup of the
locus

$\mathsf{TC}\subset X\smallsetminus(\mathsf{QP}\cup\mathsf{ML})\,,$

and take the quotient.

We indicate the loci that need to be removed from the blowup to obtain
$\mathcal{F}_{2}$. On the sextic ADE locus, in addition to the loci
$\mathsf{ML}$ and $\mathsf{QP}$ already considered, we also need to remove the
images of the double conic and double cubic locus $\mathsf{DC}$ and the
consecutive triple point locus $\mathsf{CTP}$.

Over the exceptional divisor
$[\mathbb{P}_{(2,3)}(\operatorname{Sym}^{8}W^{*}\oplus\operatorname{Sym}^{12}W^{*})/\operatorname{PSL}(W)]$,
there is an open subset corresponding to elliptic K3 surfaces. By [M], a pair
of binary forms
$(A,B)\in\operatorname{Sym}^{8}W^{*}\oplus\operatorname{Sym}^{12}W^{*}$
corresponds to an elliptic K3 surface if and only if

  * (i)

$4A^{3}+27B^{2}$ is not identically zero.

  * (ii)

For each point $q\in\mathbb{P}^{1}$, the order of vanishing of $A$ at $q$ is
at most $3$ or the order of vanishing of $B$ at $q$ is at most $5$.

This description was used in the calculations of [CK]. We note that these are
exactly the same requirements singled out in [S, Theorem 4.3]. Indeed, using
the above notation, we can view $g_{4}$ and $g_{6}$ as binary forms $A,B$ of
degrees $8$ and $12$ on $\mathbb{P}^{1}$, respectively. From the exact
description of $\widehat{S}$, we find that the branch curve takes the form
$D=\\{y^{3}+yA+B=0\\}$. Condition (i) corresponds to reduced branch curves
$D$; this is discussed in [S, Theorem 4.3, Case 2]. Condition (ii) ensures $D$
has no consecutive triple points; this was noted in the proof of [S, Theorem
4.3, Case (1.i)] and can be seen as follows: locally near $y=u=0$, the
equation $y^{3}+yA(u)+B(u)$ is in the ideal $(y,u^{2})^{3}$ if and only if
$A,B$ vanish with order at least $4$ and $6$ at the origin. The complement of
(i) corresponds to the restriction of the strict transform of the locus
$\mathsf{DC}$ to the exceptional divisor of the weighted blowup. The
complement of (ii) corresponds to the restriction of the strict transform of
$[\mathsf{CTP}/G]$ to the exceptional divisor.

In conclusion, the moduli space $\mathcal{F}_{2}$ is obtained by removing the
strict transform of (the images of) $\mathsf{DC}$ and $\mathsf{CTP}$ from the
weighted blowup.

###  3.4. Strategy

The following procedure will be used to complete the proof of parts (ii) and
(iii) of Theorem 1.

  * (a)

We start with

$\mathsf{A}_{G}^{*}(X)=\mathbb{Q}[H,c_{2},c_{3}]/(p)\,,$

where

$p=H^{28}+c_{1}^{G}(\operatorname{Sym}^{6}V^{*})H^{27}+\cdots+c_{28}^{G}(\operatorname{Sym}^{6}V^{*})\,.$

  * (b)

We impose relations by removing $\mathsf{ML}$ (the locus of multiple lines)
and $\mathsf{QP}$ (the locus of quadruple points). That is, we compute the
Chow ring $\mathsf{A}^{*}_{G}(Y)$ where

$Y=X\smallsetminus(\mathsf{ML}\cup\mathsf{QP})\,.$

This is carried out in Sections 4 and 5.

  * (c)

We perform the weighted blowup $\widetilde{Y}$ of $Y$ at $\mathsf{TC}$ (the
triple conic locus) and compute the Chow ring
$\mathsf{A}^{*}_{G}(\widetilde{Y})$ . This is carried out in Section 6.

  * (d)

Finally, we impose further relations by removing from $\widetilde{Y}$ the
strict transforms of $\mathsf{CTP}$ (the consecutive triple point locus) and
$\mathsf{DC}$ (the union of the double conic and double cubic locus). This is
carried out in Section 6.

From the above discussion of Shah’s work, steps (a)–(d) account for all of the
relations and, therefore, yield a presentation of
$\mathsf{A}^{*}(\mathcal{F}_{2})$. The calculation of the Chow Betti numbers
then proves parts (ii) and (iii) of Theorem 1.

##  4\. Relations from the locus of multiple lines

We impose here relations obtained by removing $\mathsf{ML}$. We follow the
notation from Section 3. We begin by studying the map

$m_{1}:\mathbb{P}\operatorname{Sym}^{4}V^{*}\times\mathbb{P}V^{*}\rightarrow
X=\mathbb{P}\operatorname{Sym}^{6}V^{*},$

whose image is $\mathsf{ML}$. Let $h_{1}$ and $h_{2}$ denote the hyperplane
classes of $\mathbb{P}\operatorname{Sym}^{4}V^{*}$ and $\mathbb{P}V^{*}$
pulled back to the product
$\mathbb{P}\operatorname{Sym}^{4}V^{*}\times\mathbb{P}V^{*}$. As before, let
$H$ be the hyperplane class of $X$.

For notational convenience, we will denote the map $m_{1}$ simply by $m$.
Moreover, the same symbols $m,h_{1},h_{2},H$ will be used to denote the
corresponding maps and classes on the quotients by $G$.

######  Lemma 6.

The image of

$\mathsf{A}_{G}^{*}(\mathbb{P}\operatorname{Sym}^{4}V^{*}\times\mathbb{P}V^{*})\xrightarrow{m_{*}}\mathsf{A}_{G}^{*}(X)$

is the ideal generated by the three classes

$\sum_{j=0}^{11+i}\alpha_{i,j}H^{j}$

for $0\leq i\leq 2$, where the coefficients $\alpha_{ij}\in\mathsf{A}^{*}_{G}$
are recursively given by the formula

$\alpha_{i,k}=\gamma_{*}((h_{1}+2h_{2})^{27-k}\cdot
h_{2}^{i})-\sum_{j=k+1}^{11+i}\alpha_{i,j}s^{G}_{j-k}(\operatorname{Sym}^{6}V^{*})\,.$

Here,
$\gamma_{*}:\mathsf{A}_{G}^{*}(\mathbb{P}\operatorname{Sym}^{4}V^{*}\times\mathbb{P}V^{*})\rightarrow\mathsf{A}_{G}^{*}$
denotes the equivariant pushforward to a point.

###### Proof.

Consider the pullback

$m^{*}:\mathsf{A}_{G}^{*}(X)\to\mathsf{A}_{G}^{*}(\mathbb{P}\operatorname{Sym}^{4}V^{*}\times\mathbb{P}V^{*}),$

and note that $m^{*}H=h_{1}+2h_{2}$. By the projective bundle formula [F,
Chapter 3], the $\mathsf{A}_{G}^{*}(X)$-module
$\mathsf{A}_{G}^{*}(\mathbb{P}\operatorname{Sym}^{4}V^{*}\times\mathbb{P}V^{*})$
is generated by the classes $1,h_{2},h_{2}^{2}$. Therefore, the image of

$\mathsf{A}_{G}^{*}(\mathbb{P}\operatorname{Sym}^{4}V^{*}\times\mathbb{P}V^{*})\xrightarrow{m_{*}}\mathsf{A}_{G}^{*}(X)$

is the ideal generated by the pushforwards $m_{*}(h_{2}^{i})$ for $0\leq i\leq
2$. We write

$m_{*}(h_{2}^{i})=\sum_{j=0}^{11+i}\alpha_{i,j}H^{j}\,,$

where $\alpha_{i,j}\in\mathsf{A}^{11+i-j}_{G}$. We want to determine the
coefficients $\alpha_{i,j}$. To pick out $\alpha_{i,k}$, multiply by
$H^{27-k}$. We have

$H^{27-k}\cdot m_{*}(h_{2}^{i})=\sum_{j=0}^{11+i}\alpha_{i,j}H^{j+27-k}\,.$

Consider the commutative diagram

${{\left[\left(\mathbb{P}\operatorname{Sym}^{4}V^{*}\times\mathbb{P}V^{*}\right)/G\right]}}$${{[X/G]}}$${\,\mathrm{B}G\,.}$$\scriptstyle{\gamma}$$\scriptstyle{m}$$\scriptstyle{\rho}$

Then,

$\rho_{*}(H^{27-k}\cdot
m_{*}(h_{2}^{i}))=\rho_{*}\left(\sum_{j=0}^{11+i}\alpha_{i,j}H^{j+27-k}\right)=\sum_{j=0}^{11+i}\alpha_{i,j}s^{G}_{j-k}(\operatorname{Sym}^{6}V^{*})\,.$

On the other hand, by the projection formula,

$\rho_{*}(H^{27-k}\cdot m_{*}(h_{2}^{i}))=\rho_{*}m_{*}(m^{*}H^{27-k}\cdot
h_{2}^{i})=\gamma_{*}((h_{1}+2h_{2})^{27-k}\cdot h_{2}^{i})\,.$

Thus,

$\sum_{j=0}^{11+i}\alpha_{i,j}s^{G}_{j-k}(\operatorname{Sym}^{6}V^{*})=\gamma_{*}((h_{1}+2h_{2})^{27-k}\cdot
h_{2}^{i})\,.$

Note that $s^{G}_{0}(\operatorname{Sym}^{6}V^{*})=1$ and
$s^{G}_{j-k}(\operatorname{Sym}^{6}V^{*})=0$ for $j-k<0$. Simplifying and
rearranging, we see that

(6) $\alpha_{i,k}=\gamma_{*}((h_{1}+2h_{2})^{27-k}\cdot
h_{2}^{i})-\sum_{j=k+1}^{11+i}\alpha_{i,j}s^{G}_{j-k}(\operatorname{Sym}^{6}V^{*})\,.$

∎

######  Remark 7.

Equation (6) gives a recursion for computing $\alpha_{i,k}$. The pushforwards
$\gamma_{*}(h_{1}^{a}\cdot h_{2}^{b})$ can be immediately determined via the
projective bundle geometry [F, Chapter 3]. Thus, we can express the classes
$\alpha_{i,k}$ in terms of the generators $H,c_{2},c_{3}$ of
$\mathsf{A}^{*}_{G}(X)$. As a result, the image of $m_{*}$ specified by Lemma
6, and consequently the relations obtained by removing the locus
$\mathsf{ML}$, can be made explicit.

We carried out this procedure in the Macaulay2 package Schubert2 [GS, GSSEC].
The interested reader can consult [COP1] for the implementation.

$\bullet$ For $i=0$, the polynomial $\sum_{j=0}^{11}\alpha_{0,j}H^{j}$ is
given by

$\displaystyle 1555200c_{2}^{4}c_{3}$
$\displaystyle+9552816c_{2}c_{3}^{3}+(518400c_{2}^{5}+11162448c_{2}^{2}c_{3}^{2})H+(5716656c_{2}^{3}c_{3}+56538324c_{3}^{3})H^{2}$
$\displaystyle+(712080c_{2}^{4}+8743140c_{2}c_{3}^{2})H^{3}+3852036c_{2}^{2}c_{3}H^{4}+(311700c_{2}^{3}+12450672c_{3}^{2})H^{5}$
$\displaystyle-519696c_{2}c_{3}H^{6}+107640c_{2}^{2}H^{7}+243324c_{3}H^{8}-9900c_{2}H^{9}+480H^{11}\,.$

$\bullet$ For $i=1$, the polynomial $\sum_{j=0}^{12}\alpha_{1,j}H^{j}$ is
given by

$\displaystyle 1866240c_{2}^{3}c_{3}^{2}$
$\displaystyle+15431472c_{3}^{4}+(362880c_{2}^{4}c_{3}+4968864c_{2}c_{3}^{3})H+(17280c_{2}^{5}+5732856c_{2}^{2}c_{3}^{2})H^{2}$
$\displaystyle+(1278288c_{2}^{3}c_{3}+47364588c_{3}^{3})H^{3}-(74040c_{2}^{4}+7471926c_{2}c_{3}^{2})H^{4}$
$\displaystyle+1636848c_{2}^{2}c_{3}H^{5}-(51630c_{2}^{3}-4266918c_{3}^{2})H^{6}-598968c_{2}c_{3}H^{7}$
$\displaystyle+36810c_{2}^{2}H^{8}+40392c_{3}H^{9}-2850c_{2}H^{10}+30H^{12}\,.$

$\bullet$ For $i=2$, the polynomial $\sum_{j=0}^{13}\alpha_{2,j}H^{j}$ is
given by

$\displaystyle 1259712c_{2}^{2}c_{3}^{3}$
$\displaystyle+(1765152c_{2}^{3}c_{3}^{2}+42620256c_{3}^{4})H-(565920c_{2}^{4}c_{3}-19960020c_{2}c_{3}^{3})H^{2}$
$\displaystyle+(61056c_{2}^{5}+7261812c_{2}^{2}c_{3}^{2})H^{3}-(426564c_{2}^{3}c_{3}-28062369c_{3}^{3})H^{4}$
$\displaystyle-(15404c_{2}^{4}+8744085c_{2}c_{3}^{2})H^{5}+1276371c_{2}^{2}c_{3}H^{6}-(63167c_{2}^{3}-1147635c_{3}^{2})H^{7}$
$\displaystyle-218646c_{2}c_{3}H^{8}+12903c_{2}^{2}H^{9}+5139c_{3}H^{10}-389c_{2}H^{11}+H^{13}\,.$

##  5\. Relations from quadruple points

We impose here relations obtained by removing the locus $\mathsf{QP}$ of
sextics with quadruple points. A sextic $f$ has a quadruple point if locally
analytically it lies in the ideal $(x,y)^{4}$. In this section, we will use
standard material on jet bundles, see [CL1, Proposition 3.2] for a review of
the necessary background.

Denote by $\pi:[\mathbb{P}V/G]\rightarrow\mathrm{B}G$ the universal
$\mathbb{P}^{2}$-bundle and let $z$ be the hyperplane class. Then
$\pi_{*}\mathcal{O}(6)=\operatorname{Sym}^{6}V^{*}$ as a $G$-equivariant
bundle. Its projectivization is $[X/G]$ and the projectivization of
$\pi^{*}\pi_{*}\mathcal{O}(6)$ is $\left[\left(\mathbb{P}V\times
X\right)/G\right]$. Consider the rank $10$ bundle of principal parts
$P^{3}(\mathcal{O}(6))$ relatively over $[\mathbb{P}V/G]\to\mathrm{B}G$. It
comes equipped with an equivariant evaluation map

$\pi^{*}\operatorname{Sym}^{6}V^{*}=\pi^{*}\pi_{*}\mathcal{O}(6)\rightarrow
P^{3}(\mathcal{O}(6))\,,$

which on fibers takes a sextic to its expansion along a third order
neighborhood. This evaluation map is surjective because $\mathcal{O}(6)$ is
$6$-very ample and hence also $3$-very ample. The kernel is thus an
equivariant vector bundle $K_{\operatorname{quad}}$ of rank $18$. The bundle
$K_{\operatorname{quad}}$ parametrizes pairs $(f,p)$ where $f$ is a sextic
with a quadruple or worse point at $p$. After projectivizing, the following
diagram summarizes the above discussion:

(7)
${{[\mathbb{P}K_{\operatorname{quad}}/G]}}$${{\left[\left(\mathbb{P}V\times
X\right)/G\right]}}$${{[X/G]}}$${{[\mathbb{P}V/G]}}$${\mathrm{B}G\,.}$$\scriptstyle{\rho^{\prime\prime}}$$\scriptstyle{j}$$\scriptstyle{\rho^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\rho}$$\scriptstyle{\pi}$

The image of $\pi^{\prime}\circ j$ is the locus $[\mathsf{QP}/G]$. Moreover,
since $\pi^{\prime}\circ j$ is proper, the induced map on Chow groups with
$\mathbb{Q}$-coefficients

$\mathsf{A}_{*}([\mathbb{P}K_{\operatorname{quad}}/G])\rightarrow\mathsf{A}_{*}([\mathsf{QP}/G])$

is surjective. Indeed, one sees this by applying [V, Lemma 3.8] to the map

$(\mathbb{P}K_{\operatorname{quad}}\times U)/G\rightarrow(\mathsf{QP}\times
U)/G\,,$

where $U$ is an open subset of a representation $V$ of $G$ on which $G$ acts
freely and the codimension of the complement of $U$ in $V$ is sufficiently
large as in [EG, Definition-Proposition 1]. Therefore, we can compute the
image of

$\mathsf{A}_{*}([\mathsf{QP}/G])\rightarrow\mathsf{A}_{*}([X/G])$

by computing the image of

$\mathsf{A}_{*}([\mathbb{P}K_{\operatorname{quad}}/G])\rightarrow\mathsf{A}_{*}([X/G])\,.$

######  Lemma 8.

The ideal of relations obtained from removing the locus of sextics with
quadruple points is generated by the classes

$\sum_{i=0}^{10}\rho^{*}\pi_{*}\left(z^{j}\cdot
c^{G}_{i}(P^{3}(\mathcal{O}(6)))\right)\cdot H^{10-i}\,,$

where $0\leq j\leq 2$ and $z$ is the hyperplane class of $\pi$.

###### Proof.

From the explicit calculation of the Chow ring of projective bundles it
follows that every class
$\alpha\in\mathsf{A}^{*}_{G}(\mathbb{P}K_{\operatorname{quad}})$ is a pullback
of a class $\beta\in\mathsf{A}^{*}_{G}(\mathbb{P}V\times X)$. Then, by the
projection formula,

$j_{*}\alpha=j_{*}j^{*}\beta=[\mathbb{P}K_{\operatorname{quad}}]^{G}\cdot\beta\,.$

Because $\mathbb{P}K_{\operatorname{quad}}$ is linearly embedded in
$\mathbb{P}V\times X$, its equivariant fundamental class is given by

$[\mathbb{P}K_{\operatorname{quad}}]^{G}=c^{G}_{10}(\rho^{\prime*}P^{3}(\mathcal{O}(6))\otimes\mathcal{O}_{\rho^{\prime}}(1))=\sum_{i=0}^{10}\rho^{\prime*}c^{G}_{i}(P^{3}(\mathcal{O}(6)))\cdot\pi^{\prime*}H^{10-i}\,.$

Every class $\beta\in\mathsf{A}^{*}_{G}(\mathbb{P}V\times X)$ is of the form

$\beta=\beta_{0}+\beta_{1}z+\beta_{2}z^{2}\,,$

where $\beta_{i}\in\mathsf{A}_{G}^{*}(X)$ and $z$ is the hyperplane class of
the projective bundle $\left[\left(\mathbb{P}V\times
X\right)/G\right]\rightarrow[X/G]$. Hence the ideal generated by pushforwards
of classes on $[\mathbb{P}K_{\operatorname{quad}}/G]$ is just the ideal
generated by the classes

$\pi^{\prime}_{*}\bigg{(}z^{j}\cdot\bigg{(}\sum_{i=0}^{10}\rho^{\prime*}c^{G}_{i}(P^{3}(\mathcal{O}(6)))\cdot\pi^{\prime*}H^{10-i}\bigg{)}\bigg{)}=\sum_{i=0}^{10}\rho^{*}\pi_{*}(z^{j}\cdot
c^{G}_{i}(P^{3}(\mathcal{O}(6))))\cdot H^{10-i}$

for $0\leq j\leq 2$, where to obtain the equality we have used [F, Proposition
1.7]. ∎

######  Remark 9.

The $3$ relations provided by Lemma 8 can be written explicitly in terms of
the generators $H,c_{2},c_{3}$ of $\mathsf{A}_{G}^{*}(X)$. We note:

  * (i)

The equivariant Chern classes of the jet bundle $P^{3}(\mathcal{O}(6))$ are
computed using the filtration by the vector bundles
$\mathcal{O}(6)\otimes\operatorname{Sym}^{k}\Omega_{\pi}$ for $0\leq k\leq 3$
.

  * (ii)

The pushforwards $\pi_{*}(z^{j})$ are immediately found using the projective
bundle geometry.

$\bullet$ The $j=0$ case yields

$\displaystyle-1574$ $\displaystyle
64c_{2}c_{3}^{2}-236196c_{2}^{2}c_{3}H-61020c_{2}^{3}H^{2}+434484c_{3}^{2}H^{2}+382725c_{2}c_{3}H^{3}+76545c_{2}^{2}H^{4}$
$\displaystyle-66339c_{3}H^{5}-13230c_{2}H^{6}+405H^{8}\,.$

$\bullet$ The $j=1$ case yields

$\displaystyle 518$ $\displaystyle
40c_{2}^{3}c_{3}-122472c_{3}^{3}+(17280c_{2}^{4}-539460c_{2}c_{3}^{2})H-446148c_{2}^{2}c_{3}H^{2}$
$\displaystyle-(91320c_{2}^{3}-339309c_{3}^{2})H^{3}+244215c_{2}c_{3}H^{4}+39690c_{2}^{2}H^{5}-17577c_{3}H^{6}-2880c_{2}H^{7}+30H^{9}\,.$

$\bullet$ The $j=2$ case yields

$\displaystyle 2099$ $\displaystyle
52c_{2}^{2}c_{3}^{2}+(253152c_{2}^{3}c_{3}-338256c_{3}^{3})H+(61056c_{2}^{4}-812592c_{2}c_{3}^{2})H^{2}-475308c_{2}^{2}c_{3}H^{3}$
$\displaystyle-(76460c_{2}^{3}-178632c_{3}^{2})H^{4}+104733c_{2}c_{3}H^{5}+13293c_{2}^{2}H^{6}-3267c_{3}H^{7}-390c_{2}H^{8}+H^{10}\,.$

##  6\. The weighted blowup

###  6.1. Overview

Lemmas 6 and 8 allow us to compute the Chow ring of the stack

$\left[\left(X\smallsetminus(\mathsf{ML}\cup\mathsf{QP})\right)/G\right]\,.$

The next step in the procedure is to perform the weighted blowup along the
locus $\mathsf{TC}$.

###  6.2. The Chow ring of weighted blowups

First, we discuss the intersection theory of weighted blowups in general,
following Arena and Obinna [A, AO].

We will require weighted Chern and Segre classes. Let $E\to B$ be a vector
bundle over a base $B$ equipped with a fiberwise action of $\mathbb{G}_{m}$.
We define

$P_{E}(t)=c^{\mathbb{G}_{m}}_{\text{top}}(E)\,,$

a polynomial of degree $\text{rk }E$ in the equivariant parameter $t$. In case
the $\mathbb{G}_{m}$-action is of weight $1$, $P_{E}(t)$ recovers the total
Chern polynomial of $E$. For the case of $(2,3)$-weighted blowups needed
below, $E=E_{2}\oplus E_{3}$ is split into subbundles of weights $2$, $3$
respectively. Then, we have

$P_{E}(t)=\prod_{i=1}^{\text{rk
}E_{2}}(2t+\alpha_{i})\cdot\prod_{i=1}^{\text{rk }E_{3}}(3t+\beta_{i})\,,$

where $\alpha_{i}$ and $\beta_{i}$ are the Chern roots of $E_{2}$ and $E_{3}$.
The weighted Segre class of $E$ is given by

$s^{\mathsf{wt}}(E)=\frac{1}{P_{E}(1)}\,,$

see [A, Remark 8.1.2].

For the blowup geometry, let $i:Z\hookrightarrow Y$ be a closed embedding of
codimension $d$ with $Z$ and $Y$ both nonsingular. Let $N$ be the normal
bundle weighted by a $\mathbb{G}_{m}$-action. Let $P_{N}(t)$ be the
$\mathbb{G}_{m}$-equivariant top Chern class $c_{d}^{\mathbb{G}_{m}}(N)$.

Let $f:\widetilde{Y}\rightarrow Y$ denote the corresponding weighted blowup
along $Z$, and $j:\widetilde{Z}\hookrightarrow\widetilde{Y}$ the exceptional
divisor. The exceptional divisor $g:\widetilde{Z}\to Z$ is a weighted
projective bundle. We have

$\mathsf{A}^{*}(\widetilde{Z})=\mathsf{A}^{*}(Z)[t]/(P_{N}(t))$

where $t$ is set to the hyperplane class of $g$.

By [AO, Theorem 5.5], there is a commutative diagram

${\mathsf{A}^{*}(\widetilde{Z})}$${\mathsf{A}^{*}(\widetilde{Y})}$${\mathsf{A}^{*}(Z)}$${\mathsf{A}^{*}(Y)}$$\scriptstyle{j_{*}}$$\scriptstyle{i_{*}}$$\scriptstyle{f^{!}}$$\scriptstyle{f^{*}}$

where $f^{!}(\alpha)=g^{*}\alpha\cdot\delta$. Here,

$\delta=\frac{P_{N}(t)-P_{N}(0)}{t}\,.$

Furthermore, we have the following structure theorem for the Chow groups of
weighted blowups, which we present in the simplified setting where the
pullback $\mathsf{A}^{*}(Y)\rightarrow\mathsf{A}^{*}(Z)$ is surjective [AO,
Corollary 6.5]. When all weights are $1$, Theorem 10 below recovers [Ke,
Appendix, Theorem 1].

######  Theorem 10 (Arena-Obinna).

In the above setting, if $i^{*}:\mathsf{A}^{*}(Y)\rightarrow\mathsf{A}^{*}(Z)$
is surjective, then

$\mathsf{A}^{*}(\widetilde{Y})\cong\mathsf{A}^{*}(Y)[t]/(t\cdot\ker(i^{*}),Q(t))\,,$

where $Q(t)=P_{N}(t)-P_{N}(0)+[Z]$. Furthermore, $[\widetilde{Z}]=-t$.

Let $\zeta:S\hookrightarrow Y$ be a nonsingular closed substack of dimension
$k$ such that $S\cap Z$ is also nonsingular. Let
$\widetilde{S}\subset\widetilde{Y}$ be the strict transform inside the
weighted blowup. We will use a formula for
$[\widetilde{S}]\in\mathsf{A}_{k}(\widetilde{Y})$ in terms of the total
transform $f^{*}[S]$ and a class pushed forward from the exceptional divisor
(generalizing [F, Theorem 6.7] in the unweighted case). The exceptional term
involves the weighted top Chern class $P_{N}$ of the weighted normal bundle
and the weighted Segre class of the normal bundle $N_{S/S\cap Z}$. The
following result is [A, Theorem 8.2.1].

######  Theorem 11.

With notation as above, we have

$[\widetilde{S}]=f^{*}[S]-j_{*}\\{P_{N}(1)(1+t+t^{2}+\dots)\cdot
g^{*}\zeta_{*}s^{\mathsf{wt}}(N_{S/S\cap Z})\\}_{k}\,,$

where $-t$ is the class of the exceptional divisor.

Here, the subscript $k$ denotes taking the dimension $k$ piece of the term in
brackets.

###  6.3. The weighted blowup of the locus of triple conics

We apply these results to our situation. Set
$Y=X\smallsetminus(\mathsf{ML}\cup\mathsf{QP})$. We have the diagram

${\mathsf{A}_{G}^{*}(\widetilde{\mathsf{TC}})}$${\mathsf{A}_{G}^{*}(\widetilde{Y})}$${\mathsf{A}_{G}^{*}(\mathsf{TC})}$${\mathsf{A}_{G}^{*}(Y)}$$\scriptstyle{j_{*}}$$\scriptstyle{i_{*}}$$\scriptstyle{f^{!}}$$\scriptstyle{f^{*}}$

######  Corollary 12.

With notation as above, we have

$\mathsf{A}_{G}^{*}(\widetilde{Y})\cong\mathsf{A}_{G}^{*}(Y)[t]/(t\cdot
H,t\cdot c_{3},Q(t))\,,$

where $Q(t)=P_{N}(t)-P_{N}(0)+[\mathsf{TC}]$. Furthermore,
$[\widetilde{\mathsf{TC}}]=-t$.

###### Proof.

We note that

$i^{*}:\mathsf{A}^{*}_{G}(Y)\to\mathsf{A}^{*}_{G}(\mathsf{TC}),\quad
i^{*}H=0,\quad i^{*}c_{3}=0\,.$

Indeed, for the vanishing of $i^{*}H$, we recall that
$\mathsf{TC}=\mathbb{P}^{5}\smallsetminus\Delta_{2},$ where $\Delta_{2}$ is
the divisor of singular conics. The vanishing of $i^{*}c_{3}$ is a consequence
of the fact that the stabilizer of the orbit $\mathsf{TC}$ is
$\operatorname{PSL}_{2}.$ In fact,

$\mathsf{A}_{G}^{*}(\mathsf{TC})=\mathsf{A}^{*}_{G}(G/\mathsf{PSL}_{2})=\mathsf{A}^{*}_{\mathsf{PSL}_{2}}(\text{pt})=\mathbb{Q}[c_{2}].$

Furthermore, using the map $\mathsf{PSL}_{2}\to G$, $g\to\text{Sym}^{2}g$, we
see that $i^{*}c_{2}=4c_{2}.$ These remarks show

$\ker i^{*}=\langle H,c_{3}\rangle\,$

and that $i^{*}$ is surjective. Applying Theorem 10 yields the result. ∎

######  Remark 13.

The Chow ring $\mathsf{A}^{*}_{G}(Y)$ is has a presentation with generators
$H,c_{2},c_{3}$ and 6 relations (obtained from Remarks 7 and 9). Applying
Corollary 12 then yields a complete presentation of
$\mathsf{A}^{*}_{G}(\widetilde{Y})$ by imposing $3$ additional relations.
After restriction to $\mathcal{F}_{2}$,

$t=-[\mathsf{Ell}]\,.$

The polynomial $P_{N}$, required for the third relation of Corollary 12, will
be recorded in equation (13) below. However, for degree reasons, when
restricting to $\mathcal{F}_{2}$, only the first two relations

$\left[\mathsf{Ell}\right]\cdot H=0\,,\quad\left[\mathsf{Ell}\right]\cdot
c_{3}=0$

will be meaningful.

###  6.4. A resolution of $\mathsf{CTP}$

By the discussion in Section 3, we have morphisms

$\mathcal{F}_{2}\stackrel{{\scriptstyle\zeta}}{{\longrightarrow}}[\widetilde{Y}/G]\stackrel{{\scriptstyle
f}}{{\longrightarrow}}[Y/G]\,,$

where the complement of the image of $\zeta$ is the union of
$\widetilde{\mathsf{DC}}$ and $\widetilde{\mathsf{CTP}}.$ As noted in the
previous paragraph, we have a complete description of
$\mathsf{A}^{*}_{G}(\widetilde{Y})$. To understand
$\mathsf{A}^{*}(\mathcal{F}_{2})$, we first compute relations by removing
$\widetilde{\mathsf{CTP}}$.

An additional complication arises here. Theorem 11 does not directly apply to
the blowup $\widetilde{Y}$ because $\mathsf{CTP}$ is singular. In the
following subsection, we will use Theorem 11 for a resolution of
$\mathsf{CTP}$, which we now construct.

Recall that a sextic $f$ has a consecutive triple point if analytically
locally it lies in the ideal $(x,y^{2})^{3}$. We begin by constructing the
relevant bundle of principal parts. Note that the local equation is in the
ideal $(x,y^{2})^{3}$ if and only if the coefficients in the Taylor expansion
of the monomials in the set

$S=\\{1,x,y,x^{2},xy,y^{2},x^{2}y,xy^{2},y^{3},xy^{3},y^{4},y^{5}\\}$

all vanish. To record the data of these monomial coefficients, we use the
machinery of refined principal parts bundles as in [CL1, Section 3.2]. The
universal $\mathbb{P}^{2}$ bundle is denoted by
$\pi:[\mathbb{P}V/G]\rightarrow\mathrm{B}G$. Set $T$ to be the tangent bundle
of $\mathbb{P}V$. The set $S$ is admissible in the sense of [CL1, Definition
3.7]. There is a rank $12$ bundle on the domain of
$a:[\mathbb{P}T/G]\rightarrow[\mathbb{P}V/G]$ denoted $P^{S}(\mathcal{O}(6))$
and an evaluation morphism

(8)
$a^{*}\pi^{*}\operatorname{Sym}^{6}V^{*}=a^{*}\pi^{*}\pi_{*}\mathcal{O}(6)\rightarrow
a^{*}P^{5}(\mathcal{O}(6))\rightarrow P^{S}(\mathcal{O}(6))\,,$

where the first map is the usual fifth order principal parts evaluation pulled
back to $\mathbb{P}T$, and the second map truncates the Taylor series along
the monomials in $S$. The composite is surjective, as the first map is
surjective because $\mathcal{O}(6)$ is $5$-very ample, and the second map is
surjective by definition. Let $K_{\mathrm{ctp}}$ denote the kernel of the
morphism (8):

(9) $0\to K_{\text{ctp}}\to a^{*}\pi^{*}\operatorname{Sym}^{6}V^{*}\rightarrow
P^{S}(\mathcal{O}(6))\to 0\,.$

It is a $G$-equivariant vector bundle of rank $16$ on $\mathbb{P}T\times X$,
parametrizing pairs $(f,p,v)$ where $f$ is a sextic with a consecutive triple
point at $p$ in the tangent direction $v\in T_{\,p\,}\mathbb{P}V.$

Denote by $\mathbb{P}K^{\circ}$ the restriction of
$\mathbb{P}K_{\operatorname{ctp}}$ to $\mathbb{P}T\times Y$, where
$Y=X\smallsetminus(\mathsf{QP}\cup\mathsf{ML})$. The following diagram
summarizes the situation:

(10) ${{[\mathbb{P}K^{\circ}/G]}}$${{\left[\left(\mathbb{P}T\times
Y\right)/G\right]}}$${{\left[\left(\mathbb{P}V\times
Y\right)/G\right]}}$${{[Y/G]}}$${{[\mathbb{P}T/G]}}$${{[\mathbb{P}V/G]}}$${\mathrm{B}G\,.}$$\scriptstyle{\iota}$$\scriptstyle{\rho_{3}}$$\scriptstyle{a^{\prime}}$$\scriptstyle{\rho_{2}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\rho}$$\scriptstyle{a}$$\scriptstyle{\pi}$

The image of $\pi^{\prime}\circ a^{\prime}\circ\iota$ is the locus
$\mathsf{CTP}\cap Y$, which by abuse of notation we will simply refer to as
$\mathsf{CTP}$.

We want to impose relations from removing $\widetilde{\mathsf{CTP}}$, not just
$\mathsf{CTP}$. We perform the weighted blowup of $Y$ along $\mathsf{TC}$ to
obtain $\widetilde{Y}$. Because $\pi$ is flat, the weighted blowup of
$\mathbb{P}V\times Y$ along $\pi^{\prime-1}(\mathsf{TC})$ is simply
$\mathbb{P}V\times\widetilde{Y}$. Similarly, the weighted blowup of
$\mathbb{P}T\times Y$ along $a^{\prime-1}\pi^{\prime-1}(\mathsf{TC})$ is
$\mathbb{P}T\times\widetilde{Y}$. Let $\widetilde{\mathbb{P}K^{\circ}}$ be the
strict transform of $\mathbb{P}K^{\circ}$. We have the following commutative
diagram, in which each cell in the right most $3\times 3$ square in the blowup
diagram is Cartesian:

(11)
${{[\widetilde{\mathbb{P}K^{\circ}}/G]}}$${{[\mathbb{P}T\times\widetilde{Y}/G]}}$${{[\mathbb{P}V\times\widetilde{Y}/G]}}$${{[\widetilde{Y}/G]}}$${{[\mathbb{P}K^{\circ}/G]}}$${{[\mathbb{P}T\times
Y/G]}}$${{[\mathbb{P}V\times
Y/G]}}$${{[Y/G]}}$${{[\mathbb{P}T/G]}}$${{[\mathbb{P}V/G]}}$${\mathrm{B}G\,.}$$\scriptstyle{h}$$\scriptstyle{\iota_{\mathsf{str}}}$$\scriptstyle{f_{2}}$$\scriptstyle{a^{\prime\prime}}$$\scriptstyle{f_{1}}$$\scriptstyle{\pi^{\prime\prime}}$$\scriptstyle{f}$$\scriptstyle{\iota}$$\scriptstyle{\rho_{3}}$$\scriptstyle{a^{\prime}}$$\scriptstyle{\rho_{2}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\rho}$$\scriptstyle{a}$$\scriptstyle{\pi}$

The next lemma shows we can compute relations from removing
$\widetilde{\mathsf{CTP}}$ by finding the image of the proper pushforward
$\mathsf{A}^{G}_{*}(\widetilde{\mathbb{P}K^{\circ}})\rightarrow\mathsf{A}^{G}_{*}(\widetilde{Y})$
along the morphism $\pi^{\prime\prime}\circ
a^{\prime\prime}\circ\iota_{\mathrm{str}}$.

######  Lemma 14.

The map $\pi^{\prime\prime}\circ
a^{\prime\prime}\circ\iota_{\mathrm{str}}:[\widetilde{\mathbb{P}K^{\circ}}/G]\rightarrow[\widetilde{Y}/G]$
is proper and its image is $\widetilde{\mathsf{CTP}}$. Therefore, the induced
proper pushforward map

$\mathsf{A}^{G}_{*}(\widetilde{\mathbb{P}K^{\circ}})\rightarrow\mathsf{A}^{G}_{*}(\widetilde{\mathsf{CTP}})$

is surjective.

###### Proof.

By the universal property of blowing up, $\pi^{\prime\prime}\circ
a^{\prime\prime}\circ\iota_{\mathrm{str}}$ factors through
$\widetilde{\mathsf{CTP}}$. It is proper because $\iota_{\mathrm{str}}$ is a
closed embedding and $a^{\prime\prime}$ and $\pi^{\prime\prime}$ are
projective bundles. Moreover, it is dominant because
$[\mathbb{P}K^{\circ}/G]\rightarrow[Y/G]$ surjects onto $\mathsf{CTP}$. The
first statement follows. For the surjectivity of the pushforward
$(\pi^{\prime\prime}\circ a^{\prime\prime}\circ\iota_{\mathrm{str}})_{*}$, we
apply the same argument given before Lemma 8. ∎

###  6.5. Chow classes on $\mathsf{CTP}$ and its resolution

In order to apply Lemma 14, we study the Chow rings of the the spaces in
diagram (11). We let

$z=c_{1}(\mathcal{O}_{\pi}(1)),\quad\tau=c_{1}(\mathcal{O}_{a}(1)).$

Because $a^{\prime}$ and $a^{\prime\prime}$ (respectively, $\pi^{\prime}$ and
$\pi^{\prime\prime}$) are pullbacks of $a$ (respectively, $\pi$), we will omit
pullbacks and use $\tau$ (respectively, $z$) also for the hyperplane classes
of $a^{\prime}$ and $a^{\prime\prime}$ (respectively, $\pi^{\prime}$ and
$\pi^{\prime\prime}$). By the projective bundle formula and Corollary 12, the
Chow ring of every space in the rightmost $3\times 3$ square in diagram (11)
is understood.

Moreover, because $\mathbb{P}K^{\circ}$ is open inside a projective bundle
over $\mathbb{P}T$, the Chow ring $\mathsf{A}^{*}_{G}(\mathbb{P}K^{\circ})$ is
generated over $\mathsf{A}^{*}_{G}$ by the pullbacks of $z,\tau$ and the
hyperplane class of $\rho_{3}$. Because $\mathbb{P}K_{\operatorname{ctp}}$ is
linearly embedded in
$\mathbb{P}(a^{*}\pi^{*}\operatorname{Sym}^{6}V^{*})=\mathbb{P}T\times X$, the
hyperplane class of $\rho_{3}$ is the restriction of the hyperplane class of
$\rho_{2}$, which in turn is the pullback of $H$ along
$a^{\prime}\circ\pi^{\prime}$. In particular, every class in
$\mathsf{A}^{*}_{G}(\mathbb{P}K^{\circ})$ is pulled back from
$\mathsf{A}^{*}_{G}(\mathbb{P}T\times Y)$.

To determine generators for
$\mathsf{A}^{*}_{G}(\widetilde{\mathbb{P}K^{\circ}})$, we note that
$\widetilde{\mathbb{P}K^{\circ}}$ is the weighted blowup of
$\mathbb{P}K^{\circ}$ along the incidence correspondence
$J=\mathsf{TC}\times_{\mathsf{CTP}}\mathbb{P}K^{\circ}$. Thus,
$\mathsf{A}^{*}_{G}(\widetilde{\mathbb{P}K^{\circ}})$ can be computed via
Theorem 10. The answer is recorded in Corollary 16 below.

We first note that $J$ parametrizes tuples $(g,p,v)$ such that $g$ is a
nonsingular conic with a point $p$ and a tangent vector $v$ to $g$ at $p$. We
can describe $J$ using a bundle of principal parts, similarly to the way we
formed $K_{\operatorname{ctp}}$. Let $S^{\prime}=\\{1,y\\}$ and form the
bundle of principal parts $P^{S^{\prime}}(\mathcal{O}(2))$ in the sense of
[CL1, Definition 3.7]. This is a rank $2$ bundle on $\mathbb{P}T$ equipped
with a $G$-equivariant evaluation morphism

$a^{*}\pi^{*}\operatorname{Sym}^{2}V^{*}\rightarrow
a^{*}P^{1}(\mathcal{O}(2))\rightarrow P^{S^{\prime}}(\mathcal{O}(2))\,,$

which is surjective because $\mathcal{O}(2)$ is $2$-very ample. We denote the
kernel by $U$. The space $\mathbb{P}U$ parametrizes triples $(g,p,v)$ such
that $g$ is a conic with a point $p$ and tangent vector $v$ to $g$ at $p$.
Thus, $J\subset\mathbb{P}U$ is the open subspace corresponding to when the
conic is nonsingular. Note that $J$ itself is nonsingular because it is open
inside of a projective bundle $\mathbb{P}U$ over $\mathbb{P}T$.

######  Lemma 15.

The Chow ring $\mathsf{A}_{G}^{*}(J)$ is generated by the restrictions of
$c_{2}$, $z$ and $\tau$ to $J$. In particular, the pullback map

$\mathsf{A}_{G}(\mathbb{P}K^{\circ})\rightarrow\mathsf{A}_{G}^{*}(J)$

is surjective.

###### Proof.

By definition, $\mathbb{P}U$ is a projective bundle over $\mathbb{P}T$, and so
$\mathsf{A}^{*}_{G}(\mathbb{P}U)$ is generated by (the pullbacks of)
$c_{2},c_{3},z,\tau$ and $\zeta=c_{1}(\mathcal{O}_{\mathbb{P}U}(1))$.
Moreover,
$\mathbb{P}U\rightarrow\mathbb{P}(a^{*}\pi^{*}\operatorname{Sym}^{2}V^{*})=\mathbb{P}\operatorname{Sym}^{2}V^{*}\times\mathbb{P}T$
is a linear embedding, under which $J$ maps to the locus of nonsingular conics
$(\mathbb{P}\operatorname{Sym}^{2}V^{*}\smallsetminus\Delta_{2})\times\mathbb{P}T$.
The classes $c_{2},c_{3},z,\tau,$ and $\zeta$ are all pulled back from
$\mathbb{P}\operatorname{Sym}^{2}V^{*}\times\mathbb{P}T$, and on
$(\mathbb{P}\operatorname{Sym}^{2}V^{*}\smallsetminus\Delta_{2})\times\mathbb{P}T$,
the hyperplane class and $c_{3}$ vanish. Therefore, $\mathsf{A}^{*}_{G}(J)$ is
generated by the restrictions of $c_{2},z,\tau$. ∎

######  Corollary 16.

The Chow ring $\mathsf{A}^{*}_{G}(\widetilde{\mathbb{P}K^{\circ}})$ is
generated as an $\mathsf{A}^{*}_{G}(\widetilde{Y})$ module by the restrictions
of $\tau$ and $z$. Hence, the image of the pushforward map

$\mathsf{A}_{G}^{*}(\widetilde{\mathbb{P}K^{\circ}})\rightarrow\mathsf{A}_{G}^{*}(\widetilde{Y})$

is generated by the classes

$(\pi^{\prime\prime}\circ
a^{\prime\prime})_{*}([\widetilde{\mathbb{P}K^{\circ}}]^{G}\cdot\tau^{j}z^{i})$

for $0\leq j\leq 1$ and $0\leq i\leq 2$.

###### Proof.

By Theorem 10, $\mathsf{A}^{*}_{G}(\widetilde{\mathbb{P}K^{\circ}})$ is
generated as an algebra by pullbacks from
$\mathsf{A}_{G}^{*}(\mathbb{P}K^{\circ})$ and the fundamental class of the
exceptional divisor. Because the exceptional divisor of
$\widetilde{\mathbb{P}K^{\circ}}$ is the restriction of the exceptional
divisor of $\mathbb{P}T\times\widetilde{Y}$, every class on
$\widetilde{\mathbb{P}K^{\circ}}$ is pulled back from
$\mathbb{P}T\times\widetilde{Y}$. The Chow ring
$\mathsf{A}^{*}_{G}(\mathbb{P}T\times\widetilde{Y})$ is generated as an
$\mathsf{A}^{*}_{G}(\widetilde{Y})$ module by $\tau$ and $z$. The second
statement follows from the first by the projection formula. ∎

By Lemma 14, to compute
$\mathsf{A}^{*}_{G}(\widetilde{Y}\smallsetminus\widetilde{\mathsf{CTP}})$, we
need to make explicit the classes in Corollary 16 in terms of the generators
of $\mathsf{A}^{*}_{G}(\widetilde{Y})$.

###  6.6. Relations from consecutive triple points

To compute the pushforwards in Corollary 16 in terms of the generators for
$\mathsf{A}^{*}_{G}(\widetilde{Y})$, we need a more explicit formula for the
class $[\widetilde{\mathbb{P}K^{\circ}}]^{G}$. We use Theorem 11, the weighted
blowup formula, applied to the weighted blowup
$f_{2}:\mathbb{P}T\times\widetilde{Y}\rightarrow\mathbb{P}T\times Y$ along
$\mathbb{P}T\times\mathsf{TC}$. We denote by
$i_{2}:\mathbb{P}T\times\mathsf{TC}\rightarrow\mathbb{P}T\times Y$ the
inclusion. We have the weighted blowup diagram

${\mathbb{P}T\times\widetilde{\mathsf{TC}}}$${\mathbb{P}T\times\widetilde{Y}}$${\mathbb{P}T\times\mathsf{TC}}$${\mathsf{\mathbb{P}}T\times
Y\,.}$$\scriptstyle{j_{2}}$$\scriptstyle{g_{2}}$$\scriptstyle{f_{2}}$$\scriptstyle{i_{2}}$

Let $\xi:J\rightarrow\mathbb{P}T\times\mathsf{TC}$ be the inclusion. By
Theorem 11,

(12)
$[\widetilde{\mathbb{P}K^{\circ}}]^{G}=f_{2}^{*}[\mathbb{P}K^{\circ}]^{G}-j_{2*}\\{P_{N}(1)(1+t+t^{2}+\dots)\cdot
g_{2}^{*}\xi_{*}s^{\mathsf{wt}}(N_{J/\mathbb{P}K^{\circ}})\\}_{18}\,,$

where $-t$ is the class of the exceptional divisor.

We analyze the two terms on the right hand side of equation (12) separately.
The fundamental class $[\mathbb{P}K^{\circ}]$ can be determined analogously to
the computation of $[\mathbb{P}K_{\operatorname{quad}}]$ in Section 5. Because
$\mathbb{P}K^{\circ}$ is linearly embedded in $\mathbb{P}T\times Y$, its
equivariant fundamental class is given by

$[\mathbb{P}K^{\circ}]^{G}=c^{G}_{12}(\rho_{2}^{*}P^{S}(\mathcal{O}(6))\otimes\mathcal{O}_{\rho_{2}}(1))=\sum_{k=0}^{12}\rho_{2}^{*}\,c^{G}_{k}(P^{S}(\mathcal{O}(6)))\cdot(\pi^{\prime}\circ
a^{\prime})^{*}H^{12-k}\,.$

######  Remark 17.

Using the above calculation of the equivariant fundamental class, the $6$
classes

$(\pi^{\prime\prime}\circ
a^{\prime\prime})_{*}(f_{2}^{*}[\mathbb{P}K^{\circ}]^{G}\cdot\tau^{j}z^{i})=f^{*}(\pi^{\prime}\circ
a^{\prime})_{*}([\mathbb{P}K^{\circ}]^{G}\cdot\tau^{j}z^{i})$

for $0\leq j\leq 1$ and $0\leq i\leq 2$ can be written explicitly in terms of
the $f$ pullbacks of the generators $H,c_{2},c_{3}$ of
$\mathsf{A}^{*}_{G}(Y)$. The equivariant Chern classes of
$P^{S}(\mathcal{O}(6))$ are computed using that $P^{S}(\mathcal{O}(6))$ is
filtered by tensor products of symmetric powers of the tautological bundles on
$\mathbb{P}T$ and the bundle $\mathcal{O}(6)$. More explicitly, we have the
tautological sequence

$0\rightarrow\mathcal{O}_{\mathbb{P}T}(-1)\rightarrow a^{*}T\rightarrow
Q\rightarrow 0\,.$

We set $\Omega_{x}=\mathcal{O}_{\mathbb{P}T}(1)$ and $\Omega_{y}=Q^{*}$. Then
for each monomial $x^{i}y^{j}$ in $S$, the filtration will have successive
quotients
$\operatorname{Sym}^{i}\Omega_{x}\otimes\operatorname{Sym}^{j}\Omega_{y}\otimes\mathcal{O}(6)$.
For more details on this filtration, see [CL1, Section 3.2]. In any case,

$c_{1}(\Omega_{x})=\tau\,,\quad c_{1}(\Omega_{y})=-3z-\tau\,,$

and the Chern classes of $P^{S}(\mathcal{O}(6))$ follow from here.

We have implemented the calculation in the Macaulay2 package Schubert2 [GS,
GSSEC]:

$\bullet$ For $i=0$ and $j=0$, we obtain

$\displaystyle-36288c_{2}^{3}c_{3}$
$\displaystyle-244944c_{3}^{3}-(254592c_{2}^{4}+2610792c_{2}c_{3}^{2})H-1154736c_{2}^{2}c_{3}H^{2}$
$\displaystyle+(848280c_{2}^{3}+4719870c_{3}^{2})H^{3}+883548c_{2}c_{3}H^{4}-588546c_{2}^{2}H^{5}+61236c_{3}H^{6}$
$\displaystyle+118620c_{2}H^{7}-4362H^{9}\,.$

$\bullet$ For $i=1$ and $j=0$, we obtain

$\displaystyle 34560c_{2}^{5}$
$\displaystyle+233280c_{2}^{2}c_{3}^{2}+(83376c_{2}^{3}c_{3}-2350296c_{3}^{3})H-(623856c_{2}^{4}+6328044c_{2}c_{3}^{2})H^{2}$
$\displaystyle-1198476c_{2}^{2}c_{3}H^{3}+(810240c_{2}^{3}+2435751c_{3}^{2})H^{4}-180306c_{2}c_{3}H^{5}-316413c_{2}^{2}H^{6}$
$\displaystyle+84186c_{3}H^{7}+28950c_{2}H^{8}-381H^{10}\,.$

$\bullet$ For $i=2$ and $j=0$, we obtain

$\displaystyle 70848c_{2}^{4}c_{3}$
$\displaystyle+478224c_{2}c_{3}^{3}+(254592c_{2}^{5}+2960712c_{2}^{2}c_{3}^{2})H+(525240c_{2}^{3}c_{3}-4865832c_{3}^{3})H^{2}$
$\displaystyle-(847664c_{2}^{4}+6647562c_{2}c_{3}^{2})H^{3}-93798c_{2}^{2}c_{3}H^{4}+(589876c_{2}^{3}+567756c_{3}^{2})H^{5}$
$\displaystyle-395280c_{2}c_{3}H^{6}-117822c_{2}^{2}H^{7}+26550c_{3}H^{8}+4432c_{2}H^{9}-14H^{11}\,.$

$\bullet$ For $i=0$ and $j=1$, we obtain

$\displaystyle-69120c_{2}^{5}$
$\displaystyle-466560c_{2}^{2}c_{3}^{2}-(616896c_{2}^{3}c_{3}-1032264c_{3}^{3})H+(1181376c_{2}^{4}+10152540c_{2}c_{3}^{2})H^{2}$
$\displaystyle+2659392c_{2}^{2}c_{3}H^{3}-(1697460c_{2}^{3}+5233653c_{3}^{2})H^{4}-441774c_{2}c_{3}H^{5}+623943c_{2}^{2}H^{6}$
$\displaystyle-30942c_{3}H^{7}-56070c_{2}H^{8}+831H^{10}\,.$

$\bullet$ For $i=1$ and $j=1$, we obtain

$\displaystyle 31104c_{2}^{4}c_{3}$
$\displaystyle+209952c_{2}c_{3}^{3}-(463104c_{2}^{5}+4805568c_{2}^{2}c_{3}^{2})H-(1385856c_{2}^{3}c_{3}-5604552c_{3}^{3})H^{2}$
$\displaystyle+(1718688c_{2}^{4}+12248496c_{2}c_{3}^{2})H^{3}+845640c_{2}^{2}c_{3}H^{4}-(1212072c_{2}^{3}+1992276c_{3}^{2})H^{5}$
$\displaystyle+343116c_{2}c_{3}H^{6}+225864c_{2}^{2}H^{7}-30564c_{3}H^{8}-9024c_{2}H^{9}+48H^{11}\,.$

$\bullet$ For $i=2$ and $j=1$, we obtain

$\displaystyle 69120c_{2}^{6}$
$\displaystyle+575424c_{2}^{3}c_{3}^{2}+734832c_{3}^{4}+(149472c_{2}^{4}c_{3}-4974696c_{2}c_{3}^{3})H$
$\displaystyle-(1180896c_{2}^{5}+12642480c_{2}^{2}c_{3}^{2})H^{2}-(953352c_{2}^{3}c_{3}-7605414c_{3}^{3})H^{3}$
$\displaystyle+(1698256c_{2}^{4}+7821144c_{2}c_{3}^{2})H^{4}-773712c_{2}^{2}c_{3}H^{5}-(623858c_{2}^{3}+346977c_{3}^{2})H^{6}$
$\displaystyle+263682c_{2}c_{3}H^{7}+55773c_{2}^{2}H^{8}-7110c_{3}H^{9}-896c_{2}H^{10}+H^{12}\,.$

Next, we compute the contributions from the the second term

$j_{2*}\\{P_{N}(1)(1+t+t^{2}+\dots)\cdot
g_{2}^{*}\xi_{*}s^{\mathsf{wt}}(N_{J/\mathbb{P}K^{\circ}})\\}_{18}\,$

in equation (12). First, recall that the polynomial $P_{N}$ is the top
weighted Chern class of the weighted normal bundle
$N_{\mathbb{P}T\times\mathsf{TC}/\mathbb{P}T\times
Y}=\text{pr}^{*}N_{\mathsf{TC}/Y}$. We recall from Section 3.3 that

$N_{\mathsf{TC}/Y}=\operatorname{Sym}^{8}W^{*}\oplus\operatorname{Sym}^{12}W^{*}$

with weights $2$ and $3$, where $W$ is a rank $2$ bundle with trivial first
Chern class such that $\operatorname{Sym}^{2}W=V$. In particular,
$c^{G}_{2}(W)=\frac{1}{4}c_{2}$. Therefore,

$P_{N}(t)=(\pi^{\prime}\circ
a^{\prime})^{*}\left(\left(\sum_{i=0}^{9}c^{G}_{i}(\operatorname{Sym}^{8}W^{*})(2t)^{9-i}\right)\left(\sum_{i=0}^{13}c^{G}_{i}(\operatorname{Sym}^{12}W^{*})(3t)^{13-i}\right)\right)\,.$

In fact,

(13)
$\displaystyle\,\,\,\,P_{N}(t)=816293376\,t^{22}+14375833344\,c_{2}t^{20}+106093574496\,c_{2}^{2}t^{18}$
$\displaystyle+428059655424\,c_{2}^{3}t^{16}+1034306024256\,c_{2}^{4}t^{14}+1543171803264\,c_{2}^{5}t^{12}+1416080524896\,c_{2}^{6}t^{10}$
$\displaystyle+772698973824\,c_{2}^{7}t^{8}+233467992576\,c_{2}^{8}t^{6}+34249734144c_{2}^{9}t^{4}+1791590400\,c_{2}^{10}t^{2}\,.$

Next, we calculate the weighted Segre class
$s^{\mathsf{wt}}(N_{J/\mathbb{P}K^{\circ}})$. Because $J$ is open in
$\mathbb{P}U$, $N_{J/\mathbb{P}K^{\circ}}$ is just the restriction of
$N_{\mathbb{P}U/\mathbb{P}K^{\circ}}$ to $J$. We first discuss the weight
decomposition of this bundle.

The bundle $K_{\operatorname{ctp}}$ was constructed as a subbundle

$K_{\operatorname{ctp}}\hookrightarrow
a^{*}\pi^{*}\operatorname{Sym}^{6}V^{*}$

corresponding to monomials in the set
$S=\\{1,x,y,x^{2},xy,y^{2},x^{2}y,xy^{2},y^{3},xy^{3},y^{4},y^{5}\\}$, which
are the monomials that must vanish for the sextic to lie in $(x,y^{2})^{3}$.
The filtration determining the weights is the restriction of the filtration
determining the weights on $a^{*}\pi^{*}\operatorname{Sym}^{6}V^{*}$. This
filtration was constructed in [S, Section 5]. We fix the equation $Q$ of a
nonsingular conic and obtain a filtration

$\operatorname{Sym}^{6}V^{*}\supset Q\cdot\operatorname{Sym}^{4}V^{*}\supset
Q^{2}\cdot\operatorname{Sym}^{2}V^{*}\supset Q^{3}\cdot\mathbb{C}\,.$

This corresponds to the decomposition of each sextic into pieces

$f=f_{6}+f_{4}\cdot Q+f_{2}\cdot Q^{2}+f_{0}\cdot Q^{3}.$

The relevant successive subquotients are

$\operatorname{Sym}^{6}V^{*}/\operatorname{Sym}^{4}V^{*}\cong\operatorname{Sym}^{12}W^{*},\quad\operatorname{Sym}^{4}V^{*}/\operatorname{Sym}^{2}V^{*}\cong\operatorname{Sym}^{8}W^{*},$

which correspond to picking out the terms $f_{6}|_{Q}$ and $f_{4}|_{Q}$. The
restriction of this filtration to $K_{\operatorname{ctp}}$ is denoted

$K_{\operatorname{ctp}}\supset Q\cdot K_{4}\supset Q^{2}\cdot K_{2}\supset
Q^{3}\cdot K_{0}\,.$

We can describe the pieces of the filtration more explicitly in terms of
principal parts. In local coordinates, we write the equation for the
nonsingular conic as $Q=x-y^{2}$. Then $K_{0}$ is spanned by $Q^{3}$. The
bundle $K_{2}$ is given by the degree $2$ polynomials $h$ such that

$Q^{2}h\in(x,y^{2})^{3}.$

Similarly, $K_{4}$ is given degree $4$ polynomials $h$ such that

$Qh\in(x,y^{2})^{3}.$

Therefore, sections of $K_{2}$ are degree $2$ polynomials with monomials in
the set $\\{x,x^{2},xy,y^{2}\\}$, and sections of $K_{4}$ are degree $4$
polynomials with monomials in
$\\{x^{2},x^{3},x^{2}y,xy^{2},x^{4},x^{3}y,xy^{3},x^{2}y^{2},y^{4}\\}$. In
other words, setting $S^{\prime}=\\{1,y\\}$, we have an exact sequence

(14) $0\rightarrow K_{2}\rightarrow
a^{*}\pi^{*}\operatorname{Sym}^{2}V^{*}\rightarrow
P^{S^{\prime}}(\mathcal{O}(2))\rightarrow 0\,,$

where $P^{S^{\prime}}(\mathcal{O}(2))$ is the bundle of principle parts along
$S^{\prime}$ in the sense of [CL1, Definition 3.7]. Note that this was exactly
the way in which $U$ was defined, so $K_{2}\cong U$. Similarly, we have

(15) $0\rightarrow K_{4}\rightarrow
a^{*}\pi^{*}\operatorname{Sym}^{4}V^{*}\rightarrow
P^{S^{\prime\prime}}(\mathcal{O}(4))\rightarrow 0\,,$

where $S^{\prime\prime}=\\{1,x,y,xy,y^{2},y^{3}\\}$. Therefore, the normal
bundle $N_{J/\mathbb{P}K^{\circ}}$ has the same weighted Chern class as

$K_{4}/K_{2}\oplus K_{\operatorname{ctp}}/K_{4}$

with weight $2$ on the first factor and weight $3$ on the second. Here we have
omitted the restriction to $J$ from the notation. Using the defining exact
sequences (14), (15), (9) for $K_{2},K_{4}$, and $K_{\operatorname{ctp}}$
allows us to compute the weighted Segre class
$s^{\mathsf{wt}}(N_{J/\mathbb{P}K^{\circ}})$ using standard Chern class
manipulations. The resulting expression is explicit, but a bit long, so we
record only a few terms

$s^{\mathsf{wt}}(N_{J/\mathbb{P}K^{\circ}})=\frac{1}{2^{5}\cdot
3^{7}}\left(1+(2z-5\tau)+\frac{1}{36}(-2536c_{2}+285\tau^{2}-869\tau
z-603z^{2})+\ldots\right).$

Higher order terms are calculated using Macaulay2 [COP1].

Next, we need to compute the pushforward
$\xi_{*}s^{\mathsf{wt}}(N_{J/\mathbb{P}K^{\circ}})$. Because every class on
$J$ is pulled back from $\mathbb{P}T\times\mathsf{TC}$ as was shown in the
proof of Lemma 15, the pushforward
$\xi_{*}s^{\mathsf{wt}}(N_{J/\mathbb{P}K^{\circ}})$ is given by multiplication
with the fundamental class of $J$. The fundamental class of $J$ is given by
the the top Chern class

$c^{G}_{2}(P^{S^{\prime}}(\mathcal{O}(2))\otimes\mathcal{O}_{\mathbb{P}U}(1))\,$

restricted to $J$. In fact, $\mathcal{O}_{\mathbb{P}U}(1)$ is trivial since
the hyperplane class is trivial on $\mathsf{TC}$, and thus

$[J]=c_{2}(P^{S^{\prime}}(\mathcal{O}(2)))=2z\cdot(2z+c_{1}(\Omega_{y}))=2z\cdot(2z+(-3z-\tau))\,.$

######  Remark 18.

We have now described each term in the expression

$j_{2*}\\{P_{N}(1)(1+t+t^{2}+\dots)\cdot
g_{2}^{*}\xi_{*}s^{\mathsf{wt}}(N_{J/\mathbb{P}K^{\circ}})\\}_{18}\,.$

Using Macaulay2 [COP1], we have computed the contributions coming from the
exceptional terms:

$\bullet$ For $i=0$ and $j=0$, we obtain

$-398016\,c_{2}^{4}\,\left[\mathsf{Ell}\right]+5472720\,c_{2}^{3}\,\left[\mathsf{Ell}\right]^{3}-11144448\,c_{2}^{2}\,\left[\mathsf{Ell}\right]^{5}+4477680\,c_{2}\,\left[\mathsf{Ell}\right]^{7}-279936\,\left[\mathsf{Ell}\right]^{9}\,.$

$\bullet$ For $i=1$ and $j=0$, we obtain

$1961568\,c_{2}^{4}\,\left[\mathsf{Ell}\right]^{2}-9611640\,c_{2}^{3}\,\left[\mathsf{Ell}\right]^{4}+8667504\,c_{2}^{2}\,\left[\mathsf{Ell}\right]^{6}-1474200\,c_{2}\,\left[\mathsf{Ell}\right]^{8}+23328\,\left[\mathsf{Ell}\right]^{10}\,.$

$\bullet$ For $i=2$ and $j=0$, we obtain

$398016\,c_{2}^{5}\left[\mathsf{Ell}\right]-5472720\,c_{2}^{4}\,\left[\mathsf{Ell}\right]^{3}+11144448\,c_{2}^{3}\,\left[\mathsf{Ell}\right]^{5}-4477680\,c_{2}^{2}\,\left[\mathsf{Ell}\right]^{7}+279936\,c_{2}\,\left[\mathsf{Ell}\right]^{9}\,.$

$\bullet$ For $i=0$ and $j=1$, we obtain

$-3923136\,c_{2}^{4}\,\left[\mathsf{Ell}\right]^{2}+19223280\,c_{2}^{3}\,\left[\mathsf{Ell}\right]^{4}-17335008\,c_{2}^{2}\,\left[\mathsf{Ell}\right]^{6}+2948400\,c_{2}\,\left[\mathsf{Ell}\right]^{8}-46656\,\left[\mathsf{Ell}\right]^{10}\,.$

$\bullet$ For $i=1$ and $j=1$, we obtain

$-796032\,c_{2}^{5}\,\left[\mathsf{Ell}\right]+10945440\,c_{2}^{4}\,\left[\mathsf{Ell}\right]^{3}-22288896\,c_{2}^{3}\,\left[\mathsf{Ell}\right]^{5}+8955360\,c_{2}^{2}\,\left[\mathsf{Ell}\right]^{7}-559872\,c_{2}\,\left[\mathsf{Ell}\right]^{9}\,.$

$\bullet$ For $i=2$ and $j=1$, we obtain

$3923136\,c_{2}^{5}\,\left[\mathsf{Ell}\right]^{2}-19223280\,c_{2}^{4}\,\left[\mathsf{Ell}\right]^{4}+17335008\,c_{2}^{3}\,\left[\mathsf{Ell}\right]^{6}-2948400\,c_{2}^{2}\,\left[\mathsf{Ell}\right]^{8}+46656\,c_{2}\,\left[\mathsf{Ell}\right]^{10}\,.$

These terms are to be subtracted from the terms in Remark 17 to yield the
additional $6$ relations in the Chow group of $\mathcal{F}_{2}$ coming from
the removal of $\widetilde{\mathsf{CTP}}.$

######  Remark 19.

We can compare the calculations here with the results of [CK]. In [CK, Theorem
1.2], the Chow ring of the moduli space of elliptic $K3$ surfaces has been
computed, and the ideal of relations has two generators in degrees $9$, $10$.
An easy check shows that these relations arise by restricting the relations
with $i=0$, $j=0$ and $i=1$, $j=0$ in Remark 17 and Remark 18 to the elliptic
locus.

Since the geometry of the weighted blowup and the geometry of $\mathsf{Ell}$
can be explicitly related, the matching of relations is guaranteed. In fact,
the relations in [CK, Remark 3.5] on $\mathsf{Ell}$ arise from removing
classes supported on the codimension $9$ locus $\Delta$ of elliptic fibrations
which are not stable, see [CK, Lemma 2.5(2)]. This corresponds precisely to
the exceptional divisor of the blowup $\widetilde{\mathsf{CTP}}$, as discussed
in Section 3.3.

###  6.7. Proof of Theorem 1 (iii)

The quotient presentation of $\mathcal{F}_{2}$ in Section 3 yields

$\mathsf{A}^{*}(\mathcal{F}_{2})=\mathsf{A}^{*}_{G}(\widetilde{Y}\smallsetminus(\widetilde{\mathsf{CTP}}\cup\widetilde{\mathsf{DC}}))\,.$

In the ring $\mathsf{A}_{G}^{*}(X)=\mathbb{Q}[H,c_{2},c_{3}]/(p)$, where555The
exact expression for $p$ is easy to write down. However, this will not be
needed for degree reasons.

$p=H^{28}+c_{1}^{G}(\operatorname{Sym}^{6}V^{*})H^{27}+\cdots+c_{28}^{G}(\operatorname{Sym}^{6}V^{*})\,,$

we quotient by the ideal generated by the relations from Lemmas 6 and 8
(explicitly written in Remark 7 and Remark 9) to obtain
$\mathsf{A}^{*}_{G}(Y)$. Next, we carry out the weighted blowup along
$\mathsf{TC}$, and we impose the relations in Corollary 12 to obtain
$\mathsf{A}_{G}^{*}(\widetilde{Y})$. The additional generator
$\left[\mathsf{Ell}\right]$ appears at this stage. Finally, we impose the $6$
relations given by Corollary 16 and made explicit in Remark 17 and Remark 18.
In total, there are $14$ relations that we need to account for. Using the
Macaulay2 package Schubert2 [GS, GSSEC] we find that

$\displaystyle\sum_{k=0}^{19}t^{k}\cdot\dim\mathsf{A}_{G}^{k}(\widetilde{Y}\smallsetminus\widetilde{\mathsf{CTP}})=1+$
$\displaystyle
2t+3t^{2}+5t^{3}+6t^{4}+8t^{5}+10t^{6}+12t^{7}+13t^{8}+14t^{9}+12t^{10}$
$\displaystyle+10t^{11}+8t^{12}+6t^{13}+5t^{14}+3t^{15}+2t^{16}+t^{17}\,.$

The above polynomial is precisely the polynomial in the statement of Theorem
1. The space $\mathcal{F}_{2}$ is an open substack of
$[(\widetilde{Y}\smallsetminus\widetilde{\mathsf{CTP}})/G]$ whose complement
$\widetilde{\mathsf{DC}}$ has components of codimension $17$ and $18$. By
excision, it follows that the Poincaré polynomial of $\mathcal{F}_{2}$ agrees
with that of $[(\widetilde{Y}\smallsetminus\widetilde{\mathsf{CTP}})/G]$,
except for possibly the coefficient of $t^{17}$. We know, however, that
$\dim\mathsf{A}^{17}(\mathcal{F}_{2})\geq 1$ because $\lambda^{17}\neq 0$
[GK]. Therefore, we see that $\dim\mathsf{A}^{17}(\mathcal{F}_{2})=1$. ∎

######  Remark 20.

Let $I$ be the ideal of relations in Remarks 6 and 9. Using the Macaulay2
package Schubert2 [GS, GSSEC] we find that

$\mathsf{A}^{*}_{G}(X)/I=\mathsf{A}_{G}^{*}(Y\smallsetminus\mathsf{CTP})$

has the Poincaré polynomial

(16)
$\displaystyle\sum_{k=0}^{19}t^{k}\cdot\dim\mathsf{A}_{G}^{k}(Y\smallsetminus\mathsf{CTP})=1+$
$\displaystyle
t+2t^{2}+3t^{3}+4t^{4}+5t^{5}+7t^{6}+8t^{7}+9t^{8}+9t^{9}+8t^{10}$
$\displaystyle+6t^{11}+5t^{12}+3t^{13}+3t^{14}+t^{15}+t^{16}\,.$

By excision (and the fact that we can ignore the stratum
$\widetilde{\mathsf{DC}}$), we have

(17)
$\mathsf{A}^{k-1}(\mathsf{Ell})\,\rightarrow\,\mathsf{A}^{k}(\mathcal{F}_{2})\rightarrow{\mathsf{A}}^{k}_{G}(Y\smallsetminus\mathsf{CTP})\rightarrow
0\,.$

The Chow Betti numbers of the right hand side are computed in (16), the Chow
Betti numbers of the middle term are given in Theorem 1 (iii), while the Chow
Betti numbers of $\mathsf{Ell}$ are found in [CK, Theorem 1.2 (2)]:

$\displaystyle 1+t+2t^{2}+2t^{3}+3t^{4}+3t^{5}+$ $\displaystyle
4t^{6}+4t^{7}+5t^{8}+4t^{9}+4t^{10}+3t^{11}+3t^{12}+2t^{13}+2t^{14}+t^{15}+t^{16}.$

Comparing ranks, it follows that the excision sequence (17) is exact to the
left.

In particular, the inclusion $r:\mathsf{Ell}\to\mathcal{F}_{2}$ induces an
isomorphism between the top nonvanishing Chow groups

$r_{*}:\mathsf{A}^{16}(\mathsf{Ell})\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,\mathsf{A}^{17}(\mathcal{F}_{2})\,.$

We will use the isomorphism in Section 7.

######  Remark 21.

The description of the Chow ring of $\mathcal{F}_{2}$ in terms of generators
and relations also allows us study the pairing

$\mathsf{R}^{k}(\mathcal{F}_{2})\times\mathsf{R}^{17-k}(\mathcal{F}_{2})\to\mathsf{R}^{17}(\mathcal{F}_{2})\cong\mathbb{Q}.$

For $k$ not equal to 8 or 9, the pairing is perfect. The kernel of the paring

$\mathsf{R}^{8}(\mathcal{F}_{2})\times\mathsf{R}^{9}(\mathcal{F}_{2})\to\mathsf{R}^{17}(\mathcal{F}_{2})\cong\mathbb{Q}$

has rank $1$. The computations were carried out using code available at
[COP1]. The class in $\mathsf{A}^{9}(\mathcal{F}_{2})$ generating the
Gorenstein kernel is

$\displaystyle 24$ $\displaystyle
477984\,c_{2}^{4}\,\left[\mathsf{Ell}\right]-336572280\,c_{2}^{3}\,\left[\mathsf{Ell}\right]^{3}+685383552\,c_{2}^{2}\,\left[\mathsf{Ell}\right]^{5}-275377320\,c_{2}\,\left[\mathsf{Ell}\right]^{7}+17216064\,\left[\mathsf{Ell}\right]^{9}$
$\displaystyle-14925600\,c_{2}^{4}\,H+151121700\,c_{2}^{2}\,c_{3}\,H^{2}+108694300\,c_{2}^{3}\,H^{3}-125188470\,c_{3}^{2}\,H^{3}$
$\displaystyle-305321805\,c_{2}\,c_{3}\,H^{4}-107744175\,c_{2}^{2}H^{5}+65952495c_{3}H^{6}+19658950\,c_{2}\,H^{7}-646575\,H^{9}\,.$

Finding a geometric interpretation of the kernel class is an open question.

##  7\. The cycle map

We present here the proof of part (iv) of Theorem 1. Throughout Section 7,
$H_{*}$ will denote rational Borel–Moore homology [BM]. In general, for any
scheme or Deligne-Mumford stack $M$, the group $H_{k}(M)$ carries a mixed
Hodge structure and an increasing weight filtration with weights between $-k$
and $0$. The cycle map takes values in the lowest weight piece of the Hodge
structure

$\mathsf{cl}:\mathsf{A}_{k}(M)\to W_{-2k}H_{2k}(M)\,.$

If $M$ is nonsingular, we identify cohomology and Borel–Moore homology, but
singular spaces will also enter the discussion.

We seek to show that the cycle map

$\mathsf{cl}:\mathsf{A}_{k}(\mathcal{F}_{2})\to H_{2k}(\mathcal{F}_{2})$

is an isomorphism. Using the expressions for the Poincaré polynomial
calculated in [KL2] and Appendix A together with the Chow Betti numbers from
Theorem 1 (iii), it suffices to prove that the cycle map is injective. We will
prove below the following related injectivity.

######  Lemma 22.

The cycle map

$\mathsf{cl}:{\mathsf{A}}_{k}^{G}(Y\smallsetminus\mathsf{CTP})\to
W_{-2k}H_{2k}^{G}(Y\smallsetminus{\mathsf{CTP}})$

is injective.

Assuming Lemma 22 for now, let $\alpha\in\mathsf{A}_{k}(\mathcal{F}_{2})$ be
so that $\mathsf{cl}(\alpha)=0$. We wish to show $\alpha=0.$ If $\alpha$ is
not in the kernel of the intersection pairing in
$\mathsf{A}^{*}(\mathcal{F}_{2})$, we can find a class $\alpha^{\prime}$ of
complementary degree so that $\alpha\cdot\alpha^{\prime}\neq 0.$ In
particular, we may assume $\alpha\cdot\alpha^{\prime}=\lambda^{17}$ since the
latter generates $\mathsf{A}^{17}(\mathcal{F}_{2}).$ Then,

$0=\mathsf{cl}(\alpha)\cdot\mathsf{cl}(\alpha^{\prime})=\mathsf{cl}(\lambda)^{17}.$

However, the same argument used in Chow in [GK] shows that in cohomology we
also have $\lambda^{17}\neq 0,$ yielding a contradiction.

Thus $\alpha$ must be in the kernel of the intersection paring. In particular
$\alpha$ must be the class in Remark 21, up to a multiple, but we will not use
this fact in the proof below; we only care that $k>2$. We can write

$\alpha=\zeta^{*}\beta\,.$

Here, we recall that $\zeta:\mathcal{F}_{2}\hookrightarrow[\widetilde{Y}/G]$
with complement $\widetilde{\mathsf{DC}}$ and $\widetilde{\mathsf{CTP}},$ and
$\beta$ is a Chow class on $[\widetilde{Y}/G].$

Consider the diagram

${\mathsf{A}^{G}_{k}(\widetilde{\mathsf{CTP}})}$${\mathsf{A}_{k}^{G}(\widetilde{Y})}$${\mathsf{A}^{G}_{k}(\widetilde{Y}\smallsetminus(\widetilde{\mathsf{CTP}}\cup\widetilde{\mathsf{DC}}))}$${0}$${W_{-2k}H^{G}_{2k}(\widetilde{\mathsf{CTP}})}$${W_{-2k}H_{2k}^{G}(\widetilde{Y})}$${W_{-2k}H^{G}_{2k}(\widetilde{Y}\smallsetminus(\widetilde{\mathsf{CTP}}\cup\widetilde{\mathsf{DC}}))}$${0\,.}$$\scriptstyle{\eta_{*}}$$\scriptstyle{\mathsf{cl}}$$\scriptstyle{\zeta^{*}}$$\scriptstyle{\mathsf{cl}}$$\scriptstyle{\mathsf{cl}}$$\scriptstyle{\eta_{*}}$$\scriptstyle{\zeta^{*}}$

For the second excision sequence, exactness to the right follows since we keep
track of the Hodge weights. For both exact sequences, we may ignore the two
dimensional set $\widetilde{\mathsf{DC}}$ on the left terms for dimension
reasons. Since

$\zeta^{*}\,(\mathsf{cl}(\beta))=\mathsf{cl}\,(\zeta^{*}(\beta))=\mathsf{cl}(\alpha)=0\,,$

it follows that over $\widetilde{Y}$, we have

$\mathsf{cl}(\beta)=\eta_{*}\,(\gamma)$

where $\gamma$ is an equivariant Borel–Moore homology class on the locus
$\widetilde{\mathsf{CTP}}.$ Using the blowdown map $f:\widetilde{Y}\to Y,$ we
obtain

$f_{*}\,\mathsf{cl}({\beta})=f_{*}\,\eta_{*}\,(\gamma)\,,$

where the right hand side is a Borel–Moore class on $\mathsf{CTP}.$ The
restriction $f_{*}\,\mathsf{cl}(\beta)=\mathsf{cl}\,(f_{*}\beta)$ thus
vanishes in the Borel–Moore homology of
$\left[\left(Y\smallsetminus{\mathsf{CTP}}\right)/G\right]$, so by Lemma 22,
we conclude

$f_{*}\,(\beta)=0$

in $\mathsf{A}^{G}_{k}(Y\smallsetminus\mathsf{CTP}).$ By excision, we can find
a class $\delta$ such that on $Y$ we have

$f_{*}\,(\beta)=\overline{\eta}_{*}\,(\delta)\,,$

where $\overline{\eta}:\mathsf{CTP}\cap Y\hookrightarrow Y.$ In particular

$f_{*}(\beta-f^{*}\overline{\eta}_{*}\,(\delta))=0$

in $\mathsf{A}^{G}_{k}(Y)$, hence

$\beta-f^{*}\overline{\eta}_{*}\,(\delta)$

is a class supported on the exceptional divisor of the blowup
$f:\widetilde{Y}\to Y$ by excision applied to the embedding of the exceptional
divisor in $\widetilde{Y}$. We restrict the class
$\beta-f^{*}\overline{\eta}_{*}\,(\delta)$ to $\mathcal{F}_{2}$ via $\zeta$,
and note that $f^{*}\overline{\eta}_{*}\,(\delta)$ restricts trivially since
we removed the strict transform $\widetilde{\mathsf{CTP}}.$ We conclude that

$\alpha=r_{*}\,(\epsilon)\,,$

for a class $\epsilon$ on $\mathsf{Ell},$ where as usual
$r:\mathsf{Ell}\to\mathcal{F}_{2}$ denotes the inclusion. We claim however
that in this case $\alpha$ cannot be in the kernel of the intersection pairing
unless $\alpha=0$.

To see this last statement, recall from [CK] that
$\mathsf{A}^{*}(\mathsf{Ell})$ is Gorenstein. If $\epsilon\neq 0,$ we can find
a complementary class $\epsilon^{\prime}$ with

$\epsilon\cdot\epsilon^{\prime}=\lambda^{16}\,.$

The pullback

$r^{*}:\mathsf{A}^{*}(\mathcal{F}_{2})\to\mathsf{A}^{*}(\mathsf{Ell})$

is surjective, since two of the ring generators $\lambda$, $c_{2}$ on the left
hand side are sent to the ring generators $\lambda$, $4c_{2}$ on the right
hand side. Thus, we may write

$\epsilon^{\prime}=r^{*}\xi\,.$

Since $\alpha$ is in the kernel of the pairing, we have

$0=\alpha\cdot\xi=r_{*}\,(\epsilon)\cdot\xi=r_{*}\,(\epsilon\cdot
r^{*}\xi)=r_{*}\,(\epsilon\cdot\epsilon^{\prime})=r_{*}\,(\lambda^{16})\,.$

This contradicts Remark 20. ∎Proof of Lemma 22. For simplicity, write

$Z=\mathsf{ML}\cup\mathsf{QP}\cup\mathsf{CTP}\subset X$

where as before $X$ denotes the projective space of sextics. Then
$Y\smallsetminus\mathsf{CTP}=X\smallsetminus Z.$ We need to establish the
injectivity of the map

$\mathsf{cl}:{\mathsf{A}}^{G}_{k}(X\smallsetminus Z)\to
W_{-2k}H_{2k}^{G}(X\smallsetminus Z)\,.$

Consider the following excision diagram

(18)
${\mathsf{A}^{G}_{k}(Z)}$${\mathsf{A}_{k}^{G}(X)}$${\mathsf{A}^{G}_{k}(X\smallsetminus
Z)}$${0}$${W_{-2k}H^{G}_{2k}(Z)}$${W_{-2k}H_{2k}^{G}(X)}$${W_{-2k}H^{G}_{2k}(X\smallsetminus
Z)}$${0\,.}$$\scriptstyle{\mathsf{cl}}$$\scriptstyle{\mathsf{cl}}$$\scriptstyle{\mathsf{cl}}$

We first claim that the middle cycle map is an isomorphism. Indeed, recall
that $G=\operatorname{SL}(V)$ and let $K=\operatorname{GL}(V)$. We have an
isomorphism

$\mathsf{A}^{K}_{k}(X)\to H_{2k}^{K}(X)\,.$

This follows by explicitly computing both sides. In fact, both sides agree
with the cohomology of the bundle

$X_{K}=\mathbb{P}(\operatorname{Sym}^{6}E^{*})\to BK\,,$

where $E\to\operatorname{BGL}(V)$ is the universal bundle. To go further, we
use the terminology of [To, Section 4]. There, two properties are singled out:
the weak property is the statement that the cycle map is an isomorphism, while
the strong property requires additional assumptions about odd cohomology,
which vanishes for $X_{K}$. In other words,
$\mathbb{P}(\operatorname{Sym}^{6}E^{*})\to BK$ satisfies the strong property.
To pass to the group $G$, we note that the mixed space $X_{G}\to X_{K}$ is a
$\mathbb{C}^{*}$-bundle obtained from the total space of the determinant line
bundle $F=\det\text{pr}^{*}\operatorname{Sym}^{6}E^{*}$ on
$\mathbb{P}(\operatorname{Sym}^{6}E^{*})$ and removing the zero section. By
homotopy equivalence, $F$ also satisfies the strong property since $X_{K}$
does, and the zero section satisfies it as well. The complement satisfies the
weak property by [To, Lemma 6], as claimed.

To show the rightmost cycle map is injective in the diagram (18), it suffices
to show the leftmost cycle map is surjective. For simplicity, write

$Z_{1}=\mathsf{ML}\,,\quad Z_{2}=\mathsf{QP}\,,\quad Z_{3}=\mathsf{CTP}$

for the three components of $Z$, and write $T_{1},T_{2},T_{3}$ for the
nonsingular spaces that dominate them

$T_{1}=\mathbb{P}(V^{*})\times\mathbb{P}(\operatorname{Sym}^{4}V^{*})\,,\quad
T_{2}=\mathbb{P}K_{\textrm{quad}}\,,\quad T_{3}=\mathbb{P}K_{\textrm{ctp}}\,.$

By Mayer–Vietoris in both Chow [F, Example 1.3.1(c)] and Borel–Moore homology
[BM, Theorem 3.10] and [PS, Theorem 5.35 and Remark 5.36], we have a diagram

${\mathsf{A}^{G}_{k}(Z_{1})\,\oplus\,\mathsf{A}^{G}_{k}(Z_{2})\,\oplus\,\mathsf{A}^{G}_{k}(Z_{3})}$${\mathsf{A}_{k}^{G}(Z)}$${0}$${W_{-2k}H^{G}_{2k}(Z_{1})\,\oplus
W_{-2k}H_{2k}^{G}(Z_{2})\,\oplus
W_{-2k}H^{G}_{2k}(Z_{3})}$${W_{-2k}H_{2k}^{G}(Z)}$${0\,.}$$\scriptstyle{\mathsf{cl}}$$\scriptstyle{\mathsf{cl}}$

The surjectivity of the second row follows since the $(2k-1)$st Borel–Moore
homology groups has no Hodge pieces of weight $-2k$. Therefore, to complete
the proof we need to check the surjectivity of the cycle map on the left.

For each $1\leq i\leq 3,$ we form the diagram

${\mathsf{A}^{G}_{k}(T_{i})}$${\mathsf{A}_{k}^{G}(Z_{i})}$${0}$${W_{-2k}H^{G}_{2k}(T_{i})}$${W_{-2k}H_{2k}^{G}(Z_{i})}$${0\,.}$$\scriptstyle{\mathsf{cl}}$$\scriptstyle{\mathsf{cl}}$

Surjectivity of the first row is standard (and, in fact, is not necessary for
us), while surjectivity of the second row is found in [Le, Lemma A.4] or
[Pe2]. The final step is then to prove that the cycle map on the left is
surjective. The left cycle map is an isomorphism by the same argument used for
$X$ using the explicit description of $T_{1},T_{2},T_{3}$ as iterated
projective bundles over projective spaces. ∎

##  Appendix A The Poincaré polynomial of the moduli space

###  A.1. The results of Kirwan and Lee

We discuss here the calculation of the Poincaré polynomial of
$\mathcal{F}_{2}$ in [KL1, KL2]. The value of the Poincaré polynomial given in
[KL2, Theorem 3.1] is

(19)
$1+2q^{2}+3q^{4}+5q^{6}+6q^{8}+8q^{10}+10q^{12}+12q^{14}+13q^{16}+14q^{18}+12q^{20}\\\
+10q^{22}+8q^{24}+6q^{26}+q^{27}+5q^{28}+3q^{30}+q^{31}+2q^{32}+q^{33}+3q^{35}\,.$

However, the above polynomial is incompatible with the geometry of the moduli
space. Indeed, the projective Bailey-Borel compactification

$\mathcal{F}_{2}\hookrightarrow\overline{\mathcal{F}}^{\text{\,BB}}$

has a $1$-dimensional boundary. Using this observation, it was shown in [GK]
that

(20) $\lambda^{17}\neq 0\in H^{34}(\mathcal{F}_{2}).$

In fact, intersecting two general hyperplane sections of
$\overline{\mathcal{F}}^{\text{\,BB}}$ gives a compact $17$-dimensional
subvariety of $\mathcal{F}_{2}$ on which $\lambda^{17}$ is non-zero by the
ampleness of $\lambda$. However, this contradicts the vanishing
$H^{34}(\mathcal{F}_{2})=0$ implied by (19).

The value of the Poincaré polynomial used throughout our paper is

(21)
$1+2q^{2}+3q^{4}+5q^{6}+6q^{8}+8q^{10}+10q^{12}+12q^{14}+13q^{16}+14q^{18}+12q^{20}\\\
+10q^{22}+8q^{24}+6q^{26}+q^{27}+5q^{28}+3q^{30}+q^{31}+2q^{32}+2q^{33}+q^{34}+3q^{35}\,,$

which differs from (19) by $q^{33}+q^{34}$. The correction is aligned with the
non-vanishing (20) of cohomology in degree $34.$

The main error in [KL2] is in the proof, but not the statement, of Proposition
3.2. First, in [KL2, Lemma 5.6], Kirwan and Lee claim to describe the image of
a certain map $\tau_{2}^{*}$, but actually only describe a proper subspace of
the image. This impacts the proof, but again not the statement, of [KL2, Lemma
5.7]. The inaccurate claim in the proof is used on [KL2, page 581] to study
the kernel of another map $\chi^{4}$, ultimately leading to the erroneous
Poincaré polynomial (19).

In Section A.4, we explain how to derive the correct Poincaré polynomial (21)
using the statement of [KL2, Proposition 3.2] together with the non-vanishing
(20). The latter fact was not used by Kirwan-Lee. In order to explain the
issues regarding the proof of [KL2, Proposition 3.2], a lengthier discussion
of Kirwan-Lee’s beautiful but intricate argument is required. A correct
derivation is explained in Section A.5 after we describe the geometric set-up
and a few intermediate results in Sections A.2 and A.3.

###  A.2. Kirwan’s desingularization

The approach in [KL1, KL2] starts with the GIT quotient of the space of
sextics

$\overline{\mathcal{F}}^{\text{\,GIT}}=X\sslash G\,,$

where $X=\mathbb{P}^{27}$ and $G=\operatorname{SL}_{3}$. Kirwan’s partial
desingularization

$\overline{\mathcal{F}}^{\text{\,K}}=\widetilde{X}\sslash G$

arises as a composition of four (weighted) blowups. It is obtained by first
blowing up $X^{ss}$ along the orbits whose stabilizers have the highest
dimension, deleting the unstable strata in the blowup, and then repeating the
same procedure to the resulting space. The partial desingularization
$\overline{\mathcal{F}}^{\text{\,K}}$ possesses only finite quotient
singularities, whereas the singularities of
$\overline{\mathcal{F}}^{\text{\,GIT}}$ are more complicated. One of the main
results of [KL1] is the calculation of the Betti numbers of
$\overline{\mathcal{F}}^{\text{\,K}}$:

(22)
$1+5q^{2}+11q^{4}+18q^{6}+25q^{8}+32q^{10}+40q^{12}+48q^{14}+55q^{16}+60q^{18}+60q^{20}+55q^{22}\\\
+48q^{24}+40q^{26}+32q^{28}+25q^{30}+18q^{32}+11q^{34}+5q^{36}+q^{38}\,.$

This is used in [KL2] to compute the Betti numbers of $\mathcal{F}_{2}$,
viewing the latter as an open in
$\overline{\mathcal{F}}^{\text{\,K}}.$666There are a few minor typos in the
proof of [KL1, Theorem 1.3]. On the table in [KL1, page 499], the locus
labelled $(1,0)$ corresponds to the stratum of unstable sextics of the form
$\ell^{5}m$ where $\ell,m$ are distinct lines. This stratum contributes
$\frac{q^{46}}{(1-q^{2})^{2}}\,.$ As a result, formula [KL1, Section 2.4, (1)]
should read
$\frac{1-q^{50}}{(1-q^{2})(1-q^{4})(1-q^{6})}-\frac{q^{20}-q^{28}}{(1-q^{2})^{3}}\,.$
Similarly, formula [KL1, Section 4.2, (2)] should be
$\frac{q^{22}-q^{42}}{(1-q^{2})^{2}}\,.$ Additionally, there is a misprint in
[KL1, Section 5.2, (1)] which should read
$\frac{1+q^{2}}{(1-q^{2})^{2}}(q^{16}+q^{32}-2q^{38})\,.$ There are a few
other small misprints but they do not affect the general argument.

While the Chow groups of $\overline{\mathcal{F}}^{\text{\,K}}$ are not needed
for our paper, in the spirit of Section 7, we expect that the cycle map

$\mathsf{A}^{*}(\overline{\mathcal{F}}^{\text{\,K}})\to
H^{2*}(\overline{\mathcal{F}}^{\text{\,K}})$

is an isomorphism.

###  A.3. Shah’s compactification

The first step of the desingularization procedure yields Shah’s
compactification

$\overline{\mathcal{F}}^{\text{\,Sh}}=X_{1}\sslash G\,.$

Here $X_{1}$ is the weighted blowup of the triple conic locus $\mathsf{TC}$ in
the locus of semistable sextics:

$\pi:X_{1}\to X^{ss}\,.$

Indeed, $\mathsf{TC}$ is the orbit with the largest stabilizer, namely
$R_{0}=\mathrm{SO}_{3}=\operatorname{PSL}_{2}$.

Three further (unweighted) blowups are necessary to arrive at
$\overline{\mathcal{F}}^{\text{\,K}},$ see [KL1, page 504]. The first of the
remaining three blowups has as center the orbit $G\Delta$, where the reducible
sextic $\Delta=(xyz)^{2}$ is invariant under the maximal torus $R_{1}$ in $G$.
The final two blowups have as centers the orbits $G\widehat{Z}_{R_{2}}^{ss}$
and $G\widehat{Z}_{R_{3}}^{ss},$ where $\widehat{Z}_{R_{2}}^{ss},$
$\widehat{Z}_{R_{3}}^{ss}$ are the loci of semistable points fixed by two
specific rank $1$ tori $R_{2},$ $R_{3}$:

$R_{2}=\text{diag }\langle\lambda^{-2},\lambda,\lambda\rangle,\quad
R_{3}=\text{diag
}\langle\lambda,\lambda^{-1},1\rangle,\quad\lambda\in\mathbb{C}^{*}\,.$

In fact, the locus $G\widehat{Z}_{R_{3}}^{ss}$ (which will be relevant below)
lies over the locus of products of three conics tangent at $2$ points [KL1,
Section 5.3].

There is a blowdown map

$\overline{\mathcal{F}}^{\text{\,K}}\to\overline{\mathcal{F}}^{\text{\,Sh}}\,,$

which is an isomorphism over the stable locus $X_{1}^{s}/G$ of the Shah space.
The moduli space

$\mathcal{F}_{2}\hookrightarrow X_{1}^{s}/G$

is obtained by removing the union of a line and a surface

$Z=(A\smallsetminus\Delta)\cup(B\smallsetminus(B\cap D))\,.$

The locus $A$ is a projective line passing through the point
$\Delta=(xyz)^{2}$ and corresponds to the double cubic locus in $X$. The
surface $B$ corresponds to the double conic + conic locus, and $B\cap D$ is a
curve in $B$.

The Poincaré polynomial of $X_{1}^{s}/G$ is computed in [KL2, Proposition
3.2]:

(23)
$1+2q^{2}+3q^{4}+5q^{6}+6q^{8}+8q^{10}+10q^{12}+12q^{14}+13q^{16}+14q^{18}+12q^{20}\\\
+10q^{22}+8q^{24}+6q^{26}+q^{27}+5q^{28}+3q^{30}+q^{31}+2q^{32}+q^{33}+q^{34}+q^{35}.$

This calculation is very important for the overall argument.777The
intersection homology of the Shah compactification was computed in [KL1,
Theorem 1.2]. On general grounds, see [K3, Remark 3.4], the Betti numbers of
$X_{1}^{s}/G$ agree with the intersection homology Betti numbers in degree
less than roughly the dimension (up to a correction dictated by the unstable
strata). In our case, this confirms the Poincaré polynomial of $X_{1}^{s}/G$
in degrees $\leq 16$. However, the remaining Betti numbers cannot be
immediately derived from [KL1].

###  A.4. The Poincaré polynomial of $\mathcal{F}_{2}$

We confirm the Poincaré polynomial (21) relying on equation (23). Just as on
[KL2, page 580], we use the relative homology sequence for the pair
$(X_{1}^{s}/G,\mathcal{F}_{2})$. The difference is that we take into account
that $H^{34}(\mathcal{F}_{2})\neq 0$, thus leading to the different result
(21).

First, we note the Gysin isomorphism

$H_{i}(X_{1}^{s}/G,\mathcal{F}_{2})=H^{38-i}_{c}(Z)\,.$

In fact, we have

$H_{c}^{0}(Z)=0\,,\quad H_{c}^{2}(Z)=\mathbb{Q}\oplus\mathbb{Q}\,,\quad
H_{c}^{4}(Z)=\mathbb{Q}\,,$

see [KL2, (6.3)]. The relative homology sequence

$\ldots\to H_{i}(\mathcal{F}_{2})\to H_{i}(X_{1}^{s}/G)\to H_{c}^{38-i}(Z)\to
H_{i-1}(\mathcal{F}_{2})\to\ldots$

immediately yields isomorphisms

$H_{i}(\mathcal{F}_{2})=H_{i}(X_{1}^{s}/G),\quad i\leq 32,\quad i=37,\quad
i=38\,.$

Expressions (19), (21), (23) all agree in degrees $\leq 32$ and $i=37$,
$i=38$. Furthermore,

$0\to H_{36}(\mathcal{F}_{2})\to
H_{36}(X_{1}^{s}/G)\to\mathbb{Q}\oplus\mathbb{Q}\to H_{35}(\mathcal{F}_{2})\to
H_{35}(X_{1}^{s}/G)\to 0\,.$

Using (23), we have

$H_{36}(X_{1}^{s}/G)=0\,,\quad H_{35}(X_{1}^{s}/G)=\mathbb{Q}\,.$

Therefore

$H_{36}(\mathcal{F}_{2})=0\,,\quad
H_{35}(\mathcal{F}_{2})=\mathbb{Q}\oplus\mathbb{Q}\oplus\mathbb{Q}\,,$

also in agreement with both (19) and (21).

However, discrepancies appear in degrees $33$ and $34$. We have

$0\to H_{34}(\mathcal{F}_{2})\to H_{34}(X_{1}^{s}/G)\to\mathbb{Q}\to
H_{33}(\mathcal{F}_{2})\to H_{33}(X_{1}^{s}/G)\to 0\,.$

Using $H_{34}(X_{1}^{s}/G)=\mathbb{Q}$ from (23) and the fact that
$H_{34}(\mathcal{F}_{2})\neq 0$ as noted above, it follows that the first map
must be an isomorphism and

$H_{34}(\mathcal{F}_{2})=\mathbb{Q}\,.$

Using (23) one more time, we have $H_{33}(X_{1}^{s}/G)=\mathbb{Q}$, hence

$H_{33}(\mathcal{F}_{2})=\mathbb{Q}\oplus\mathbb{Q}\,.$

This confirms equation (21).

###  A.5. Further discussion

To give further credence to (21), we also identify the faulty reasoning in
[KL2]. To this end, we need to zoom in on the argument. We will explain that
while (23) records the correct Poincaré polynomial of $X_{1}^{s}/G$, there are
some errors in the derivation. The necessary corrections impact the last page
of [KL2], and thus the final result.

The strategy used to establish (23) is as follows:

  * (i)

a lower bound on the Betti numbers of $X_{1}^{s}/G$ is obtained from the
Poincaré polynomial of $\overline{\mathcal{F}}^{\text{\,K}}$ in (22) together
with the relative homology sequence for the pair

$X_{1}^{s}/G\hookrightarrow\overline{\mathcal{F}}^{\text{\,K}}\,.$

The resulting lower bounds for Betti numbers of $X_{1}^{s}/G$ are recorded in
[KL2, (3.11), (3.12)].

  * (ii)

Matching upper bounds are obtained in [KL2, Sections 4, 5]. The outcome is
[KL2, Corollary 5.11].

In fact, steps (i) and (ii) are only carried out in degrees less or equal than
$23$, while the higher terms are determined in [KL2, Section 6].

Step (i) requires the calculation of the Poincaré polynomial of the
complement888The top term of the Poincaré polynomial of $Q$ in [KL2, (3.8)]
should be $3q^{36}$, taking into account the correction $3q^{30}$ versus
$3q^{20}$ in [KL2, (3.5)] and using the correct sign for the contribution of
$E_{T,2}\sslash G$. Similarly, there is a misprint in the first formula in
[KL2, page 569] which requires the coefficient $6$ for $q^{26}$. [KL2, Section
3]:

$Q=\overline{\mathcal{F}}^{\text{\,K}}\smallsetminus X_{1}^{s}/G\,.$

Step (ii) examines the kernel of the restriction map

$\chi^{*}:H^{*}(\overline{\mathcal{F}}^{\text{\,K}})\to H^{*}(Q)\,.$

This kernel is identified with the kernel of the restriction

$\rho^{*}:H^{*}(\overline{\mathcal{F}}^{\text{\,K}})\to
H^{*}(\widehat{E}_{1}\sslash G)\oplus H^{*}(E_{2}\sslash G)\oplus
H^{*}(E_{3}\sslash G)\,.$

Here, $\widehat{E}_{1}\sslash G$ is the strict transform of the exceptional
divisor $E_{1}\sslash G$ of the second blowup (at $\Delta$), and $E_{2}\sslash
G$ and $E_{3}\sslash G$ are the exceptional divisors of the third and fourth
blowup [KL2, page 567]. The correct identification of the kernel in
codimension $4$ is needed in [KL2, Section 6] to determine the Betti numbers
of $\mathcal{F}_{2}$ in high degrees.

Before reviewing the analysis of the kernel of $\rho^{*}$ in [KL2, Sections 4,
5], we need a few standard preliminaries. Consider the general setting of a
$G$-equivariant blowup

$p:\widetilde{M}\to M$

of a nonsingular quasiprojective $M$ along a nonsingular equivariant center
$N$ of codimension $c$, with exceptional divisor $E$. Note that the natural
sequence

(24) $0\to H_{G}^{*}(M)\to H_{G}^{*}(\widetilde{M})\to
H_{G}^{*}(E)/H_{G}^{*}(N)\to 0$

induces an additive identification [KL1, page 505]:

(25) $H_{G}^{*}(\widetilde{M})=p^{*}H_{G}^{*}(M)\oplus
H_{G}^{*}(E)/H_{G}^{*}(N).$

Furthermore, $H_{G}^{*}(E)/H_{G}^{*}(N)$ has the additive basis

$\zeta^{k}\cdot p^{*}\alpha,\quad 1\leq k\leq c-1\,,$

for classes $\alpha$ giving a basis of $H_{G}^{*}(N)$, and $\zeta$ denoting
the hyperplane class of the projective bundle $E\to N$. For (25), the
splitting

$H_{G}^{*}(E)/H_{G}^{*}(N)\to H_{G}^{*}(\widetilde{M})$

of the natural restriction map in $\eqref{blowupseq}$ is not explicitly stated
in [KL1]. However, a splitting can be specified on the additive basis

(26) $\zeta^{k}\cdot p^{*}\alpha\mapsto E^{k-1}\cdot j_{!}(p^{*}\alpha),\quad
1\leq k\leq c-1,$

with $j_{!}$ denoting the Gysin map for the closed immersion
$E\to\widetilde{M}.$ This convention is standard and is used for instance in
[K3, page 495].

As an approximation of $\rho,$ one constructs spaces dominating the cohomology
groups of the domain and target of $\rho$. For the domain, the space
$\overline{\mathcal{F}}^{\text{\,K}}$ arises as a $4$-step blowup, and each
blowup contributes to cohomology via (25). Thus, the cohomology of
$\overline{\mathcal{F}}^{\text{\,K}}$ has $5$ natural pieces, yielding
generators999This uses Kirwan surjectivity [K1]; we only obtain generators
after the unstable loci are deleted.

(27) $p^{*}:F^{*}_{1}\oplus F^{*}_{2}\oplus F^{*}_{3}\oplus F^{*}_{4}\oplus
F^{*}_{5}\twoheadrightarrow H^{*}(\overline{\mathcal{F}}^{\text{\,K}}).$

Similarly, there is a surjection

(28) $q^{*}:G^{*}_{1}\oplus G^{*}_{2}\oplus G^{*}_{3}\twoheadrightarrow
H^{*}(\widehat{E}_{1}\sslash G)\oplus H^{*}(E_{2}\sslash G)\oplus
H^{*}(E_{3}\sslash G).$

The interested reader can consult [KL2, Section 4] for a more detailed
discussion and notation. There is an induced map on generators

$\sigma^{*}:F^{*}_{1}\oplus F^{*}_{2}\oplus F^{*}_{3}\oplus F^{*}_{4}\oplus
F^{*}_{5}\to G^{*}_{1}\oplus G^{*}_{2}\oplus G^{*}_{3}\,,$

which is an approximation of $\rho^{*}.$ The kernel of $\sigma^{*}$ is
calculated first.

By [KL2, page 576], the kernel of $\sigma^{*}$ consists of pairs

$(a,b)\in F^{*}_{1}\oplus
F^{*}_{2},\quad\tau_{1}^{*}(a)+\tau_{2}^{*}(b)=0,\quad\tau_{4}^{*}(a)=0,\quad\tau_{6}^{*}(a)=0\,.$

Here,

$F_{1}^{*}=H^{*}(X)\otimes H^{*}(\operatorname{BSL}_{3}),\quad
F_{2}^{*}=\widetilde{H}^{*}(\mathbb{P}^{21})\otimes
H^{*}(\operatorname{BSO}_{3})\,.$

The space $F^{*}_{1}$ is the equivariant cohomology of the space of plane
sextics. Additively, $F^{*}_{2}$ can be identified with the equivariant
cohomology of the exceptional divisor of the first blowup, modulo the
equivariant cohomology of the center of the blowup. Indeed, the codimension of
the triple conic orbit is $27-5=22$, and the normalizer is $\mathrm{SO}_{3}.$
The maps

$\tau_{1}^{*}:F_{1}^{*}\to G_{3}^{*},\quad\tau_{2}^{*}:F_{2}^{*}\to G_{3}^{*}$

are introduced in [KL2, page 571], while $\tau_{4}^{*},\tau_{6}^{*}$ are
constructed in [KL2, pages 572–573]. The target of $\tau_{1}^{*}$ and
$\tau_{2}^{*}$ is the equivariant cohomology of the last exceptional divisor
$G_{3}^{*}=H_{G}^{*}(E_{3})$. However, the discussion of [KL2, (4.6)] shows
that these maps factor through the equivariant cohomology
$\widetilde{G}_{3}^{*}=H^{*}_{G}(G\widehat{Z}_{R_{3}}^{ss})$ of the center of
the last blowup, followed by pullback:

$\tau_{1}^{*}:F_{1}^{*}\to\widetilde{G}_{3}^{*},\quad\tau_{2}^{*}:F_{2}^{*}\to\widetilde{G}_{3}^{*}\,.$

A key step is to show

(29)
$\tau_{1}^{*}(a)+\tau_{2}^{*}(b)=\tau_{4}^{*}(a)=\tau_{6}^{*}(a)=0\implies\tau_{1}^{*}(a)=\tau_{2}^{*}(b)=0,$

see [KL2, page 576]. In turn, this relies on [KL2, Lemma 5.6] which specifies
the image of $\tau_{2}^{*}$.

It is important to understand the map $\tau_{2}^{*}$. This map arises from
blowing up the codimension $18$ orbit $G\widehat{Z}_{R_{3}}^{ss}$ after all
the other blowups have been carried out. To explain the notation,
$\widehat{Z}_{R_{3}}^{ss}$ consists of the semistable points fixed by the
torus

$R_{3}=\text{diag }\langle\lambda,\lambda^{-1},1\rangle\subset
G=\operatorname{SL}_{3}\,.$

This blowup is described in [KL1, Section 5.3]. It is noted in [KL2, page 570]
that the cohomology of the center of the blowup is

$\widetilde{G}_{3}^{*}=H^{*}_{G}(G\widehat{Z}_{R_{3}}^{ss})\cong
H_{N(R_{3})}^{*}(\widehat{Z}_{R_{3}}^{ss})\cong
H^{*}(\widehat{Z}_{R_{3}}^{ss}\sslash N(R_{3}))\otimes H^{*}(BN_{2})\,.$

Here, $N(R_{3})$ is the normalizer of $R_{3}$ in $G$ inducing a residual
action on $\widehat{Z}_{R_{3}}^{ss}$, and $N_{2}$ is the normalizer of the
maximal torus in $\operatorname{SL}_{2}.$ The first isomorphism is a general
fact which follows from [K2, Corollary 5.6], while the second isomorphism is
explained in [KL1, Section 5.3].

The map $\tau_{2}^{*}$ can be described as taking classes on the exceptional
divisor of the first blowup, viewing them as classes on the first blowup under
the construction (26), then restricting to $\widehat{Z}_{R_{3}}^{ss}$, while
switching from $G$-equivariance to $N(R_{3})$-equivariance [KL2, page 571].
Now recall that the surface $\widehat{Z}_{R_{3}}^{ss}\sslash N(R_{3})$ carries
an exceptional divisor $\theta$ obtained by blowing up the triple conic
$\delta$ in $Z_{R_{3}}^{ss}\sslash N(R_{3})$, see [KL2, page 576]. Lemma 5.6
in [KL2] states that

$\text{Im
}\tau_{2}^{*}\subset\widetilde{G}_{3}^{*}=H^{*}(\widehat{Z}_{R_{3}}^{ss}\sslash
N(R_{3}))\otimes H^{*}(BN_{2})$

equals $\\{\theta\\}\otimes H^{*}(BN_{2}).$ This is incorrect, and it should
be replaced by classes supported on $\theta$, not the class of $\theta$
itself. This comes from the fact that in (26) self-intersections of the
exceptional divisor also arise; see also (31) below. As a result, the
conclusion

$\tau_{1}^{*}(a)=\tau_{4}^{*}(a)=\tau_{6}^{*}(a)=\tau_{2}^{*}(b)=0$

in [KL2, page 576] does not hold.

However, the strategy of the argument is sound, and [KL2, Lemma 5.10] can be
salvaged. We will give the details below. We first consider the equations

$\tau_{4}^{*}(a)=\tau_{6}^{*}(a)=0\,,$

where we may assume $a$ has degree $\leq 23,$ since the lemma only concerns
such degrees. By [KL2, Lemma 5.4], the intersection $\text{Ker
}\tau_{4}^{*}\cap\text{Ker }\tau_{6}^{*}$ in the polynomial ring

$F^{*}_{1}=H^{*}(\mathbb{P}^{27})\otimes
H^{*}(\operatorname{BSL}_{3})=\mathbb{Q}[H,c_{2},c_{3}]/(H^{28})$

is generated by two classes of degrees $4$ and $14$, namely

$H^{2}\,\,\,\text{and}\,\,\,\alpha=H\cdot(4c_{2}^{3}+27c_{3}^{2})\,.$

We can explicitly list all classes in the intersection of the two kernels of
$\tau_{4}^{*}$ and $\tau_{6}^{*}$. The ranks of $\text{Ker
}\tau_{4}^{*}\cap\text{Ker }\tau_{6}^{*}$ are given in each degree by

$q^{4}+q^{6}+2q^{8}+3q^{10}+4q^{12}+6q^{14}+7q^{16}+9q^{18}+11q^{20}+13q^{22}\,,$

in agreement with [KL2, page 577]. For instance, in degree $4$ (the simplest
case), we have the unique class $H^{2}$. In degree $20$ (the most involved
case), we have the $11$ classes

$H^{10},H^{8}c_{2},H^{7}c_{3},H^{6}c_{2}^{2},H^{5}c_{2}c_{3},H^{4}c_{3}^{2},H^{4}c_{2}^{3},H^{3}c_{2}^{2}c_{3},H^{2}c_{2}^{4},H^{2}c_{2}c_{3}^{2},\alpha
c_{3}\,,$

and the classes $a$ of degree $20$ lie in the span of these terms. Similarly,

$b\in F^{*}_{2}=\widetilde{H}^{*}(\mathbb{P}^{21})\otimes
H^{*}(\operatorname{BSO}_{3})\,.$

Let $\zeta$ be the hyperplane class on the first exceptional divisor. The
ranks of $F^{*}_{2}$ in degrees $\leq 23$ are immediately calculated to be

$q^{2}+q^{4}+2q^{6}+2q^{8}+3q^{10}+3q^{12}+4q^{14}+4q^{16}+5q^{18}+5q^{20}+6q^{22}\,.$

For example, in degree $4,$ we have the class $\zeta^{2}$. In degree $20$, we
have the $5$ classes

$\zeta^{10},\,\zeta^{8}c_{2},\,\zeta^{6}c_{2}^{2},\,\zeta^{4}c_{2}^{3},\,\zeta^{2}c_{2}^{4}\,,$

and the classes $b$ of degree $20$ lie in the span of these terms. By the
proof of [KL2, Lemma 5.4], the homomorphism

$\tau_{1}^{*}:F^{*}_{1}\to\widetilde{G}^{*}_{3}=H^{*}(\widehat{Z}_{R_{3}}^{ss}\sslash
N(R_{3}))\otimes H^{*}(BN_{2})$

is given by

(30) $\tau_{1}^{*}(H)=C\otimes
1,\quad\tau_{1}^{*}(c_{2})=-1\otimes\xi+n([\text{pt}]\otimes
1),\quad\tau_{1}^{*}(c_{3})=C^{\prime}\otimes\xi,$

where $n$ is an integer, $\xi$ is the degree $4$ generator of $H^{*}(BN_{2})$,
and $C,C^{\prime}$ are curves on the surface $\widehat{Z}_{R_{3}}^{ss}\sslash
N(R_{3}).$ In fact, by construction $C^{2}=1$. Similarly,

(31) $\tau^{*}_{2}(\zeta)=\theta\otimes
1,\quad\tau^{*}_{2}(c_{2})=-1\otimes\xi\implies\tau^{*}_{2}(\zeta^{2})=-[\text{pt}]\otimes
1,\quad\tau^{*}_{2}(\zeta^{k})=0,\quad k\geq 3.$

The last vanishing can be seen using the construction (26) and the fact that
we restrict to a surface $\widehat{Z}_{R_{3}}^{ss}\sslash N(R_{3}).$ Now we
can solve (29) in each degree using (30) and (31). For instance, in degree
$4$, we need

$a\cdot\tau^{*}_{1}(H^{2})+b\cdot\tau^{*}_{2}(\zeta^{2})=0\implies a-b=0$

so the kernel is spanned by $H^{2}+\zeta^{2}$. In degree $20$, we write

$\displaystyle a$ $\displaystyle=a_{1}\cdot H^{10}+a_{2}\cdot
H^{8}c_{2}+a_{3}\cdot H^{7}c_{3}+a_{4}\cdot H^{6}c_{2}^{2}+a_{5}\cdot
H^{5}c_{2}c_{3}+a_{6}\cdot H^{4}c_{3}^{2}+a_{7}\cdot H^{4}c_{2}^{3}$
$\displaystyle\hskip 36.135pt\hskip 36.135pt\hskip 36.135pt\hskip
36.135pt\hskip 36.135pt+a_{8}\cdot H^{3}c_{2}^{2}c_{3}+a_{9}\cdot
H^{2}c_{2}^{4}+a_{10}\cdot H^{2}c_{2}c_{3}^{2}+a_{11}\cdot\alpha c_{3}\,,$
$\displaystyle b$
$\displaystyle=b_{1}\cdot\zeta^{10}+b_{2}\cdot\zeta^{8}c_{2}+b_{3}\cdot\zeta^{6}c_{2}^{2}+b_{4}\cdot\zeta^{4}c_{2}^{3}+b_{5}\cdot\zeta^{2}c_{2}^{4}\,.$

From here, using (30), (31), we find

$\tau_{1}^{*}(a)+\tau_{2}^{*}(b)=0\iff
a_{9}-4a_{11}\cdot(C.C^{\prime})-b_{5}=0\,.$

This yields a $15$-dimensional solution space. After solving (29) in each
degree, we find the ranks of $\text{Ker }\sigma^{*}$ in degree $\leq 23$ are
accurately recorded by [KL2, Lemma 5.7]:

$q^{4}+2q^{6}+3q^{8}+5q^{10}+6q^{12}+8q^{14}+10q^{16}+12q^{18}+15q^{20}+17q^{22}\,.$

However, the description of the kernel of $\sigma^{*}$ in degree $4$ needs to
be corrected. As already mentioned, this impacts the argument in [KL2, page
581].

At this stage, thanks to [KL2, Lemma 5.7], we have complete knowledge of the
kernel of $\sigma^{*}$ in degree $\leq 23$. The next results [KL2, 5.9 - 5.11]
concern $\text{Ker }\rho^{*}$ which is required in part (ii) above. No
correction to the statements in [KL2] is needed here. However, the derivation
of [KL2, Lemma 5.10] crucially uses [KL2, (5.1)]. This derivation requires a
few modifications to the values in [KL2]. Up to order $23$, we have:

(32) $\displaystyle\text{Ker }p^{*}=q^{16}+5q^{18}+14q^{20}+28q^{22}\,,$
$\displaystyle\text{Ker }q^{*}_{11}=q^{18}+3q^{20}+6q^{22}\,,$
$\displaystyle\text{Ker }q^{*}_{2}=q^{16}+3q^{18}+5q^{20}+8q^{22}\,,$

while

$\text{Ker }q^{*}_{3}=q^{18}+5q^{20}+10q^{22}$

is correct in [KL2]. Here, $p^{*}$ is introduced in (27), and
$q^{*}_{11},q^{*}_{2},q^{*}_{3}$ are certain components of the morphism (28).
Thus, using [KL2, (5.9)], the expression

$\text{Ker }q^{*}_{11}+\text{Ker }q^{*}_{2}+\text{Ker }q^{*}_{3}+\text{Ker
}\sigma^{*}-\text{Ker }p^{*}$

yields the upper bound for $\text{Ker }\rho^{*}=\text{Ker }\chi^{*}$ in [KL2,
Lemma 5.10] to be

$q^{4}+2q^{6}+3q^{8}+5q^{10}+6q^{12}+8q^{14}+10q^{16}+12q^{18}+14q^{20}+13q^{22}\,.$

This completes step (ii), and also confirms [KL2, Proposition 3.2] and
equation (23) along with it.

The method of computing of the ranks of $\text{Ker }p^{*}$ and $\text{Ker
}q^{*}_{2}$ is described in [KL2, (5.1)], but the details are suppressed and
the results are recorded imprecisely. For instance, $\text{Ker }p^{*}$
receives the following $6$ contributions:

  * •

from the domain of $p^{*}$, the term $F^{*}_{1}=H^{*}(\mathbb{P}^{27})\otimes
H^{*}(\operatorname{BSL}_{3})$ contributes

$\frac{1-q^{56}}{1-q^{2}}\cdot\frac{1}{(1-q^{4})(1-q^{6})};$

  * •

next, $F^{*}_{2}=\widetilde{H}^{*}(\mathbb{P}^{21})\otimes
H^{*}(\operatorname{BSO}_{3})$ contributes

$\frac{q^{2}-q^{44}}{1-q^{2}}\cdot\frac{1}{1-q^{4}};$

  * •

the remaining pieces of the domain of $p^{*}$ are found in [KL2, page 571]. We
have $F_{3}^{*}=\widetilde{H}^{*}(\mathbb{P}^{20})\otimes H^{*}(BN)$ which
contributes

$\frac{q^{2}-q^{42}}{1-q^{2}}\cdot\frac{1}{(1-q^{4})(1-q^{6})}\,.$

Here $N$ is the normalizer of the maximal torus $R_{1}$ in $G$ (this is
denoted $N_{3}$ in [KL2]);

  * •

similarly for $F_{4}^{*}=H^{*}(\widehat{Z}_{R_{2}}\sslash N(R_{2}))\otimes
H^{*}(B\mathbb{C}^{*})\otimes\widetilde{H}^{*}(\mathbb{P}^{18})$ we get the
contribution

$(1+q^{2})\cdot\frac{1}{1-q^{2}}\cdot\frac{q^{2}-q^{38}}{1-q^{2}}\,.$

The first term is computed in [KL1, Section 5.1];

  * •

for $F_{5}^{*}=H^{*}(\widehat{Z}_{R_{3}}\sslash N(R_{3}))\otimes
H^{*}(BN_{2})\otimes\widetilde{H}^{*}(\mathbb{P}^{17})$ we get

$(1+3q^{2}+q^{4})\cdot\frac{1}{1-q^{4}}\cdot\frac{q^{2}-q^{36}}{1-q^{2}}\,.$

The first term was computed in [KL1, Section 5.3];

  * •

for the target of $p^{*}$, the contribution of
$H^{*}(\overline{\mathcal{F}}^{\text{\,K}})$ is recorded in (22).

Putting these contributions together, we find that $\text{Ker }p^{*}$ is given
by (32), as claimed. The discrepancy with the value in [KL2, (5.1)] is
$3q^{22}\mod q^{24}$.

Next, we examine $q_{2}^{*}:G_{2}^{*}\twoheadrightarrow H^{*}(E_{2}\sslash
G).$ The dimension of the target is recorded in [KL2, (3.5)]:

$(1+q^{2})(1+2q^{2}+3q^{4}+4q^{6}+5q^{8}+6q^{10}+7q^{12}+8q^{14}+8q^{16}+8q^{18}+8q^{20}+7q^{22}),$

up to order $23$. By [KL2, (4.3)], the domain is

$G_{2}^{*}=H^{*}(\widehat{Z}_{R_{2}}\sslash N(R_{2}))\otimes
H^{*}(B\mathbb{C}^{*})\otimes H^{*}(\mathbb{P}^{18})\,,$

whose contribution equals

$(1+q^{2})\cdot\frac{1}{1-q^{2}}\cdot\frac{1-q^{38}}{1-q^{2}}\,.$

Subtracting the two series above, we find the dimension of $\text{Ker
}q^{*}_{2}$ matching the last equation in (32). The value recorded [KL2,
(5.1)] is different. The misprint likely originates with [KL1, Section 5.2].

Finally, we consider the map

$q^{*}_{1}:G^{*}_{1}\to H^{*}(\widehat{E}_{1}\sslash G)$

discussed in [KL2, (4.4)]. Here, $\widehat{E}_{1}$ is a blowup of $E_{1}^{ss}$
described in [KL2, page 570]. As noted in [KL2, page 578], the map $q^{*}_{1}$
has three components $q^{*}_{11},$ $q^{*}_{12},$ $q^{*}_{13}$, where

$q^{*}_{11}:H_{G}^{*}(E_{1})\to H^{*}(\widehat{E}_{1}\sslash G)\,.$

Since $\widehat{E}_{1}$ has no strictly semistable points, we have

$H^{*}(\widehat{E}_{1}\sslash G)=H_{G}^{*}(\widehat{E}_{1}^{ss})\,,$

see also [KL2, page 570]. We factor $q_{11}^{*}$ as the composition

$H^{*}_{G}(E_{1})\stackrel{{\scriptstyle
f^{*}}}{{\longrightarrow}}H^{*}_{G}(E_{1}^{ss})\stackrel{{\scriptstyle
g^{*}}}{{\longrightarrow}}H^{*}_{G}(\widehat{E}_{1})\stackrel{{\scriptstyle
h^{*}}}{{\longrightarrow}}H^{*}_{G}(\widehat{E}_{1}^{ss})\,.$

The first map $f^{*}$ is surjective on general grounds [K1]. We will compute
its kernel below. The middle map $g^{*}$ is a pullback induced by a blowup so
it is injective. The third map $h^{*}$ removes unstable strata from the blowup
$\widehat{E}_{1}$ to arrive at $\widehat{E}_{1}^{ss}$.

The calculation of the kernel of the surjection $f^{*}:H^{*}_{G}(E_{1})\to
H^{*}_{G}(E_{1}^{ss})$ is a matter of recording dimensions.

  * (a)

For the domain, we have $H^{*}_{G}(E_{1})=H^{*}(BN)\otimes
H^{*}(\mathbb{P}^{20})$ [KL2, page 570]. The first factor comes from center of
the blowup using the isomorphism $G\Delta=G/N$. We recall that $N$ stands for
the normalizer of the maximal torus in $G$. The projective space
$\Sigma=\mathbb{P}^{20}$ corresponds to the projectivization of the normal
bundle at $\Delta$ of the orbit $G\Delta$ in $X$. This contributes

$\frac{1}{(1-q^{4})(1-q^{6})}\cdot\frac{1-q^{42}}{1-q^{2}}\,.$

  * (b)

For the target, on general grounds we have

$H^{*}_{G}(E_{1}^{ss})=H^{*}_{G}(G\times_{N}(\mathbb{P}^{20})^{ss})=H^{*}_{N}((\mathbb{P}^{20})^{\text{ss}})\,.$

The $N$-equivariant Poincaré series of $(\mathbb{P}^{20})^{ss}$ was calculated
in [KL1, Section 6.5], equation (1). Expanding up to order $23$ we find

$1+q^{2}+2q^{4}+3q^{6}+4q^{8}+5q^{10}+7q^{12}+8q^{14}+10q^{16}+11q^{18}+11q^{20}+10q^{22}\,.$

The kernel of $f^{*}$ up to order $23$ is determined from here by subtracting
the two expressions (a) and (b). The answer reproduces the value claimed on
the second line of (32).

Since $g^{*}$ is injective, we have

$\text{Ker }g^{*}\circ f^{*}=\text{Ker }f^{*}\,.$

We furthermore claim this agrees with the kernel of $q_{11}^{*}=h^{*}\circ
g^{*}\circ f^{*}.$ To this end, it suffices to show

(33) $\text{Ker }h^{*}\cap\text{Im }g^{*}=0.$

Indeed, we need to rule out the situation that classes supported on unstable
strata of $\widehat{E}_{1}$ might equal a class pulled back from $E_{1}^{ss}$.
Should this happen, removing the unstable stratum from $\widehat{E}_{1}$ will
also kill additional classes on $E_{1}^{ss}$, thus increasing the kernel of
$q_{11}^{*}$ when compared to the kernel of $f^{*}$. The discussion might have
been clear to the authors of [KL1, KL2] and it was not recorded explicitly,
but we indicate here a possible argument.

Let $\Sigma=\mathbb{P}^{20}$. By the above remarks (a) and (b), the maps
$f^{*}$, $g^{*}$ and $h^{*}$ can be rewritten as

$H_{N}^{*}(\Sigma)\stackrel{{\scriptstyle
f^{*}}}{{\longrightarrow}}H_{N}^{*}\left(\Sigma^{ss}\right)\stackrel{{\scriptstyle
g^{*}}}{{\longrightarrow}}H_{N}^{*}(\widehat{\Sigma})\stackrel{{\scriptstyle
h^{*}}}{{\longrightarrow}}H_{N}^{*}(\widehat{\Sigma}^{ss})$

where $\widehat{\Sigma}$ is the blowup of $\Sigma^{ss}$ along the $N$-orbits
of the $R_{2}$-fixed and $R_{3}$-fixed loci. It is explained in [KL1, Section
4.3] that

$\Sigma=\mathbb{P}(W)=\mathbb{P}^{20}$

where $W$ is the subspace of sextics spanned by the $21$ monomials
$x^{i}y^{j}z^{6-i-j}$ for $i,j\geq 0$, $i+j\leq 6$ and

$(i,j)\not\in\\{(2,2),(2,1),(2,3),(3,1),(3,2),(1,3),(1,2)\\}\,.$

Using the terminology of [KL1, page 508], these monomials are obtained from
the “Hilbert diagram” in [KL1, page 497] by removing the middle hexagon.
Recalling $R_{2}=\text{diag }\langle\lambda^{-2},\lambda,\lambda\rangle,$ it
follows that the $R_{2}$-fixed locus is the projective line

$\widehat{Z}_{R_{2}}=\mathbb{P}\langle x^{2}y^{4},x^{2}z^{4}\rangle\,.$

Let $Q_{2}$ is the normalizer of $R_{2}$ in $N,$ which is easily computed to
be isomorphic to the normalizer of the maximal torus in
$\operatorname{GL}_{2}$. We have
$Q_{2}/R_{2}=\mathbb{C}^{*}\rtimes\mathbb{Z}/2\mathbb{Z}$. It is easy to see
that $R_{2}$ acts trivially on $\widehat{Z}_{R_{2}}^{ss}=\mathbb{P}^{1}$, and
the $\mathbb{C}^{*}$-factor of $Q_{2}/R_{2}$ acts with equal opposite weights.
On general grounds $\widehat{Z}_{R_{2}}^{ss}$ consists in the
$Q_{2}$-semistable points of $\widehat{Z}_{R_{2}}$, see [K2, Remark 5.5]. It
follows that the unstable points are $x^{2}y^{4}$ and $x^{2}z^{4}$, so
$\widehat{Z}_{R_{2}}^{ss}=\mathbb{C}^{*}$ Thus, the equivariant cohomology of
the orbit is

$H^{*}_{N}(N\widehat{Z}_{R_{2}}^{ss})=H^{*}_{Q_{2}}(\widehat{Z}_{R_{2}}^{ss})=H^{*}(BR_{2})=H^{*}(B\mathbb{C}^{*})\,.$

This is in agreement with [KL2, page 570].

We consider the blowup of $\Sigma^{ss}$ along the orbit of
$\widehat{Z}_{R_{2}}^{ss}.$ The exceptional divisor $F$ of the blowup is a
$\mathbb{P}^{18}$-bundle over the base. We need to identify the unstable locus
in the exceptional divisor. The weights of the representation of $R_{2}$ on
$\mathbb{P}^{18}$ can be lifted from [KL1, Section 5.2]. Up to an overall
factor of $-3$, they are $-2,-1,1,2,3,4$ with multiplicities $7,4,2,3,2,1.$
Thus, in suitable coordinates, the action is given by

$\lambda\cdot[x:y:z:w:t:s]=[\lambda^{-2}x:\lambda^{-1}y:\lambda
z:\lambda^{2}w:\lambda^{3}t:\lambda^{4}s]\,,$

where

$(x,y,z,w,t,s)\in\mathbb{C}^{7}\oplus\mathbb{C}^{4}\oplus\mathbb{C}^{2}\oplus\mathbb{C}^{3}\oplus\mathbb{C}^{2}\oplus\mathbb{C}\,$

are not all zero. The unstable locus is easily seen to be the union

$\mathbb{P}^{10}\sqcup\mathbb{P}^{7}\,,$

corresponding to $z=w=t=s=0$ and $x=y=0$ respectively. This conclusion is in
agreement with the Poincaré polynomial calculation in [KL1, Section 5.2 (1)].
Letting $\epsilon$ denote the equivariant parameter for $R_{2}$, and letting
$H$ denote the hyperplane class on $\mathbb{P}^{18}$, we compute the
$R_{2}$-equivariant classes

$\big{[}\mathbb{P}^{10}\big{]}=(H+\epsilon)^{2}(H+2\epsilon)^{3}(H+3\epsilon)^{2}(H+4\epsilon),\quad\big{[}\mathbb{P}^{7}\big{]}=(H-2\epsilon)^{7}(H-\epsilon)^{4}\,.$

Similar expressions hold for the classes of all equivariant linear subspaces
of $\mathbb{P}^{10}$ or $\mathbb{P}^{7}$: the monomials in $H$ and $\epsilon$
above will have different exponents. By inspection, nonzero combinations of
such classes never come from the base of the blowup
$H^{*}(BR_{2})=\mathbb{Q}[\epsilon]$ by pullback. This is the key to
establishing (33).

To this end, the reader may find the following diagram useful:

${H_{N}^{*}(\widehat{S})}$${H_{N}^{*}(\widehat{S}\cap
F)}$${0}$${H^{*}_{N}(\Sigma^{ss})}$${H^{*}_{N}(\widehat{\Sigma})}$${H^{*}_{N}(F)\,.}$${H_{N}^{*}(\widehat{\Sigma}^{ss})\,}$$\scriptstyle{\overline{j}^{*}}$$\scriptstyle{i_{!}}$$\scriptstyle{\overline{i}_{!}}$$\scriptstyle{g^{*}}$$\scriptstyle{j^{*}}$$\scriptstyle{h^{*}}$

Here, $\widehat{S}$ is the unstable locus of $\widehat{\Sigma}.$ The three-
term column of the diagram is the Gysin sequence for the closed subvariety
$\widehat{S}\subset\widehat{\Sigma}.$ For the first term, the cohomology is
shifted by codimension, but the notation does not indicate this explicitly. In
fact, $\widehat{S}$ is not pure dimensional, the individual connected
components need to be considered separately.

On general grounds [K1], the unstable locus admits a stratification by locally
closed nonsingular subvarieties

$\widehat{S}=\bigsqcup_{\beta}\widehat{S}_{\beta}\,.$

The intersection

$\widehat{S}\cap F=\bigsqcup_{\beta}(\widehat{S}_{\beta}\cap F)$

is the union of unstable strata of the exceptional divisor $F$, see the proof