# Full exceptional collections for
anticanonical log del Pezzo surfaces

Giulia Gugiatti  GG: Math Section
ICTP
Leonardo Da Vinci Building, Strada Costiera 11
34151 Trieste, Italy<EMAIL_ADDRESS>and  Franco Rota  FR: School of
Mathematics and Statistics
University of Glasgow
Glasgow G12 8QQ, United Kingdom<EMAIL_ADDRESS>
###### Abstract.

Motivated by homological mirror symmetry, this paper constructs explicit full
exceptional collections for the canonical stacks associated with the series of
log del Pezzo surfaces constructed by Johnson and Kollár in [JK01]. These
surfaces have cyclic quotient, non-Gorenstein, singularities. The construction
involves both the $\operatorname{GL}(2,\mathbb{C})$ McKay correspondence, and
the study of the minimal resolutions of the surfaces, which are birational to
degree two del Pezzo surfaces. We show that a degree two del Pezzo surface
arises in this way if and only if it admits a generalized Eckardt point, and
in the course of the paper we classify the blow-ups of $\mathbb{P}^{2}$ giving
rise to them. Our result on the adjoints of the functor of Ishii–Ueda [IU15]
applies to any finite small subgroup of $\operatorname{GL}(2,\mathbb{C})$.

###### Key words and phrases:

Derived categories, canonical smooth Deligne–Mumford stacks, McKay
correspondence, log del Pezzo surfaces, mirror symmetry

###### 2020 Mathematics Subject Classification:

Primary 14F08, 14A20, 14E16, 14J45; secondary 14J33

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Computation of left adjoint functor
  4. 4 The minimal resolution of the Johnson-Kollár surfaces
  5. 5 The derived category of $\mathcal{X}$

##  1\. Introduction

A log del Pezzo surface is a projective surface with quotient singularities
and ample anticanonical class. For each positive integer $k$, consider the
family of hypersurfaces of degree $8k+4$ in weighted projective space

(1) $X_{8k+4}\subset\mathbb{P}(2,2k+1,2k+1,4k+1).$

With the exception of 22 sporadic cases, all quasismooth and wellformed log
del Pezzo weighted hypersurfaces $X\subset\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$
which are anticanonical (that is, with $-K_{X}\sim\mathcal{O}_{X}(1)$) arise
in this way [JK01].

Fix an integer $k>0$ and let $X$ be a general surface of the family (1). Then
$X$ has isolated cyclic quotient log-terminal singularities. Motivated by
homological mirror symmetry, summarized in Section 1.1 below, we study the
derived category of coherent sheaves of the canonical stack of $X$. This is
the unique smooth Deligne-Mumford stack with coarse moduli space $X$ and
isomorphic to $X$ outside a locus of codimension 2 [Vis89, (2.8),(2.9)][FMN10,
§4.1]111Sometimes, the expression “canonical stack” refers to the index 1
cover stack of $X$ [KM98, §5.2]. This is a separate notion, which does not
appear in the present paper.. The main result is the following.

######  Theorem 1.1 (= (5.2)).

Let $\mathcal{X}$ be the canonical stack associated with $X$. Then
$D^{b}(\operatorname{Coh}(\mathcal{X}))$ admits full exceptional collections,
given explicitly in Section 5.

The exceptional collections of Theorem 1.1 are formed by $10+12k$ objects. In
Section 5.2 we also exhibit a collection whose nontrivial morphisms are only
in degree $0$ and $1$, which in turn implies the existence of a tilting
complex on $\mathcal{X}$ (see Section 5.3).

The starting point of our construction is the
$\operatorname{GL}(2,\mathbb{C})$ McKay correspondence, as formulated in
[IU15]. For $X$ a surface with finite quotient singularities, denote by
$\widetilde{X}$ its minimal resolution. The canonical stack $\mathcal{X}$ of
$X$ can also be interpreted as a stacky resolution:

${\widetilde{X}}$${\mathcal{X}}$${X}$

The correspondence states that there is a fully faithful functor $\Phi\colon
D^{b}(\operatorname{Coh}(\widetilde{X}))\to
D^{b}(\operatorname{Coh}(\mathcal{X}))$. Moreover, there exists a
semiorthogonal decomposition (SOD)

(2)
$D^{b}(\operatorname{Coh}(\mathcal{X}))=\left\langle\textbf{e}_{1},...,\textbf{e}_{N},\Phi\left(D^{b}(\operatorname{Coh}(\widetilde{X}))\right)\right\rangle,$

where $(\textbf{e}_{1},...,\textbf{e}_{N})$ is an exceptional collection. If
$X$ has only cyclic singularities, $\widetilde{X}$ coincides locally with the
moduli space of clusters on $X$, $\Phi$ is the integral functor of the
corresponding universal family, and the singularities determine the collection
combinatorially (see Section 2.3).

The exceptional sequences of Theorem 1.1 are constructed by applying the above
decomposition to the surfaces (1), together with:

  1. (i)

a detailed description of $D^{b}(\operatorname{Coh}(\widetilde{X}))$;

  2. (ii)

a computation of the images of $\textbf{e}_{1},...,\textbf{e}_{N}$ through the
left adjoint functor $\Psi$ to $\Phi$.

To obtain (i) and (ii), it is necessary to prove two results which may be of
independent interest. We give a brief overview here.

The first result describes the geometry of the minimal resolution
$\widetilde{X}$ of the surfaces (1).

######  Theorem 1.2 (= (4.4)).

Let $\widetilde{X}$ be the minimal resolution of $X$. Then $\widetilde{X}$ is
obtained from a smooth degree 2 del Pezzo surface $X^{\prime}$ via a sequence
of $1+4k$ blow-ups.

As an immediate consequence, full exceptional collections for $\widetilde{X}$,
and hence $\mathcal{X}$, exist. In fact, this can be shown in general for log
del Pezzo surfaces since they are rational (see Remark 4.3). In the case at
hand, we describe them explicitly combining a detailed study of
$\widetilde{X}$ and $X^{\prime}$ with Theorem 1.4 below.

The surface $X^{\prime}$ appearing in Theorem 4.4 does not depend on the
integer $k$ and it admits a generalized Eckardt point, i.e. a point of
intersection of four $(-1)$-curves. Del Pezzo surfaces of degree 2 with a
generalized Eckardt point form a stratum of codimension two in the moduli
space of all del Pezzo surfaces [Dol12, Table 8.9]. Repeating the construction
of Theorem 1.2 in families, we obtain the following result about the moduli
space $M_{k}$ of quasismooth and wellformed hypersurfaces of the form (1) (we
define $M_{k}$ in Section 4.3).

######  Proposition 1.3 (= (4.10)).

For all $k>0$ integer, $M_{k}$ is isomorphic to the moduli space of del Pezzo
surfaces with a generalized Eckardt point.

In Proposition 4.13 we classify all configurations of seven points on
$\mathbb{P}^{2}$ which, once blown up, give rise to a del Pezzo surface with a
generalized Eckardt point. This classification is then used to describe the
Hom-spaces of the exceptional collections of $\mathcal{X}$.

Our second result computes the image via $\Psi$ 222Here $\Psi$ is the left
adjoint to the (local version of) the functor $\Phi$ defined above. In the
rest of the paper, we will distinguish locally and globally defined functors
by denoting them with lower-case and upper-case letters, respectively. of
simple sheaves $\mathcal{O}_{0}\otimes\rho\in D([\mathbb{C}^{2}/G])$, where
$G$ is any finite small subgroup of $\operatorname{GL}(2,\mathbb{C})$ and
$\rho$ is an irreducible representation of $G$. Consider the minimal
resolution of $\mathbb{C}^{2}/G$, let $E_{1},\dots,E_{m}$ be the exceptional
curves and let $E$ be the fundamental cycle. Write $\rho_{i}$, with
$i=0,...,n$, for the irreducible representations of $G$, and assume that
$\rho_{i}$, with $i=1,...,m$, is the special representation corresponding to
$E_{i}$ via the McKay correspondence (see Section 2.3).

######  Theorem 1.4 (= (3.1)).

Let $\rho_{\mathrm{nat}}$ be the natural representation of
$G\subset\operatorname{GL}(2,\mathbb{C})$ and
$\rho_{\mathrm{det}}\coloneqq\Lambda^{2}\rho_{\mathrm{nat}}$. For
$i\in\\{1,\dots,m\\}$ let $-\alpha_{i}$ be the self intersection of the
exceptional curve $E_{i}$. Then we have:

(3)
$\displaystyle\Psi(\mathcal{O}_{0}\otimes\rho)\simeq\begin{cases}\mathcal{O}_{E}(E)[1]&\mbox{
if }\rho=\rho_{\mathrm{det}}^{\ast}\\\
\mathcal{O}_{E_{i}}(-\alpha_{i}+1)&\mbox{ if
}\rho\otimes\rho_{\mathrm{det}}=\rho_{i}\mbox{ for some }1\leq i\leq m\\\
0&\mbox{ otherwise}.\end{cases}$

In Section 3.1, we apply Theorem 1.4 to give an alternative proof of [IU15,
Proposition 1.1], which states that the essential image of $\Phi$ is generated
by $\\{\mathcal{O}_{\mathbb{C}^{2}}\otimes\rho_{i}\\}_{i=1}^{m}$, and its
right orthogonal by
$\\{\mathcal{O}_{\mathbb{C}^{2}}\otimes\rho_{i}\\}_{i=m+1}^{n}$.

To describe morphism spaces of the SOD (2), one can compute either the images
through $\Phi$ of objects in $D(\widetilde{X})$, or the images
$\Psi(\textbf{e}_{i})$ and use adjunction. The latter approach is simpler and
gives a more general answer. It is simpler since the computations take place
on the smooth surface $\widetilde{X}$ and not on $\mathcal{X}$. On the other
hand, our Theorem 1.4 shows that the objects
$\Psi(\mathcal{O}_{0}\otimes\rho)$ do not depend on the global properties of
$\widetilde{X}$. In fact, the formula (3) determines $\Psi(\mathbf{e}_{i})$
for all surfaces with quotient singularities by finite small groups, since the
$\textbf{e}_{i}$ all have a composition series whose factors are simple
sheaves $\mathcal{O}_{0}\otimes\rho$ [IU15, Section 3].

###  1.1. Related works and further problems

The main motivation for this work comes from mirror symmetry for Fano
varieties – also known as Fano/Landau–Ginzburg (LG) correspondence – from the
point of view of Kontsevich’s homological mirror symmetry (HMS) [Kon95]. The
correspondence predicts that the mirror of a $n$-dimensional Fano orbifold
$\mathcal{X}$ is a LG model $(Y,w)$ of the same dimension333In mirror
symmetry, it is important to think of a Fano variety $X$ with quotient
singularities as a canonical smooth Deligne-Mumford stack $\mathcal{X}$. This
allows one to define Gromov–Witten invariants for $\mathcal{X}$. See [OP18,
Section 3] for a concise summary of the quantum period of a Fano orbifold.,
and can be interpreted at different levels.

Its Hodge-theoretic formulation identifies the regularised quantum
differential operator of $\mathcal{X}$ and the Picard–Fuchs operator of
$(Y,w)$ [Gol07, Gol08, Prz07]. Recent works suggest that the Hodge-theoretic
mirrors of Fano orbifolds with a $\mathbb{Q}$-Gorenstein (qG) degeneration444
We refer the reader to [KSB88, Hac04, ACC+16] for the notion of
$\mathbb{Q}$-Gorenstein (qG) deformation of varieties with quotient
singularities. to a toric variety should be LG models restricting to specific
Laurent polynomials built out of the combinatorics of the toric variety. This
perspective, first laid out in [Gol07, Prz11, CCG+13], agrees with the
Givental/Hori–Vafa mirror construction [Giv98, HV00] for toric complete
intersections and the intrinsic mirror symmetry program [GS18] for log
Calabi–Yau pairs.

One formulation of HMS555A parallel formulation, which translates the Hodge-
theoretic version of mirror symmetry mentioned above, is an equivalence
between a suitable Fukaya category of $\mathcal{X}$ and the category of matrix
factorisations of $(Y,w)$. predicts an equivalence
$D^{b}(\operatorname{Coh}(\mathcal{X}))\equiv
D\mathrm{Lag}_{\mathrm{vc}}(Y,w)$ between the bounded derived category of
coherent sheaves of $\mathcal{X}$ and the derived category of Lagrangian
vanishing cycles of $(Y,w)$ [Sei08]666A rigorous definition of this category
has been proposed by Seidel in the case where $(Y,w)$ is a Lefschetz
fibration.. It is expected that HMS implies Hodge-theoretic mirror symmetry
(see [Kon95], and [GPS15]), even though the relation between these two
formulations is not proven in full generality. Unlike Hodge-theoretic mirror
symmetry, HMS has been proven systematically only for few classes of Fano
varieties; among them, smooth del Pezzo surfaces [AKO06], and weighted
projective planes [AKO08].

The focus of our work is on the surfaces (1), which we think of as canonical
stacks. These surfaces have isolated cyclic quotient singularities which are
qG-rigid, that is, which are rigid with respect to qG-deformations. As such,
they fit into the context of [ACC+16]. However, since their anticanonical
linear system is empty, they do not admit a qG-degeneration to a toric variety
(see [CG21, Remark 2.7]), thus they are out of the mirror symmetry setting of
[ACC+16, Conjecture A]. The work [CG21] contains the only known mirror
construction for the surfaces (1). The authors show that regularised quantum
differential operator of the surfaces is hypergeometric, and use
hypergeometric motives to build their Hodge-theoretic mirrors: these are
pencils of hyperelliptic curves which do not restrict to Laurent polynomials.

This paper is the first step towards the study of HMS for the surfaces (1), as
it describes exceptional collections of
$D^{b}(\operatorname{Coh}(\mathcal{X}))$. A construction of other exceptional
collections for $\mathcal{X}$ using other methods appears to be challenging:
for example, neither $X$ nor $\widetilde{X}$ are toric, so the Kawamata
construction [Kaw06] does not apply. Nevertheless, the surfaces $X$ are
hypersurfaces in weighted projective spaces, so they inherit a — not full —
exceptional collection from the ambient space. This collection appears in
Elagin [Ela07]. There, the author uses it to build a full exceptional
collection for the stack of a surface with one log-terminal singularity via an
ad hoc mutation of a set of generators for the residual category. There seems
to be no systematic way of extending the construction [Ela07] to other cases.

### Structure of the paper

Section 2 recalls definitions and basic properties. We set some toric geometry
notation in 2.1. Then we recall properties of log del Pezzo surfaces in 2.2
and the $\operatorname{GL}(2,\mathbb{C})$ McKay correspondence in 2.3.

Section 3 contains the computation of the adjoint functor $\Psi$, and the
alternative proof of [Ish02, Prop. 1.1] in 3.1. In Section 3.2, we rephrase
Theorem 1.4 in the cyclic singularities setting, and reprove it using toric
geometry.

Section 4 is dedicated to the minimal resolutions and contains the proof of
Theorem 1.2. In 4.4 we classify degree 2 del Pezzo surfaces admitting a
generalized Eckardt point.

In Section 5 we show the existence of full exceptional collections for
$\widetilde{X}$ and $\mathcal{X}$ and describe them explicitly. We discuss the
existence of a tilting complex for $\widetilde{X}$ and $\mathcal{X}$ in 5.3.

### Notation and conventions

We work over the field of complex numbers. We use
$D(-)=D^{b}(\mathrm{Coh}(-))$ to denote the bounded derived category of
coherent sheaves and $D_{G}(-)$ to denote the bounded derived category of
$G$-equivariant coherent sheaves. We write $\mathcal{O}(n)$ to denote the line
bundle of degree $n$ in the (weighted) projective space $\mathbb{P}$. For
$C\subset\mathbb{P}$ a rational curve, we write $\mathcal{O}_{C}(1)$ for its
line bundle of degree one, so that $\mathcal{O}(1)_{|C}=\mathcal{O}_{C}(\deg
C)$. We recollect here some of the notation used in the paper:

Notation | Description | Local alternative
---|---|---
$X$ | Singular surface | $\mathbb{C}^{2}/G$
$\widetilde{X}$ | minimal resolution of $X$ | $Y$
$\mathcal{X}$ | canonical stack of $X$ | $[\mathbb{C}^{2}/G]$
$\Phi$ | functor $\Phi\colon D(\widetilde{X})\to D(\mathcal{X})$ | $\phi$
$\Psi$ | left adjoint to $\Phi$ | $\psi$

### Acknowledgements

We are thankful to Arend Bayer, Alessio Corti, and Michael Wemyss for valuable
discussions and advice. We thank Akira Ishii for his remarks on Section 3.1.
We are grateful to Dominic Bunnett and to Stefano Filipazzi for helpful
correspondence. The second author was partially supported by EPSRC grant
EP/R034826/1.

##  2\. Preliminaries

This section is organised as follows: in 2.1 we set up our notation for toric
varieties and we state some well-known facts which we will use repeatedly; in
2.2 we recall the notion of anticanonical log del Pezzo surface, the
classification [JK01], and some basic facts about smooth del Pezzo surfaces of
degree two; 2.3 is a summary of the $\operatorname{GL}(2,\mathbb{C})$ McKay
correspondence, with a focus on the case of cyclic quotient singularities.

###  2.1. Toric setup

For details on the content of this Section, we refer the reader to [CLS11,
Chapters 1, 3 and 11] and [KM92, §10].

For a free abelian group of finite rank $L$, we denote by
$L_{\mathbb{R}}=L\otimes_{\mathbb{Z}}\mathbb{R}$ the associated real vector
space. For a torus $\mathbb{T}$, we denote by
$M=\mathrm{Hom}(\mathbb{T},\mathbb{C}^{\times})$ the character group and we
write $N=\mathrm{Hom}(M,\mathbb{Z})$. Given vectors $v_{1},\dots,v_{k}\in
L_{\mathbb{R}}$, we write $\langle v_{1},\dots,v_{k}\rangle$ for the cone they
generate.

For a fan $\Sigma\subset N_{\mathbb{R}}$, we denote by $G_{\Sigma}$ the
associated toric variety. The toric variety is proper if and only if $\Sigma$
is a complete fan, that is, its support $|\Sigma|$ is all $N_{\mathbb{R}}$.
The toric variety has finite quotient singularities if and only if the fan is
simplicial. We write $\Sigma(r)$ for the set of $r$-dimensional cones of the
fan $\Sigma$. We denote by $\rho_{i}$ the primitive generators of the
$1$-dimensional cones (or _rays_) of $\Sigma$, and by $D_{\rho_{i}}\subset
G_{\Sigma}$ the corresponding divisors under the toric orbit-cone
correspondence [CLS11, §3.2].

Below we recall the construction of star subdivision of a fan, as given in
[CLS11, §11.1].

######  Lemma/Definition 2.1 ([CLS11, Lemma 11.1.3]).

Let $\Sigma\subset N_{\mathbb{R}}$ be a fan with rays
$\rho_{1},\dots,\rho_{n}$ and let $v\in|\Sigma|\cap N$ be a primitive element.
Define _the star subdivision_ of $\Sigma$ at $v$, denoted by
$\Sigma^{\ast}(v)$, as the set of the cones:

  * (1)

$\sigma$, where $v\notin\sigma\in\Sigma$;

  * (2)

$\mathrm{cone}(\tau,v)$, where $v\notin\tau\in\Sigma$ and there exists
$\sigma\in\Sigma$ such that $\\{v\\}\cup\tau\subset\sigma$.

The set $\Sigma^{\ast}(v)$ is a refinement of $\Sigma$, thus induces a toric
morphism $G_{\Sigma^{\ast}(v)}\to G_{\Sigma}$.

In Section 4 we will use toric morphisms coming from star subdivisions to
resolve singularities of toric varieties. In particular, we will consider
situations where $\Sigma$ contains a cone $\tau$ with rays
$\rho_{i_{1}},\dots,\rho_{i_{r}}$ such that the corresponding affine chart
$U_{\tau}$ is of the form:

$U_{\tau}\simeq\mathbb{C}^{r}/{\mathbb{Z}_{m}(a_{1},\dots,a_{r})}\times(\mathbb{C}^{\times})^{n-r}\subset
G_{\Sigma}$

where $m,a_{i}$ are positive integers such that $a_{i}<m$ for all $i$. Then
the toric variety $G_{\Sigma^{\ast}(v)}$ associated to the star subdivision
$\Sigma^{\ast}(v)$, with

(4) $v=\frac{1}{m}\sum_{i=1}^{r}a_{j}\;\rho_{i_{j}}$

is the weighted blow-up of $G_{\Sigma}$ along $V(\tau)$ with weights
$\frac{1}{m}(a_{1},\dots,a_{r})$ [KM92, (10.3)], where we write $V(\tau)$ for
the closure of the $\mathbb{T}_{N}$-orbit in $G_{\Sigma}$ corresponding to
$\tau$. As such, the variety $G_{\Sigma^{\ast}(v)}$ fits in a fiber square:

${\mathcal{E}}$${G_{\Sigma^{\ast}(v)}}$${V(\tau)}$${\ G_{\Sigma}}$

where the exceptional divisor $\mathcal{E}$ is a
$\mathbb{P}(a_{1},\dots,a_{r})$-bundle over $V(\tau)$.

###  2.2. Log del Pezzo weighted hypersurfaces

For positive integers $a_{i}$, $i=1,\dots,m$, we denote by
$S=S(a_{1},\dots,a_{n})$ the graded polynomial ring
$\mathbb{C}[x_{1},\dots,x_{n}]$ where $\deg x_{i}=a_{i}$, and by
$\mathbb{P}(a_{1},\dots,a_{m})$ the weighted projective space with weights
$a_{1},\dots,a_{m}$. We simply write $\mathbb{P}$ if the weights and $m$ are
clear from the context. We write $\mathbb{P}_{\mathrm{sing}}$ for the singular
locus of $\mathbb{P}$. The space $\mathbb{P}$ is said _wellformed_ if the
quotient stack
$\boldsymbol{\mathbb{P}}=[\operatorname{Spec}{S}\setminus\\{0\\}/\mathbb{C}^{\times}]$
is canonical, that is, the natural morphism
$s\colon\boldsymbol{\mathbb{P}}\to\mathbb{P}$ is an isomorphism in codimension
$1$.

A subvariety $X\subset\mathbb{P}$ is called _quasismooth_ of dimension $m$ if
its affine cone $C_{X}\subset\mathbb{C}^{n+1}$ is smooth of dimension $m+1$
outside of its vertex $\underline{0}$. This implies that the singularities of
$X$ are only due to the $\mathbb{C}^{\times}$-action defining $\mathbb{P}$,
hence are cyclic quotient singularities. A subvariety $X\subset\mathbb{P}$ of
codimension $c$ is called _wellformed_ if $\mathbb{P}$ is wellformed and
$\mathrm{codim}_{X}(\mathbb{P}_{\mathrm{sing}}\cap X)\geq 2$. Observe that, if
$X=\operatorname{Proj}(S/I)\subset\mathbb{P}$, where $I$ is a homogeneous
ideal of $S$, is quasismooth and wellformed, then $X$ is the coarse moduli
space of the canonical stack
$[\operatorname{Spec}(S/I)\setminus\\{0\\}/\mathbb{C}^{\times}]$.

For $X=X_{d}\subset\mathbb{P}(a_{1},\dots,a_{m})$ a quasismooth and wellformed
hypersurface of degree $d$, the adjunction formula for the canonical sheaf
$K_{X}$ [IF00, Section 6.14] states that

$K_{X}=\mathcal{O}_{X}(d-\sum_{i=1}^{m}a_{i}).$

By definition $X$ is Fano if $-K_{X}$ is ample, i.e., if and only if
$\sum_{i=1}^{m}a_{i}-d>0$. $X$ is called _anticanonical_ if
$-K_{X}=\mathcal{O}_{X}(1)$, i.e., if and only if $\sum_{i=1}^{m}a_{i}-d=1$.

A Fano surface with quotient singularities is called a _log del Pezzo
surface_. Johnson and Kollár [JK01] classify all anticanonical quasismooth
wellformed log del Pezzo surfaces in weighted projective $3$-spaces. They are
the general members of either the series (1), where $k\in\mathbb{N},\ k>0$, or
of one of $22$ sporadic families. The sporadic families are all listed in
[JK01, Theorem 8]. General surfaces of the form (1) have isolated log-terminal
singularities (see Section 4.1). On the other hand, there are three sporadic
families for which quasismoothness actually coincides with smoothness, namely
cubic surfaces $X_{3}\subset\mathbb{P}^{3}$, degree-two del Pezzo surfaces
$X_{4}\subset\mathbb{P}(1,1,1,2)$, and degree-one del Pezzo surfaces
$X_{6}\subset\mathbb{P}(1,1,2,3)$.

####  2.2.1. Degree-two del Pezzo surfaces

We will prove in Section 4 that the geometry of the series (1) is related to
the geometry of some special degree-two smooth del Pezzo surfaces. Below we
recall some basic facts about degree-two smooth del Pezzo surfaces which we
will use repeatedly.

Let $S$ be a smooth del Pezzo surface of degree two. Then $S$ is isomorphic to
the blow-up of $\mathbb{P}^{2}$ at $7$ distinct points in general position
[Man86, IV.24.4][Dem80, Thm 1]. Conversely, the choice of a set
$\mathcal{L}=\\{l_{1},\dots,l_{7}\\}$ of $7$ disjoint $(-1)$-curves on $S$ is
equivalent to the choice of a birational morphism $\pi\colon
S\to\mathbb{P}^{2}$ which expresses $S$ as the blowup of $\mathbb{P}^{2}$ at
the $7$ points $x_{i}=\pi(l_{i})$. We refer to such a set $\mathcal{L}$ as to
an exceptional set for $S$.

On the other hand, $S$ is expressed by its anticanonical series as a double
cover $\sigma\colon S\to\mathbb{P}^{2}$ branched along a quartic curve
$Q=(f_{4}(x_{1},x_{2},x_{3})=0)\subset\mathbb{P}^{2}$ [Kol96, III.3.5] Hence,
$S$ can be defined by an equation of the form

$(x_{4}^{2}-f_{4}(x_{1},x_{2},x_{3})=0)\subset\mathbb{P}(1,1,1,2).$

The surface $S$ contains $56$ $(-1)$-curves, which can be described in terms
of the morphisms $\sigma$ and $\pi$. On the one hand, the $56$ $(-1)$-curves
are the preimages through $\sigma$ of the $28$ lines bitangent to $Q$. Each
preimage is a reducible curve of the form $m\cup m^{\prime}$ where
$m,m^{\prime}$ are $(-1)$-curves on $S$ satisfying $m+m^{\prime}\sim-K_{S}$
(we refer to $m$ and $m^{\prime}$ as _dual_ curves). Thus any two of the $56$
$(-1)$-curves are either disjoint, intersect at one point, or intersect at two
points precisely if they are dual to each other. On the other hand, given an
exceptional set $\mathcal{L}=\\{l_{1},\dots,l_{7}\\}$ for $S$ and the
corresponding blow-down morphism $\pi\colon S\to\mathbb{P}^{2}$, the $28$
pairs of $(-1)$-curves described above are the strict transforms through $\pi$
of:

  * •

the line through two points $x_{i},x_{j}$ and the conic through the remaining
5 points (21 pairs);

  * •

the cubic with a node at one of the $x_{i}$ and passing through the remaining
6 points, paired with the exceptional divisor $l_{i}$ (7 pairs).

By the proof of [TVAV09, Lemma 4.1], at most four bitangents to $Q$ can meet
at one point, thus at most four (-1)-curves on $S$ can meet at one point of
$S$. Such points on $S$ are called generalized Eckardt points.

Denote by $I^{1,7}$ the lattice $\mathbb{Z}^{8}$ together with the symmetric
bilinear form defined by the diagonal matrix with diagonal entries
$(1,-1,\dots,-1)$ with respect to the standard basis

$e_{0}=(1,0,\dots,0)^{T},\quad
e_{1}=(1,0\dots,0)^{T},\quad\dots\quad,e_{7}=(0,0\dots,1)^{T}.$

The Picard lattice $\operatorname{Pic\,}(S)$ is isomorphic to the lattice
$I^{1,7}$[Dol12, Section 8.2.6]. The set of $(-1)$-curves on $S$ is in
bijection with the set of exceptional vectors in $I^{1,7}$, that is, the
vectors $v\in I^{1,7}$ such that $v\cdot k=-1$, where $k=(3,-1,\dots,-1)^{T}$,
and $v^{2}=-1$. These vectors are listed in [Dol12, Proposition 8.2.19].

###  2.3. $\operatorname{GL}(2,\mathbb{C})$ McKay correspondence

Suppose $G$ is a finite subgroup of $\operatorname{GL}(2,\mathbb{C})$ which is
small, i.e. it acts freely on $\mathbb{C}^{2}/\\{0\\}$. Let $G$ act naturally
on $S=\mathbb{C}[x,y]$ and set $R=S^{G}$. We can associate two resolutions to
the singularity $X=\operatorname{Spec}R$: its minimal resolution $f\colon Y\to
X$, and a non-commutative crepant resolution ([vdB04a, Definition 4.1]) given
by the skew group algebra $S\ast G$. The category of finitely generated $S\ast
G$-modules is equivalent to the category of $G$-equivariant sheaves on
$\mathbb{C}^{2}$, and therefore to the category of coherent sheaves over the
quotient stack $[\mathbb{C}^{2}/G]$:

$\mathrm{mod}\;S\ast G\simeq\operatorname{Coh}[\mathbb{C}^{2}/G].$

There are bijective correspondences among the sets of irreducible
representations of $G$, indecomposable Cohen-Macaulay $R$-modules,
indecomposable projective $S\ast G$-modules, and indecomposable full sheaves
on $Y$ [Esn85, Remark 1.4, Lemma 2.2]. More explicitly, to an irreducible
representation $\rho$ of $G$, we associate a free $S$-module777By [Aus86,
Lemma 1.1], a $S\ast G$-module is projective precisely if it is free as a
$S$-module. Then, the $S\otimes\rho$ for $\rho\in\mathrm{Irrep}(G)$ are all
the indecomposable projectives. $S\otimes\rho$, the CM $R$-module

$M_{\rho}\coloneqq(S\otimes\rho^{*})^{G}$

and the coherent sheaf $\widetilde{M_{\rho}}\coloneqq
f^{*}(M_{\rho})/\mathrm{tors.}$ on $Y$ (sheaves arising this way are called
full).

Moreover, irreducible exceptional curves of $Y$ are in bijection with special
representations of $G$, i.e. irreducible representations $\rho$ for which
$H^{1}\left(\widetilde{M_{\rho}}^{\vee}\right)=0$. Let $E_{1},...,E_{m}$ be
the exceptional curves on $Y$: Wunram [Wun88, Theorem 1.2] shows that there
are exactly $m+1$ special representations $\rho_{0},\rho_{1},...,\rho_{m}$ of
$G$, where $\rho_{0}$ is the trivial representation and the other satisfy the
multiplication formula

$\mathrm{c}_{1}(\widetilde{M}_{\rho_{i}})\cdot E_{j}=\delta_{ij}.$

A refined version of the formula is given by Ishii in [Ish02], and we briefly
recall it here.

Consider the equivariant Koszul complex of $0\in\mathbb{C}^{2}$:

(5) $0\to
S\otimes\rho_{\mathrm{det}}\xrightarrow{(y,-x)^{T}}S\otimes\rho_{\mathrm{nat}}\xrightarrow{(x,y)}S\to\mathcal{O}_{0}\to
0,$

where $\rho_{\mathrm{nat}}$ is the representation induced by
$G\subset\operatorname{GL}(2,\mathbb{C})$ and
$\rho_{\mathrm{det}}\coloneqq\Lambda^{2}\rho_{\mathrm{nat}}$. Tensoring with
$\rho^{*}$ and taking invariants we obtain the Auslander-Reiten (AR) sequences

(6) $\begin{split}A_{\rho}\coloneqq\left[0\to\tau(M_{\rho})\to N_{\rho}\to
M_{\rho}\to 0\right],\end{split}$

where $N_{\rho}\coloneqq(S\otimes\rho_{\mathrm{nat}}\otimes\rho^{*})^{G}$ and
$\tau(M_{\rho})\coloneqq(S\otimes\rho_{\mathrm{det}}\otimes\rho^{*})^{G}$. The
$A_{\rho}$ are exact except for $\rho=\rho_{0}$, in which case

$0\to\tau(R)\to N_{\rho_{0}}\to R\to\mathcal{O}_{0}\to 0$

is exact.

Recall that the fundamental cycle of $Y\to\operatorname{Spec}R$ is
$E=\sum_{i=1}^{m}r_{i}E_{i}$, where $r_{i}\geq 1$ are the unique smallest
integers satisfying $E\cdot E_{i}\leq 0$ for all $i=1,...,m$ [Art66]. An
immediate consequence of the proof of [Ish02, Theorem 5.1] is:

######  Proposition 2.2.

Let $\rho\in\mathrm{Irrep}(G)$. Denote by $E$ the fundamental cycle of
$f\colon Y\to X$ and by $\widetilde{A_{\rho}}\in D^{b}(Y)$ the complex formed
by the torsion-free pull-back of the terms of $A_{\rho}$. Then

(7)
$\displaystyle\widetilde{A_{\rho}}\simeq\begin{cases}\mathcal{O}_{E}&\mbox{ if
}\rho=\rho_{0},\\\ \mathcal{O}_{E_{i}}(-1)[1]&\mbox{ if }\rho=\rho_{i}\mbox{
for }1\leq i\leq m,\\\ 0&\mbox{ otherwise.}\end{cases}$

####  2.3.1. Derived McKay correspondence

In the above setting, there is a full and faithful functor $\phi\colon D(Y)\to
D[\mathbb{C}^{2}/G]$ constructed as follows. One observes that $Y$ can be
constructed as the $G$-Hilbert scheme parametrizing $G$-clusters of
$\mathbb{C}^{2}$ (i.e. finite, $G$-invariant subschemes
$Z\subset\mathbb{C}^{2}$ such that $H^{0}(\mathcal{O}_{Z})$ is isomorphic as a
$\mathbb{C}[G]$-module to the regular representation of $G$) [Ish02, Theorem
3.1, Theorem 8.1]. Thus, $Y\times\mathbb{C}^{2}$ admits a universal family
$Z$, and one can define the diagram

(8)
${Z}$${Y\times\mathbb{C}^{2}}$${\mathbb{C}^{2}}$${Y}$${X}$${\subseteq}$$\scriptstyle{\pi_{\mathbb{C}^{2}}}$$\scriptstyle{\pi_{Y}}$$\scriptstyle{f}$

where we let $G$ act trivially on $X$ and $Y$, so that all morphisms are
equivariant. Define the Fourier–Mukai functor

(9)
$\phi(-)\coloneqq\operatorname{\mathbf{R}\\!}\pi_{\mathbb{C}^{2}*}\left(\pi_{Y}^{*}(-\otimes\rho_{0})\otimes^{\operatorname{\mathbf{L}\\!}}\mathcal{O}_{Z}\right)\colon
D(Y)\to D([\mathbb{C}^{2}/G]).$

The functor $\phi$ is an equivalence if and only if $Y$ is a crepant
resolution, which happens if and only if
$G\subset\operatorname{SL}(2,\mathbb{C})$, equivalently $R$ is Gorenstein.
Even in the $G\subset\operatorname{GL}(2,\mathbb{C})$ case, $\phi$ admits left
and right adoints. So its essential image is admissible and there is a
semiorthogonal decomposition:

(10)
$D([\mathbb{C}^{2}/G])=\left\langle\phi(D(Y))^{\perp},\phi(D(Y))\right\rangle.$

The essential image of $\phi$ and its right orthogonal are described as
follows. Since the functor $(-)^{G}$ of taking invariant part defines an
equivalence of categories between projective $S\ast G$-modules and CM
$R$-modules [Aus86, Sec. 2], the
$\\{S\otimes\rho\\}_{\rho\in\mathrm{Irrep}(G)}$ are projective generators of
$\mathrm{mod-}S\ast G$ and hence they generate $D([\mathbb{C}^{2}/G])$.

######  Proposition 2.3.

[IU15, Prop. 1.1] 888In Section 3.1 we provide an alternative proof of
Proposition 2.3 based on our Theorem 3.1. The essential image of $\phi$ in
$D([\mathbb{C}^{2}/G])$ is generated by the modules $\\{S\otimes\rho\\}$ where
$\rho$ is a special representation. The right orthogonal of $\phi(D(Y))$ in
$D([\mathbb{C}^{2}/G])$ is generated by sheaves $\mathcal{O}_{0}\otimes\rho$
where $\mathcal{O}_{0}=S/(x,y)$ and $\rho$ is not special.

######  Remark 2.4.

Let $\mu$ be an irreducible representation of $G$ and write $a_{\mu\nu}$ for
the multiplicity of $\nu\in\mathrm{Irrep}(G)$ in the decomposition of
$\mu\otimes\rho_{\mathrm{nat}}$ into irreducible representations. Then by (5)
we have

$\begin{split}&\dim\operatorname{Hom}(\mathcal{O}_{0}\otimes\mu,\mathcal{O}_{0}\otimes\nu)=\delta_{\mu\nu}\\\
&\dim\operatorname{Ext}^{1}(\mathcal{O}_{0}\otimes\mu,\mathcal{O}_{0}\otimes\nu)=a_{\mu\nu}\\\
&\dim\operatorname{Ext}^{2}(\mathcal{O}_{0}\otimes\mu,\mathcal{O}_{0}\otimes\nu)=\dim\operatorname{Hom}(\mathcal{O}_{0}\otimes\nu,\mathcal{O}_{0}\otimes\mu\otimes\rho_{\mathrm{det}})\end{split}$

This information is summarised in the _McKay quiver of $G$_, whose vertices
represent the irreducible representation of $G$, and whose solid (resp.
dashed) arrows from $\mu$ to $\nu$ denote a basis of
$\operatorname{Ext}^{1}(\mathcal{O}_{0}\otimes\mu,\mathcal{O}_{0}\otimes\nu)$
(resp.
$\operatorname{Ext}^{2}(\mathcal{O}_{0}\otimes\mu,\mathcal{O}_{0}\otimes\nu)$).
We give two examples in Figure 2 and Figure 2.

####  2.3.2. Cyclic singularities

For relatively prime integers $1\leq a<r$, define the group
$G=\frac{1}{r}(1,a)$ as the cyclic small subgroup of
$\operatorname{GL}(2,\mathbb{C})$ generated by

$\zeta\coloneqq\begin{pmatrix}\epsilon&0\\\ 0&\epsilon^{a}\end{pmatrix}$

where $\epsilon$ is a primitive $r$-th root of unity. We use the same notation
$\frac{1}{r}(1,a)$ for the corresponding cyclic quotient singularity
$\operatorname{Spec}R$.

######  Definition 2.5.

For integers $1\leq a<r$ as above, set $i_{0}=r$, $i_{1}=a$, and define
$i_{t}$ inductively by

(11) $i_{t}\coloneqq\alpha_{t-1}i_{t-1}-i_{t-2}$

where $\alpha_{t-1}\geq 2$ and $0\leq i_{t}<i_{t-1}$, until $i_{n}=1$ and
$i_{n+1}=0$. The $I$-series $I(r,a)$ is the sequence
$I(r,a)\coloneqq\\{i_{0}=r,...,i_{n+1}=0\\}$.

The equations (11) can be rearranged and spliced together to give the
Hirzebruch–Jung continued fraction expansion of $\frac{r}{a}$, i.e. the
expression:

(12)
$\frac{r}{a}=\alpha_{1}-\frac{1}{\alpha_{2}-\frac{1}{\alpha_{3}-\frac{1}{(...)}}}\eqqcolon[\alpha_{1},...,\alpha_{n}]$

It is well-known [Rie74] that the dual graph of the minimal resolution $Y$ of
$\frac{1}{r}(1,a)$ is the labeled diagram

$-\alpha_{1}$$-\alpha_{2}$$\cdots$$-\alpha_{n-1}$$-\alpha_{n}$

For $l\in\mathbb{Z}$, denote by $\rho_{l}$ the irreducible representation of
$G$ acting with weight $l$. Then the modules

$M_{l}\coloneqq(\mathbb{C}[x,y]\otimes\rho_{l}^{*})^{G}=\left\\{f\in\mathbb{C}[x,y]\mid\zeta\cdot
f=\epsilon^{l}f\right\\}$

are the indecomposable CM $R$-modules999Clearly, $\rho_{l}=\rho_{m}$ and
$M_{l}=M_{m}$ if $l\equiv m\mod r$., and $M_{l}$ is special if $l\in I(r,a)$
[Wun87]. In [Wun88, Appendix A1]) it is shown that their torsion free pull-
backs $\widetilde{M}_{l}$ satisfy

(13) $c_{1}(\widetilde{M}_{l})\cdot E_{t}=\delta_{l,i_{t}}.$

Moreover, since in this case $\rho_{\mathrm{nat}}=\rho_{1}\oplus\rho_{a}$ and
$\rho_{\mathrm{det}}=\rho_{a+1}$, the AR-sequences (6) specialise as follows:

(14) $\begin{split}A_{l}\coloneqq\left[0\to M_{l-a-1}\to M_{l-1}\oplus
M_{l-a}\to M_{l}\to 0\right],\end{split}$

and by (13) Proposition 2.2 rewrites as:

######  Proposition 2.6.

Let $l\in\\{0,\dots,r-1\\}$. Denote by $E$ the fundamental cycle of $f\colon
Y\to X$ and by $\widetilde{A_{l}}\in D^{b}(Y)$ the torsion-free pull-back of
the AR-sequence ending in $M_{l}$. Then we have

(15) $\displaystyle\widetilde{A_{l}}\simeq\begin{cases}\mathcal{O}_{E}&\mbox{
if }l=0\\\ \mathcal{O}_{E_{t}}(-1)[1]&\mbox{ if }l=i_{t}\in
I(r,a)\setminus\\{0\\}\\\ 0&\mbox{ otherwise.}\end{cases}$

In the cyclic case, the right orthogonal $\phi(D(Y))^{\perp}$ admits a full
exceptional collection. Every non-special representation $\rho$ contributes
one exceptional sheaf, which has finite length and socle
$\mathcal{O}_{0}\otimes\rho$ [IU15, Sec. 3]. We do not recall here the
construction of [IU15], but we point out below that the exceptional sheaves
are simple in the two cases we consider in this paper, namely
$G=\frac{1}{n}(1,1)$, $n>1$ integer, and $G=\frac{1}{2k+1}(1,k)$, $k\geq 1$
integer.

######  Example 2.7.

Let $G=\frac{1}{n}(1,1)$, for $n>1$ integer. Then $I(n,1)=\\{n,1,0\\}$ and
$\frac{n}{1}=[n]$. It follows that the minimal resolution
$Y=\mathrm{Tot}\left(\mathcal{O}_{\mathbb{P}^{1}}(-n)\right)$ has a single
exceptional $(-n)$-curve, $\rho_{0}$ and $\rho_{1}$ are the only special
representations, and the exceptional collection computed in [IU15] reads

(16)
$\phi(D(Y))^{\perp}=\left(\mathcal{O}_{0}\otimes\rho_{2},...,\mathcal{O}_{0}\otimes\rho_{n-1}\right).$

For all $i$, the McKay quiver of $G$ has two solid arrows from $\rho_{i}$ to
$\rho_{i+1}$, and one dashed arrow from $\rho_{i}$ to $\rho_{i+1}$ (see Figure
2).

######  Example 2.8.

Let $G=\frac{1}{2k+1}(1,k)$, for $k\geq 1$ integer. One computes

$\frac{2k+1}{k}=[3,\underbrace{2,...,2}_{k-1\text{ times}}],$

and the relevant terms of the $i$-series are $i_{t}=k+1-t$ for $t=1,...,k+1$.
It follows that the minimal resolution contains one $(-3)$-curve $E_{1}$ and
$k-1\times(-2)$-curves $E_{i}$, $i\in\\{2,\dots k\\}$, and the special
representations are $\rho_{i_{t}}=\rho_{k+1-t}$, $t=1,...,k+1$. The
exceptional collection computed in [IU15] reads

$\phi(D(Y))^{\perp}=\left(\mathcal{O}_{0}\otimes\rho_{k+1},...,\mathcal{O}_{0}\otimes\rho_{2k}\right).$

For all $i$, the McKay quiver of $G$ has a solid arrow from $\rho_{i}$ to
$\rho_{i+1}$, a solid arrow from $\rho_{i}$ to $\rho_{i+k}$, and a dashed
arrow from $\rho_{i}$ to $\rho_{i+k+1}$.

Figure 2 and 2 depict the McKay quiver of $\frac{1}{5}(1,1)$ and of
$\frac{1}{7}(1,3)$ , with objects and morphisms in $\phi(D(Y))^{\perp}$
colored in blue.

$\rho_{2}$$\rho_{3}$$\rho_{4}$$\rho_{0}$$\rho_{1}$ Figure 1.  The case
$\frac{1}{5}(1,1)$.

$\rho_{3}$$\rho_{4}$$\rho_{5}$$\rho_{6}$$\rho_{0}$$\rho_{1}$$\rho_{2}$ Figure
2.  The case $\frac{1}{7}(1,3)$.

####  2.3.3. Global case

Suppose $X$ is a complex quasi-projective surface with isolated quotient
singularities, i.e. every singular point of $X$ is étale-locally isomorphic to
a quotient $\mathbb{C}^{2}/G$, where $G$ is a finite subgroup of
$\operatorname{GL}(2,\mathbb{C})$. Let $\tilde{X}$ be the minimal resolution
of $X$, and denote by $\mathcal{X}$ the canonical stack associated to $X$. By
[IU15, Theorem 1.4], there is a fully faithful functor $\Phi\colon
D(\tilde{X})\to D(\mathcal{X})$.

Suppose that $X$ has only cyclic quotient singularities and consider the
diagram

${\mathcal{\mathcal{Z}}}$${\tilde{X}\times\mathcal{X}}$${\mathcal{X}}$${\tilde{X}}$${X}$${\subseteq}$$\scriptstyle{\pi_{\mathcal{X}}}$$\scriptstyle{\pi_{\tilde{X}}}$$\scriptstyle{s}$$\scriptstyle{f}$

where $\mathcal{Z}$ is the substack
$(\tilde{X}\times_{X}\mathcal{X})_{\mathrm{red}}$. In this case, [IU15, Prop.
8.1] shows that the Fourier-Mukai functor
$\Phi=\pi_{\mathcal{X}*}(\pi_{\tilde{X}}^{*}(-)\otimes\mathcal{O}_{\mathcal{Z}})$
is fully faithful. Over the smooth locus of $X$, $\mathcal{Z}$ is the graph of
$s$. On a singular chart of $X$, $\mathcal{Z}$ restricts to the quotient
$[Z/G]$ of the universal subscheme of diagram (8). This yields a
semiorthogonal decomposition

$D(\mathcal{X})=\left\langle\textbf{e}_{1},...,\textbf{e}_{N},\Phi(D(\widetilde{X}))\right\rangle,$

where $(\textbf{e}_{1},...,\textbf{e}_{N})$ is an exceptional collection,
containing the local contributions described in Section 2.3.2.

##  3\. Computation of left adjoint functor

Consider the fully faithful functor $\phi\colon D(Y)\to D([\mathbb{C}^{2}/G])$
of Equation (9) and the semiorthogonal decomposition (10). By Proposition 2.3,
$\phi(D(Y))^{\perp}$ is generated by simple sheaves of the form
$\mathcal{O}_{0}\otimes\rho$, where $\rho$ is nonspecial. In this Section we
introduce the functor $\psi\colon D([\mathbb{C}^{2}/G])\to D(Y)$ left adjoint
to $\phi$, and compute the images $\psi(\mathcal{O}_{0}\otimes\rho)$ for all
$\rho$.

We follow the same setup and notation of Section 2.3. The left adjoint of
$\phi$ is:

(17)
$\psi(-)=[\operatorname{\mathbf{R}\\!}\pi_{Y*}\left(\pi_{\mathbb{C}^{2}}^{*}(-\otimes
K_{\mathbb{C}^{2}})\otimes^{\operatorname{\mathbf{L}\\!}}\mathcal{O}_{\mathcal{Z}}^{\vee}\right)]^{G}[2]$

where $K_{\mathbb{C}^{2}}=\mathcal{O}_{\mathbb{C}^{2}}\otimes\rho_{\det}$ is
the $G$-equivariant canonical sheaf of $\mathbb{C}^{2}$.

######  Theorem 3.1.

For $i\in\\{1,\dots,m\\}$ let $-\alpha_{i}$ be the self intersection of the
exceptional curve $E_{i}$. Then we have:

$\displaystyle\psi(\mathcal{O}_{0}\otimes\rho)\simeq\begin{cases}\mathcal{O}_{E}(E)[1]&\mbox{
if }\rho=\rho_{\mathrm{det}}^{\ast}\\\
\mathcal{O}_{E_{i}}(-\alpha_{i}+1)&\mbox{ if
}\rho\otimes\rho_{\mathrm{det}}=\rho_{i}\mbox{ for some }1\leq i\leq m\\\
0&\mbox{ otherwise}.\end{cases}$

###### Proof.

By definition of $\psi$ we have:

(18)
$\begin{split}\psi(\mathcal{O}_{0}\otimes\rho)&=\left[\operatorname{\mathbf{R}\\!}\pi_{Y\ast}\left(\pi_{\mathbb{C}^{2}}^{\ast}(\mathcal{O}_{0})\otimes^{\operatorname{\mathbf{L}\\!}}\mathcal{O}_{\mathcal{Z}}^{\vee}\otimes\rho\otimes\rho_{\mathrm{det}}[2]\right)\right]^{G}\\\
&=\left[\operatorname{\mathbf{R}\\!}\pi_{Y\ast}\left(\mathcal{O}_{Y\times\\{0\\}}\otimes^{\operatorname{\mathbf{L}\\!}}\mathcal{O}_{\mathcal{Z}}^{\vee}\otimes\rho\otimes\rho_{\mathrm{det}}[2]\right)\right]^{G}\\\
&=\left[\operatorname{\mathbf{R}\\!}\pi_{Y\ast}\bigl{(}\operatorname{\mathbf{R}\\!}\mathcal{H}om\left(\mathcal{O}_{\mathcal{Z}},\mathcal{O}_{Y\times\\{0\\}}\otimes\rho\otimes\rho_{\mathrm{det}}[2]\right)\bigr{)}\right]^{G}\end{split}$

Consider the resolution obtained from pulling back (5) to
$Y\times\mathbb{C}^{2}$:

$0\to\mathcal{O}_{Y\times\mathbb{C}^{2}}\otimes\rho_{\mathrm{det}}\xrightarrow{(y,-x)^{T}}\mathcal{O}_{Y\times\mathbb{C}^{2}}\otimes\rho_{\mathrm{nat}}\xrightarrow{(x,y)}\mathcal{O}_{Y\times\mathbb{C}^{2}}\to\mathcal{O}_{Y\times\\{0\\}}\to
0.$

By Grothendieck–Verdier duality

(19)
$\operatorname{\mathbf{R}\\!}\pi_{Y\ast}\operatorname{\mathbf{R}\\!}\mathcal{H}om\left(\mathcal{O}_{\mathcal{Z}},\mathcal{O}_{Y\times\mathbb{C}^{2}}\otimes\rho_{\mathrm{det}}[2]\right)=\operatorname{\mathbf{R}\\!}\mathcal{H}om(\operatorname{\mathbf{R}\\!}\pi_{Y\ast}\mathcal{O}_{\mathcal{Z}},\mathcal{O}_{Y})$

Denote by $\mathcal{R}$ the vector bundle
$\mathcal{R}=\operatorname{\mathbf{R}\\!}\pi_{Y\ast}\mathcal{O}_{\mathcal{Z}}$.
Then the right hand side of the equation above is
$\mathcal{R}^{\vee}=\operatorname{\mathbf{R}\\!}\mathcal{H}om(\mathcal{R},\mathcal{O}_{Y})$.
Combining (18) and (19) we find that

(20)
$\psi(\mathcal{O}_{0}\otimes\rho)=\left[\mathcal{R}^{\vee}\otimes(\rho_{\mathrm{det}}\otimes\rho)\xrightarrow{}\mathcal{R}^{\vee}\otimes(\rho_{\mathrm{nat}}\otimes\rho)\xrightarrow{}\mathcal{R}^{\vee}\otimes\rho\right]^{G}.$

The vector bundle $\mathcal{R}$ decomposes as a sum

(21) $\mathcal{R}=\oplus_{\rho}\mathcal{R}_{\rho}\otimes\rho$

according to the decomposition of the regular representation. Moreover, for
all $\rho\in\mathrm{Irrep}(G)$, we have
$(\mathcal{R}^{\vee}\otimes\rho)^{G}=(\mathcal{R}_{\rho}^{\vee}\otimes\rho^{*}\otimes\rho)=\mathcal{R}_{\rho}^{\vee}$.
Then by exactness of the functor $(-)^{G}$:

(22)
$\psi(\mathcal{O}_{0}\otimes\rho)=\left[\mathcal{R}_{\rho_{\mathrm{det}}\otimes\rho}^{\vee}\xrightarrow{}\mathcal{R}_{\rho_{\mathrm{nat}}\otimes\rho}^{\vee}\xrightarrow{}\mathcal{R}_{\rho}^{\vee}\right].$

By [Ish02, Corollary 3.2], $\mathcal{R}_{\rho}$ is the torsion free pull-back
$\mathcal{R}_{\rho}=\widetilde{M_{\rho}}$. Therefore,
$\psi(\mathcal{O}_{0}\otimes\rho)$ is dual to the torsion-free pull-back of
$\widetilde{A}_{\rho\otimes\rho_{\mathrm{det}}}$:

$\operatorname{\mathbf{R}\\!}\mathcal{H}om(\widetilde{A}_{{\rho\otimes\rho_{\mathrm{det}}}},\mathcal{O}_{Y})[2]=\psi(\mathcal{O}_{0}\otimes\rho),$

since all summands of $\mathcal{R}$ are locally free. The statement then
follows from Proposition 2.2. ∎

######  Remark 3.2.

Let $\operatorname{Coh}_{G}(\mathbb{C}^{2})_{0}$ be the finite length category
of $G$-equivariant sheaves supported at $0\in\mathbb{C}^{2}$, and let
$\\!{}^{0}\mathrm{Per}(Y/X)$ be the category of $0$-perverse sheaves on $Y$
(as defined in [Bri02]).

The complexes $\tilde{A}_{\rho}$, for $\rho$ special, are quasi-isomorphic to
simple $(-1)$-perverse sheaves. Dualizing them (with the functor
$\operatorname{\mathbf{R}\\!}\mathcal{H}om(-,K_{Y})[2]$), we obtain simple
$0$-perverse sheaves ([VdB04b, Prop. 3.5.8]). Theorem 3.1 shows that $\psi$
sends simple sheaves supported at $0$ to simple $0$-perverse sheaves twisted
by $-K_{Y}$. It follows that $\psi\otimes K_{Y}$ restricts to an exact functor
of abelian categories
$\operatorname{Coh}_{G}(\mathbb{C}^{2})_{0}\to\\!^{0}\mathrm{Per}(Y/X)$.

###  3.1. Alternative proof of Prop. 2.3 (or [Ish02, Prop. 1.1])

As an application of Theorem 3.1, we give an alternative proof of [IU15, Prop.
1.1].

Let $\theta$ be the right adjoint to $\phi$:

$\begin{split}\theta(-)&:=[\operatorname{\mathbf{R}\\!}\pi_{Y\ast}\left(\pi_{\mathbb{C}^{2}}^{\ast}(-)\otimes^{\operatorname{\mathbf{L}\\!}}\mathcal{O}_{\mathcal{Z}}^{\vee}\otimes\pi_{Y}^{\ast}(K_{Y})[2]\right)]^{G}\\\
&\
=[\operatorname{\mathbf{R}\\!}\pi_{Y\ast}\left(\pi_{\mathbb{C}^{2}}^{\ast}(-)\otimes^{\operatorname{\mathbf{L}\\!}}\mathcal{O}_{\mathcal{Z}}^{\vee}\right)]^{G}\otimes
K_{Y}[2].\end{split}$

Note that

(23) $\theta(-\otimes K_{\mathbb{C}^{2}})=\psi(-)\otimes K_{Y}.$

###### Proof of Prop. 2.3.

Consider the semiorthogonal decomposition (10). Let
$\mathcal{C}=\ker(\theta)$, i.e. the full triangulated subcategory of
$D^{b}_{G}(\mathbb{C}^{2})$ containing objects $C$ such that $\theta(C)=0$. By
the adjunction between $\phi$ and $\theta$ one has
$\mathcal{C}=\phi(D(Y))\\!^{\perp}$, thus
$\\!{}^{\perp}\mathcal{C}\simeq\phi(D(Y))$.

#####  Step 1:

we show that the category $\mathcal{C}$ is generated by the simple sheaves
$\mathcal{O}_{0}\otimes\rho$, with $\rho$ not special.

Suppose $E$ is a complex of $G$-equivariant sheaves on $\mathbb{C}^{2}$ which
is not supported on the origin. Then by [BM02, Lemma 5.3] there exists a point
in $X\setminus\\{0\\}$, equivalently a free orbit $Gx$ in $\mathbb{C}^{2}$,
and an integer $i$ such that

$\operatorname{Ext}^{i}_{\mathbb{C}^{2}}(\mathcal{O}_{Gx},E)^{G}\neq 0.$

The functor $\phi$ maps points of $Y\setminus F$ to free orbits, so we may
write $\mathcal{O}_{Gx}\simeq\phi(\mathcal{O}_{p})$ for some $p\in Y\setminus
F$. Then,

$\operatorname{Ext}^{i}(\phi(\mathcal{O}_{p}),E)^{G}=\operatorname{Ext}^{i}_{[\mathbb{C}^{2}/G]}(\phi(\mathcal{O}_{p}),E)=\operatorname{Ext}^{i}_{Y}(\mathcal{O}_{p},\theta(E))\neq
0,$

thus $E\notin\mathcal{C}$. Therefore, all objects of $\mathcal{C}$ are
supported at the origin.

Theorem (3.1), combined with (23), implies that
$\mathcal{O}_{0}\otimes\rho\in\mathcal{C}$ if and only if $\rho$ is not
special. Then sheaves in $\mathcal{C}$ have a composition series with factors
the non-special simples. Since $\theta$ is exact on
$\operatorname{Coh}_{G}(\mathbb{C}^{2})_{0}$ by Remark 3.2 and (23), a complex
$E\in D_{G}(\mathbb{C}^{2})$ lies in $\mathcal{C}$ precisely if its
cohomologies do, which proves the claim.

##### Step 2:

we show that the sheaf $\mathcal{O}_{\mathbb{C}^{2}}\otimes\rho$ is in the
essential image of $\Phi$ for $\rho$ special.

One readily checks that, for two irreducible representations $\sigma$ and
$\tau$,

$\operatorname{\mathbf{R}\\!}\operatorname{Hom}(\mathcal{O}_{\mathbb{C}^{2}}\otimes\sigma,\mathcal{O}_{0}\otimes\tau)=0$

unless $\sigma=\tau$ (see Remark 2.4). Then by Step 1 we have that
$\mathcal{O}_{\mathbb{C}^{2}}\otimes\rho\in~{}^{\perp}\mathcal{C}=\operatorname{Im}\Phi$
for $\rho$ special.

##### Step 3:

we show that the sheaves $\mathcal{O}_{\mathbb{C}^{2}}\otimes\rho$ for $\rho$
special generate the essential image of $\Phi$.

We have

(24)
$\begin{split}\psi(\mathcal{O}_{\mathbb{C}^{2}}\otimes\rho)&=\left[\operatorname{\mathbf{R}\\!}\pi_{Y\ast}\operatorname{\mathbf{R}\\!}\mathcal{H}om\left(O_{\mathcal{Z}},\mathcal{O}_{Y\times\mathbb{C}^{2}}\otimes\rho\otimes\rho_{\mathrm{det}}[2]\right)\right]^{G}\\\
&=\left[\operatorname{\mathbf{R}\\!}\mathcal{H}om(\operatorname{\mathbf{R}\\!}\pi_{Y\ast}\mathcal{O}_{\mathcal{Z}},\mathcal{O}_{Y})\otimes\rho\right]^{G}\\\
&=\left[\mathcal{R}^{\vee}\otimes\rho\right]^{G}=\mathcal{R}_{\rho}^{\vee}\end{split}$

by (19) and (21). Then by Step 2, Equation (24), and the fact that

$\phi\circ\psi_{|{\phi(D(\operatorname{Coh}(Y)))}}=\mathrm{id}_{\phi(D(\operatorname{Coh}(Y)))}$

we get that
$\Phi(\mathcal{R}_{\rho}^{\vee})=\mathcal{O}_{\mathbb{C}^{2}}\otimes\rho$ for
all $\rho$ special. Since the $\mathcal{R}_{\rho}^{\vee}$ are generators for
$D^{b}(Y)$ by [VdB04b, Theorem B], the special
$\\{\mathcal{O}_{\mathbb{C}^{2}}\otimes\rho\\}$ generate
$\Phi(D^{b}(\operatorname{Coh}(Y)))$. ∎

###  3.2. Cyclic singularities case

Suppose we are in the cyclic setting of Section 2.3.2. Then we can give a more
precise statement of Theorem 3.1:

######  Proposition 3.3.

We have:

(25)
$\displaystyle\psi(\mathcal{O}_{0}\otimes\rho_{i})\simeq\begin{cases}\mathcal{O}_{E}(E)[1]&\mbox{
if }a+1+i=0\\\ \mathcal{O}_{E_{t}}(-\alpha_{t}+1)&\mbox{ if }a+1+i=i_{t}\in
I(r,a)\setminus\\{0,r\\}\\\ 0&\mbox{ otherwise}.\end{cases}$

###### Proof.

The proof goes as the proof of Theorem 3.1. Recall that in this case
$\rho_{\det}=\rho_{a+1}$, and that Proposition 2.6 computes the torsion-free
pullbacks $\widetilde{A_{l}}$. ∎

######  Example 3.4.

Suppose $G=\frac{1}{5}(1,3)$. We have $\frac{5}{3}=[2,3]$ and
$I(5,3)=\\{5,3,1,0\\}$. The SOD (2) from [IU15] reads

$D^{b}(\mathcal{X})=\left\langle
G,\mathcal{O}_{0}\otimes\rho_{4},\phi(D^{b}(\operatorname{Coh}(Y)))\right\rangle$

where $G=\mathbb{C}[x,y]/(x,y^{2})\otimes\rho_{4}$ fits in an extension
$\mathcal{O}_{0}\otimes\rho_{2}\xrightarrow{y}G\to\mathcal{O}_{0}\otimes\rho_{4}$.
By Proposition 3.3 there is a short exact sequence

$\mathcal{O}_{E_{2}}(-2)\to\psi(G)\to\mathcal{O}_{E_{1}}(-1).$

It is straightforward to check that $\psi(G)$ can also be written as an
extension of $\mathcal{O}_{p}$ by $\mathcal{O}_{E}(E)$, where $p\in E_{2}$.

######  Example 3.5.

Let $G=\frac{1}{n}(1,1)$, $n>1$ integer. Let $E$ the single exceptional
$(-n)$-curve in the minimal resolution of $X$ (see Example 2.7). Then Theorem
3.3 reads

(26) $\psi(\mathcal{O}_{0}\otimes\rho_{i})\simeq\begin{cases}0&\mbox{ if
}i=0,...,n-3\\\ \mathcal{O}_{E}(-n)[1]&\mbox{ if }i=n-2\\\
\mathcal{O}_{E}(-n+1)&\mbox{ if }i=n-1.\end{cases}$

######  Example 3.6.

Let $G=\frac{1}{2k+1}(1,k)$, with $k\geq 1$. Let $E_{1},\dots,E_{k}$ be the
$k$ exceptional curves in $Y$, where $E_{1}^{2}=-3$ and $E_{i}^{2}=-2$ for
$i=2,...,k$ (see Example 2.8). The fundamental cycle is
$E=\sum_{i=1}^{k}E_{i}$. Theorem 3.3 reads

(27) $\psi(\mathcal{O}_{0}\otimes\rho_{i})\simeq\begin{cases}0&\mbox{ if
}i=0,...,k-1\\\ \mathcal{O}_{E}(E)[1]&\mbox{ if }i=k\\\
\mathcal{O}_{E_{2k+1-i}}(-\alpha_{2k+1-i}+1)&\mbox{ if
}i=k+1,...,2n.\end{cases}$

####  3.2.1. Alternative (toric) approach to Proposition 3.3

Here we present another possible approach, based on toric geometry, to the
computation in Proposition 3.3.

First, we briefly recall how to describe the singularity $X=\frac{1}{r}(1,a)$
and its minimal resolution torically. The singularity $X$ is the affine toric
surface $U_{\sigma}$ associated to the cone

(28) $\sigma=\langle re_{1}-ae_{2},e_{2}\rangle\subset
N_{\mathbb{R}}\simeq\mathbb{R}^{2}$

Its minimal resolution can be obtained by performing $n$ consecutive weighted
blow-ups with weights $\frac{1}{{i}_{t-1}}(1,i_{t})$, where $t=1,\dots,n$.
More explicitly, write $\rho_{0}=e_{2}$ and $\rho_{n+1}=re_{1}-ae_{2}$. Then
$Y$ is the smooth affine toric surface with fan $\Sigma$, where $\Sigma$ is
the refinement of $\sigma$ obtained by adding the rays generated by the
vectors

(29)
$\rho_{t}=\frac{1}{{i}_{t-1}}\left(\rho_{n+1}+i_{t}\cdot\rho_{t-1}\right)\quad
t=1,\dots,n$

Indeed, if ${\sigma}_{t-1}=\langle\rho_{n+1},\rho_{t-1}\rangle$, the
associated surface is the cyclic singularity
$U_{{\sigma}_{t-1}}=\frac{1}{{i}_{t-1}}(1,i_{t})$. The surface associated to
the refinement of ${\sigma}_{t-1}$ obtained by adding the ray $\rho_{t}$ has
two charts: the chart corresponding to
$\tau_{t}=\langle\rho_{t-1},\rho_{t}\rangle$ is $U_{\tau_{t}}=\mathbb{C}^{2}$,
and the chart corresponding to
${\sigma}_{t}=\langle\rho_{n+1},\rho_{t}\rangle$ is
$U_{{\sigma}_{t}}=\frac{1}{{i}_{t}}(1,i_{t+1})$ if $t<n$, while is
$\mathbb{C}^{2}$ if $t=n$. Note that the vectors $\rho_{t}$ satisfy the
relation $\rho_{t-1}+\rho_{t+1}=\alpha_{t}\cdot\rho_{t}$, which implies that
the self-intersection of the exceptional curve $E_{t}$ associated to the ray
$\rho_{t}$ is $-\alpha_{t}$ (see [CLS11, Theorem 10.4.4]).

The map $f\colon Y\to X$ can be described locally as follows. Let
$u_{t},v_{t}$ be the coordinates on $U_{\tau_{t}}$, and, for $t<n$, let
$x_{t},y_{t}$ be the coordinates on $\mathbb{C}^{2}$ such that
$U_{{\sigma}_{t}}=\operatorname{Spec}\mathbb{C}[x_{t},y_{t}]^{G}$,
$G=\frac{1}{{i}_{t}}(1,i_{t+1})$. Then:

(30) $\begin{cases}x_{t-1}^{i_{t-1}}=u_{t}\\\
y_{t-1}^{i_{t-1}}=u_{t}^{i_{t}}v_{t}^{i_{t-1}}\end{cases}\quad\text{and}\quad\begin{cases}x_{t-1}^{i_{t-1}}=x_{t}^{i_{t-1}}y_{t}\\\
y_{t-1}^{i_{t-1}}=y_{t}^{i_{t}}\end{cases}$

For $t=n$, the same expressions hold, $x_{n},y_{n}$ being the coordinates on
$U_{{\sigma}_{n}}=\mathbb{C}^{2}$.

######  Remark 3.7.

Any 2-dimensional strictly convex cone can be written in the form (28), called
_normal form_. There is an ambiguity for normal forms of 2-dimensional cones:
if $\sigma=\langle re_{1}-ae_{2},e_{2}\rangle$ and $\sigma^{\prime}=\langle
r^{\prime}e_{1}^{\prime}-a^{\prime}e_{2}^{\prime},e_{2}^{\prime}\rangle$ are
lattice equivalent cones, then $r=r^{\prime}$ and either $a=a^{\prime}$ or
$aa^{\prime}=1$ mod $r$. In terms of the associated singularity, this
corresponds to writing $X$ as either $\frac{1}{r}(1,a)$ or
$\frac{1}{r}(a^{\prime},1)$. Note that, if $r/a=[\alpha_{1},...,\alpha_{n}]$,
then $r/a^{\prime}=[\alpha_{n},...,\alpha_{1}]$, that is, $E_{1},\dots,E_{n}$
appear in reverse order when resolving $\frac{1}{r}(a^{\prime},1)$ as
indicated above. Define the $j$-series $J(r,a)$ to be the sequence
$J(r,a)\coloneqq\\{j_{0},j_{1},\dots,j_{n},j_{n+1}\\}$, where
$j_{0}=0,j_{1}=1$, and $j_{t}=j_{t-1}\alpha_{t-1}-j_{t-2}$. Then,
$I(r,a^{\prime})=\\{j_{n+1},j_{n},\dots,j_{1},j_{0}\\}$.

The Picard group of the minimal resolution $Y$ is generated by divisors
$D_{1},...,D_{n}$ determined by the condition $D_{i}\cdot E_{j}=\delta_{ij}$.
For $t=1,\dots,n$, formula (13) implies
$\widetilde{M}_{\rho_{i_{t}}}=\mathcal{O}_{Y}(D_{t})$. Now let
$l\in\\{0,\dots,r-1\\}$. By [Wun88, §A.1, Theorem a)] the summand
$\mathcal{R}_{\rho_{l}}=\widetilde{M}_{\rho_{l}}$ appearing in (21) is the
line bundle $\mathcal{O}_{Y}(\sum_{i}d^{(l)}_{i}D_{i})$, where the
$d^{(l)}_{i}\geq 0$ are uniquely determined by the I-series $I(r,a)$ by
requiring

$l=d_{1}i_{1}+\dots+d_{n}i_{n}\quad\text{and}\quad
0\leq\sum_{t>t_{0}}d_{t}i_{t}<i_{t_{0}}\ \text{for any $t_{0}$}.$

We can then interpret the sequences (22) by studying the $D_{i}$’s and the
divisors associated to the invariant coordinates $x^{r}$ and $y^{r}$ on $X$,
without recurring to Proposition 2.2.

Consider the divisors on $X$ defined by $x^{r}$ and $y^{r}$. By (30), their
strict transforms under $f\colon Y\to X$ are nonreduced curves
$A^{\prime}=rA$, $B^{\prime}=rB$; by construction, the curve $A$ (resp. $B$)
meets the curve $E_{n}$ (resp. $E_{1}$) at exactly one point (see top of
Figure 3). In what follows, we write the pull-backs $f^{*}(x^{r})$,
$f^{*}(y^{r})$ and the $D_{i}$’s as linear combinations of $A,B$, and
$E_{1},...,E_{n}$.

For $s\in\\{0,1,...,n,n+1\\}$ define a sequence
$H(s)\coloneqq\\{h^{(s)}_{0},...,h^{(s)}_{n+1}\\}$ by letting $h^{(s)}_{s}=0$,
$h^{(s)}_{s-1}=h^{(s)}_{s+1}=1$ and

$\begin{split}h^{(s)}_{t}=h^{(s)}_{t+1}\alpha_{t+1}-h^{(s)}_{t+2}&\ \ \mbox{
for }0\leq t<s-1,\\\ h^{(s)}_{t}=h^{(s)}_{t-1}\alpha_{t-1}-h^{(s)}_{t-2}&\ \
\mbox{ for }s+1<t\leq n+1.\end{split}$

Observe that $H(0)=J(r,a)$ and $H(n+1)=I(r,a)$. More generally, the truncation
$H{(s)}_{\geq s}$ is the $j$-series $J(i_{s},i_{s+1})$ and the truncation
$H{(s)}_{\leq s}$ is the $i$-series $I(j_{s},j_{s-1}^{\prime})$, with
$j_{s-1}j_{s-1}^{\prime}=1\mod j_{s}$.

By (30), the terms of $H(0)$ and $H(n+1)$ are the coefficients of the pull-
back of the Cartier divisors $(x^{r}=0)$ and $(y^{r}=0)$ to $Y$:

$\begin{split}(f^{*}(x^{r})=0)=\sum_{t=1}^{n}h^{(0)}_{t}E_{t}+h^{(0)}_{n+1}A\qquad\
(f^{*}(y^{r})=0)=h^{(n+1)}_{0}B+\sum_{t=1}^{n}h^{(n+1)}_{t}E_{t}.\end{split}$

Moreover, one checks that there are numerical equivalences 101010In this
setting, linear and numerical equivalence coincide. In fact, since we are in a
toric setting, they coincide up to torsion of the divisor class group, which
is zero because $Y$ is smooth. Again given the toric setting, it is not
surprising that working up to equivalence does not suffice to compute the
cohomologies of the sequences (22), which is why the expressions (31) are
relevant to our computation.

(31) $D_{s}\equiv\sum_{t=s+1}^{n}h^{(s)}_{t}E_{t}+h^{(s)}_{n+1}A\equiv
h^{(s)}_{0}B+\sum_{t=1}^{s-1}h^{(s)}_{t}E_{t}\qquad\qquad s=1,...,n.$

An alternative way to obtain the expressions (31) is to look at the fibers of
maps $Y\to\mathbb{P}^{1}$ constructed as follows. For each $s$, contract all
the exceptional curves on $Y$ except for $E_{s}$, to obtain a morphism
$\tau_{s}\colon Y\to T_{s}$, where $T_{s}$ is the total space of an orbibundle
over the projective line $\mathbb{P}(j_{s-1},i_{s+1})\simeq\mathbb{P}^{1}$
(see Figure 3). Project further $\sigma_{s}\colon T_{s}\to\mathbb{P}^{1}$.
Then, by (30), the two expressions for $D_{s}$ written in (31) are the fibers
of $\sigma_{s}\circ\tau_{s}\colon Y\to\mathbb{P}^{1}$ over the points $(0:1)$
and $(1:0)$ in $\mathbb{P}^{1}$.

…$(x^{r}=0)$$(y^{r}=0)$$X$$Y$$B$$A$$E_{n}$$E_{1}$$E_{2}$$E_{3}$$E_{4}$$T_{3}$$\mathbb{P}^{1}$$\frac{1}{i_{3}}(1,i_{4})$$\frac{1}{j_{3}}(j_{2},1)$
Figure 3. A picture of the surface $Y$ and the maps $f\colon Y\to X$,
$p_{3}\circ\tau_{3}\colon Y\to\mathbb{P}^{1}$. The toric surface $T_{3}$ has
two singular points of type $\frac{1}{i_{3}}(1,i_{4})$ and
$\frac{1}{j_{3}}(j_{2},1)$.

This setup can be used to interpret the maps in the complexes (22). Indeed,
invariant functions pull-back to maps:

$\mathcal{O}_{Y}\xrightarrow{f^{*}(x^{r})}\mathcal{O}_{Y}\ \ \qquad\ \
\mathcal{O}_{Y}\xrightarrow{f^{*}(y^{r})}\mathcal{O}_{Y}$

which must factor through rational maps

(32)
$x_{(l)}\colon\mathcal{O}_{Y}(\sum_{i}d^{(l)}_{i}D_{i})\simeq\mathcal{R}_{l}\xrightarrow{}\mathcal{R}_{l+1}\simeq\mathcal{O}_{Y}(\sum_{i}d^{(l+1)}_{i}D_{i})$

(and similarly for $y$) satisfying $(x_{(l)})\leq(f^{*}(x^{r}))$. This last
condition uniquely determines a divisor in the linear series
$|\sum_{i}d^{(l)}_{i}D_{i}|$, which allows one to compute the cohomoologies of
(22) without ambiguity: the maps $x_{(l)}$ and $y_{(l)}$ are precisely those
appearing in (22) (after taking duals). We illustrate this in an example.

######  Example 3.8.

Suppose $G=\frac{1}{5}(1,2)$. The continued fraction expansion is
$\frac{7}{3}=[3,2]$ and the $i$-series $I(7,3)=\\{5,2,1,0\\}$. Equation (27)
reads

(33) $\psi(\mathcal{O}_{0}\otimes\rho_{i})\simeq\begin{cases}0&\mbox{ if
}i=0,1,\\\ \mathcal{O}_{F}(F)[1]&\mbox{ if }i=2\\\
\mathcal{O}_{E_{5-i}(-\alpha_{5-i}+1)}&\mbox{ if }i=3,4.\end{cases}$

We verify it using the approach of Section 3.2.1. Applying [Wun88, §A.1] we
have

$\mathcal{R}_{i}=\begin{cases}\mathcal{O}_{Y}&i=0\\\
\mathcal{O}_{Y}(D_{3-i})&i=1,2\\\
\mathcal{O}_{Y}(D_{1}+D_{5-i})&i=3,4\end{cases}$

From Section 3.2.1 we get the following equalities:

$\displaystyle(f^{*}x^{5}=0)=E_{1}+3E_{2}+5A,$ $\displaystyle D_{1}\equiv
B\equiv E_{2}+2A,$ $\displaystyle(f^{*}y^{5}=0)=5B+2E_{1}+E_{2},$
$\displaystyle D_{2}\equiv 3B+E_{1}\equiv A.$

The maps $x_{(l)}$ are uniquely determined by the above. Indeed, denote by
$a,b,e_{1},e_{2}$ the sections defining the divisors $A,B,E_{1},E_{2}$. Then
we have:

$\displaystyle x_{(0)}\colon$
$\displaystyle\mathcal{R}_{0}\xrightarrow{a}\mathcal{R}_{1}$ $\displaystyle
y_{(0)}\colon\mathcal{R}_{0}\xrightarrow{b}\mathcal{R}_{2}$ $\displaystyle
x_{(1)}\colon$
$\displaystyle\mathcal{R}_{1}\xrightarrow{e_{2}a}\mathcal{R}_{2}$
$\displaystyle y_{(2)}\colon\mathcal{R}_{2}\xrightarrow{b}\mathcal{R}_{4}$
$\displaystyle x_{(2)}\colon$
$\displaystyle\mathcal{R}_{2}\xrightarrow{a}\mathcal{R}_{3}$ $\displaystyle
y_{(4)}\colon\mathcal{R}_{4}\xrightarrow{be_{1}}\mathcal{R}_{1}$
$\displaystyle x_{(3)}\colon$
$\displaystyle\mathcal{R}_{3}\xrightarrow{e_{2}a}\mathcal{R}_{4}$
$\displaystyle y_{(1)}\colon\mathcal{R}_{1}\xrightarrow{b}\mathcal{R}_{3}$
$\displaystyle x_{(4)}\colon$
$\displaystyle\mathcal{R}_{4}\xrightarrow{e_{1}e_{2}a}\mathcal{R}_{0}$
$\displaystyle
y_{(3)}\colon\mathcal{R}_{3}\xrightarrow{be_{1}e_{2}}\mathcal{R}_{0},$

so that $f^{*}x^{5}=x_{(4)}x_{(3)}x_{(2)}x_{(1)}x_{(0)}$ (and similarly for
$y$). The dual maps $x_{(l)}^{\vee}$ (resp. $y_{(l)}^{\vee}$) to $x_{(l)}$
(resp $y_{(l)}$) correspond to multiplication by the same sections. Then,
consider for example the sequence:

$\displaystyle\psi(\mathcal{O}_{0}\otimes\rho_{2})=\left[\mathcal{R}_{0}^{\vee}\xrightarrow{(-y_{(3)}^{\vee},x_{(4)}^{\vee})^{T}}\mathcal{R}_{3}^{\vee}\oplus\mathcal{R}_{4}^{\vee}\xrightarrow{(x_{(2)}^{\vee},y_{(2)}^{\vee})}\mathcal{R}_{2}^{\vee}\right].$

The kernel of the map $(x_{(2)}^{\vee},y_{(2)}^{\vee})=(a,b)$ is

$\mathcal{O}_{Y}(-2D_{1}-D_{2})\xrightarrow{(-b,a)^{T}}\mathcal{R}_{3}^{\vee}\oplus\mathcal{R}_{4}^{\vee}.$

We have $(-y_{(3)}^{\vee},x_{(4)}^{\vee})^{T}=e_{1}e_{2}(-b,a)^{T}$, so

$\psi(\mathcal{O}_{0}\otimes\rho_{2})=\operatorname{coker}\left(\mathcal{R}_{0}\xrightarrow{e_{1}e_{2}}\mathcal{O}(-2D_{1}-D_{2})\right)[1]\simeq\mathcal{O}_{F}(F)[1].$

Similarly, consider:

$\displaystyle\psi(\mathcal{O}_{0}\otimes\rho_{3})=\left[\mathcal{R}_{1}^{\vee}\xrightarrow{(-y_{(4)}^{\vee},x_{(0)}^{\vee})^{T}}\mathcal{R}_{4}^{\vee}\oplus\mathcal{R}_{0}^{\vee}\xrightarrow{(x_{(3)}^{\vee},y_{(3)}^{\vee})}\mathcal{R}_{3}^{\vee}\right].$

with $(-y_{(4)}^{\vee},x_{(0)}^{\vee})^{T}=(-be_{1},a)^{T}$ and
$(x_{(3)}^{\vee},y_{(3)}^{\vee})=(e_{2}a,be_{1}e_{2})$. This shows that the
complex is exact in the middle. However, the cokernel of $(-be_{1},a)^{T}$ is
$(a,be_{1})$, and hence

$\psi(\mathcal{O}_{0}\otimes\rho_{3})=\operatorname{coker}\left(\mathcal{O}(-2D_{1}+D_{2})\xrightarrow{e_{2}}\mathcal{R}_{3}^{\vee}\right)\simeq\mathcal{O}_{E_{2}}(-1).$

One proves the remaining cases in the same way.

##  4\. The minimal resolution of the Johnson-Kollár surfaces

For $k>0$ integer, let $X^{(k)}$ be a sufficiently general log del Pezzo
surface of the family (1). The surface $X^{(k)}$ is the vanishing locus of a
generic section of the sheaf $\mathcal{O}_{\mathbb{P}}(8k+4)$ on
$\mathbb{P}=\mathbb{P}(2,2k+1,2k+1,4k+1)$. Let
$f^{(k)}\colon\widetilde{X}^{(k)}\to X^{(k)}$ be the minimal resolution. In
this Section we describe in details the geometry of the surface
$\widetilde{X}^{(k)}$. We will omit the superscript $-^{(k)}$ from our
notation whenever the context allows it.

###  4.1. Basic properties

Fix $k>0$ integer. We denote by $x,y_{1},y_{2},z$ the weighted homogeneous
coordinates of $\mathbb{P}$. The singular locus $\mathbb{P}_{\mathrm{sing}}$
of $\mathbb{P}$ has three components:

  * 1.

the point $(1:0:0:0)$, which is a singularity of type $1/2(1,1,1)$;

  * 2.

the line $(x=z=0)$, whose points all have a neighbourhood analytically
isomorphic to $\mathbb{A}^{1}\times 1/(2k+1)(1,k)$;

  * 3.

the point $(0:0:0:1)$, which is a singularity of type $1/(4k+1)(4,1,1)$.

It is easy to check that a section of the sheaf
$\mathcal{O}_{\mathbb{P}}(8k+4)$ is of the form:

(34)
$h^{(k)}(x,y_{1},y_{2},z)=ax^{4k+2}+h_{4}(y_{1},y_{2})+bxz^{2}+x^{2k+1}h_{2}(y_{1},y_{2})+x^{k+1}h_{1}(y_{1},y_{2})z$

where $a,b\in\mathbb{C}$ and $h_{i}(y_{1},y_{2})$ is a homogeneous polynomial
of degree $i$ in the variables $y_{1},y_{2}$.

Let $X$ be the vanishing locus of $h=h^{(k)}$111111 If $h$ is irreducible,
then $X$ is wellformed. Indeed, if $h$ is irreducible, then
$h_{4}(y_{1},y_{2})\neq 0$, that is, $X$ does not contain the line $(x=z=0)$.
By [IF00, Corollary 8.5], $X$ is quasismooth if and only if $a\neq 0$, $b\neq
0$, and for $i=1,2$ the polynomial $h_{4}(y_{1},y_{2})$ has a monomial of the
form $y_{i}^{m}y_{j}$, $m>0$, $j\in\\{1,2\\}$ (these conditions imply that $h$
is irreducible).. For a generic choice of $h$, $(1:0:0:0)\notin X$, and
$(0:0:0:1)\in X$ is a singularity of type $1/(4k+1)(1,1)$. Moreover, since
$h(0,y_{1},y_{2},0)=h_{4}(y_{1},y_{2})$, $X$ intersects the line $(x=z=0)$ in
four points, each of which is a singularity of type $1/(2k+1)(1,k)$. We write
$p=(0:0:0:1)$ and denote by $p_{1},p_{2},p_{3},p_{4}$ the four points in
$X\cap(x=z=0)$.

By Equation (34) the curve $(x=0)\cap X$ is the union of four lines
$c_{i}\in|\mathcal{O}_{\mathbb{P}}(2)|\oplus|\mathcal{O}_{\mathbb{P}}(2k+1)|$,
$i=1,2,3,4$; for each $i$, the line $c_{i}$ passes through the points $p_{i}$
and $p$ (see the bottom left of Figure 6).

######  Remark 4.1.

By the general theory of surface cyclic quotient singularities, the surface
$\widetilde{X}$ is obtained from $X$ by a sequence of $1+4k$ weighted blow-
ups: a single weighted blow-up at $p$, and a sequence of $k$ weighted blow-ups
centered at each $p_{i}$ (see Examples 2.7 and 2.8).

We denote by $F$ the $-(4k+1)$\- exceptional curve over $p$, and for
$i=1,2,3,4$ we denote by $C_{i,0}$ the $(-3)$-exceptional curve, and by
$C_{i,j}$, $j=1,\dots,k-1$, the $(-2)$-exceptional curves over $p_{i}$.
Moreover, for $i=1,2,3,4$ we denote by $C_{i,k}$ the strict transform in
$\widetilde{X}$ of the line $c_{i}\subset X$ (see the top of Figure 6).

######  Remark 4.2.

The minimal resolution $f\colon\widetilde{X}\to X$ is noncrepant. More
precisely, the singularities of $X$ are strictly log-terminal:

(35) $K_{\widetilde{X}}=f^{\ast}(K_{X})+d\cdot
F+\sum_{i=1}^{4}\sum_{j=1}^{k}d_{j}\cdot C_{ij}$

where

(36) $d=-\frac{4k-1}{4k+1}\quad\text{and}\quad d_{j}=-\frac{k-(j-1)}{2k+1}$

######  Remark 4.3 (Rationality and existence of exceptional collections).

Note that, for any log del Pezzo surface $X$, its minimal resolution
$\widetilde{X}$ is rational. Indeed, by Castelnuovo’s criterion [KSC04,
Theorem 3.2], it suffices to show that

$H^{1}\left(\widetilde{X},\mathcal{O}_{\widetilde{X}}\right)=0,\quad\mbox{and}\quad
H^{0}\left(\widetilde{X},\mathcal{O}_{\widetilde{X}}(2K_{\widetilde{X}})\right)=0.$

By the Leray spectral sequence (and since $X$ has rational singularities), we
have
$H^{1}(\widetilde{X},\mathcal{O}_{\widetilde{X}})=H^{1}\left(X,\mathcal{O}_{X}\right)$,
which vanishes by the Kawamata-Viehweg vanishing theorem [Laz04, Theorem
9.1.18] combined with the del Pezzo assumption. For the second vanishing, we
may argue as follows. Since the singularities are log-terminal, we may write

$f^{*}K_{X}=K_{\widetilde{X}}+\Delta$

where $\Delta$ is effective. Since $-f^{*}K_{X}$ is big, so is
$-K_{\widetilde{X}}=-f^{*}K_{X}+\Delta$. This forces the vanishing of all
groups $H^{0}(\widetilde{X},\mathcal{O}_{\widetilde{X}}(mK_{\widetilde{X}}))$,
$m>0$. Thus $\widetilde{X}$ is a smooth rational surface.

It follows that both $\widetilde{X}$ and the canonical stack $\mathcal{X}$
admit full exceptional collections. Indeed, $\widetilde{X}$ is obtained
through a sequence of blow-ups from a minimal rational surface (i.e.
$\mathbb{P}^{2}$, $\mathbb{P}^{1}\times\mathbb{P}^{1}$, or $\mathbb{F}_{n}$
with $n\geq 2$). Since minimal rational surfaces admit full exceptional
collections, so does $\widetilde{X}$ by Orlov’s formula [Orl92]. Finally, in
virtue of the decomposition (2), $D(\mathcal{X})$ admits a full exceptional
collection if $D(\widetilde{X})$ does.

###  4.2. Explicit geometry of the minimal resolution

Fix a generic polyonomial $h=h^{(k)}$, equivalently,
$a,b\in\mathbb{C}\setminus\\{0\\}$, and polynomials $h_{4},h_{2},h_{1}$ as in
Equation (34). Let $X$ be the corresponding log del Pezzo surface and
$f\colon\widetilde{X}\to X$ its minimal resolution. Let
$X^{\prime}\subset\mathbb{P}(2,1,1,1)$ be the smooth del Pezzo surface of
degree two defined by the polynomial $h^{(0)}$, i.e.

(37)
$h^{(0)}(x,y_{1},y_{2},z)=ax^{2}+h_{4}(y_{1},y_{2})+x\left(bz^{2}+h_{2}(y_{1},y_{2})+h_{1}(y_{1},y_{2})z\right)$

where $x,y_{1},y_{2},z$ are now weighted homogeneous coordinates on
$\mathbb{P}(2,1,1,1)$. The hyperplane $(x=0)\subset\mathbb{P}(2,1,1,1)$
intersects $X^{\prime}$ in four lines $c_{i,0}$, $i=1,2,3,4$, which meet at
the point $q\coloneqq(0:0:0:1)$. The next Theorem shows that $\widetilde{X}$
is obtained from $X^{\prime}$ by a sequence of $4k+1$ smooth blow-ups.

######  Theorem 4.4.

There is a birational morphism $\tau\colon\widetilde{X}\to X^{\prime}$, which
contracts the $4k+1$ curves $C_{i,k},\dots,C_{i,1},$ $i=1,2,3,4$, and $F$ to
the point $q$, as pictured in Figure 4.

$\widetilde{X}$$X^{\prime}$$q$$c_{3,0}$$c_{4,0}$F$c_{1,0}$$c_{2,0}$$C_{2,k}$$C_{3,2}$$C_{4,2}$$C_{1,1}$$C_{3,0}$$C_{2,1}$$C_{1,2}$$C_{3,k}$$C_{4,k}$$C_{3,1}$$C_{4,1}$$C_{1,0}$$C_{4,0}$$C_{2,0}$$C_{1,k}$$C_{1,k-1}$$C_{2,k-1}$$C_{3,k-1}$$C_{4,k-1}$$C_{2,2}$
Figure 4. A picture of the morphism $\tau\colon\widetilde{X}\to X^{\prime}$.
The surface $\widetilde{X}$ is obtained from the surface $X^{\prime}$ by a
sequence of $4k+1$ smooth blow-ups.

###### Proof.

Let $\Delta\subset N_{\mathbb{R}}\simeq\mathbb{R}^{3}$ be the fan of
$\mathbb{P}$ and let $\rho_{1},\rho_{2},\rho_{3},\rho_{4}$ be the primitive
generators of the rays of $\Delta$. One may choose for instance:

$\rho_{1}=(k,k,4k+1)^{T}\quad\rho_{2}=(1,0,0)^{T}\quad\rho_{3}=(0,1,0)^{T}\quad\rho_{4}=(-1,-1,-2)^{T}$

Consider the star subdivision $\Sigma=\Delta^{\ast}(v)$, where

(38) $v=\frac{1}{4k+1}\left(4\rho_{1}+\rho_{2}+\rho_{3}\right)$

The toric variety $G=G_{\Sigma}$ is the weighted blow-up of $\mathbb{P}$ at
$p$ with weights $1/(4k+1)(4,1,1)$. Note that the ray $\rho_{1}$ belongs to
the cone generated by $\rho_{4}$ and $v$ (see Figure 5); more precisely,

$\rho_{1}=\frac{1}{2}(\rho_{4}+(2k+1)v)$

Let $Y\subset G$ be the strict transform of $X$. The exceptional divisor of
the morphism $G\to\mathbb{P}$ is the torus-invariant divisor $D_{v}\subset G$.
By construction, the exceptional curve $D_{v}\cap Y\subset Y$ over the point
$p\in X$ is a $-(4k+1)$-curve. The line $D_{\rho_{1}}\cap D_{\rho_{4}}\subset
G$ is a line of $1/(2k+1)(1,k)$ points, and the surface $Y$ intersects this
line in four points (see the left of Figure 6).

Now consider the sequence of $k$ star-subdivisions of the fan $\Sigma$ defined
by setting $\Sigma_{0}=\Sigma^{\ast}(v_{0})$, where

(39) $v_{0}=\frac{1}{2k+1}(\rho_{1}+k\rho_{4})$

and, for $i=1,\dots,k-1$, $\Sigma_{i}=\Sigma_{i-1}^{\ast}(v_{i})$, where

(40) $v_{i}=\frac{1}{k-(i-1)}(\rho_{1}+(k-i)v_{i-1})$

and let $G_{i}=G_{\Sigma_{i}}$ (see Figure 5). The variety
$G_{0}=G_{\Sigma_{0}}$ is the weighted blow-up of $G$ along the line
$D_{\rho_{1}}\cap D_{\rho_{4}}$ with weights $1/(2k+1)(1,k)$. For
$i=1,\dots,k-1$, the line $D_{\rho_{1}}\cap D_{v_{i-1}}\subset G_{i-1}$ is a
line of $\frac{1}{k-(i-1)}(1,k-i)$ points, and the variety $G_{i}$ is the
weighted blow-up of $G_{i-1}$ along this line.

$0$$\rho_{1}$$\rho_{2}$$\rho_{3}$$\rho_{4}$$v_{0}$$v_{1}$$v$ Figure 5. A
picture of the fan $\widetilde{\Sigma}$ for $k=2$. The segments colored in
blue all belong to the same plane. The three star-subdivisions of the fan
$\Delta$ are represented by the segments in red. (For sake of clarity, the
picture is out of scale.)

Write $\widetilde{G}=G_{k-1}$ and $\widetilde{\Sigma}=\Sigma_{k-1}$, and let
$\widetilde{X}$ be the strict transform of $X$ in $\widetilde{G}$. By
construction, the minimal resolution $f\colon\widetilde{X}\to X$ is the
restriction of the birational morphism $\widetilde{G}\to\mathbb{P}$. The curve
$D_{v}\cap\widetilde{X}\subset\widetilde{X}$ is the $-(4k+1)$-exceptional
curve $F$ over the point $p\in X$, and the intersection
$\widetilde{X}\cap(D_{v_{0}}\cup D_{v_{1}}\cup\dots\cup D_{v_{k-1}})$ is given
by the union of the four chains

$C_{i}=\bigcup_{j=0}^{k-1}C_{i,j}\quad i=1,2,3,4$

where $C_{i,j}$ is the $j$th exceptional curve over the point $p_{i}\in X$
(see Remark 4.1 and the top of Figure 6). The intersection $\widetilde{X}\cap
D_{\rho_{1}}\subset\widetilde{X}$ is the union of the proper transforms
$C_{i,k}$ of the lines $c_{i}\subset X$, $i=1,2,3,4$. Each $C_{i,k}$ has self-
intersection $-1$; indeed $K_{X}.c_{i}=-\frac{1}{(2k+1)(4k+1)}$ since
$c_{i}\in|\mathcal{O}_{\mathbb{P}}(2)|\oplus|\mathcal{O}_{\mathbb{P}}(2k+1)|$,
thus by Remark 4.2

$K_{\widetilde{X}}.C_{i,k}=K_{X}.c_{i}-\frac{4k-1}{4k+1}-\frac{1}{2k+1}=-1$

and by adjunction $C_{i,k}^{2}=-1$.

Now, observe that the vectors $v_{1},\dots,v_{k-1},\rho_{1}$ all belong to the
cone $\langle v_{0},v\rangle$ (see Figure 5). In particular, it follows from
(38) and (39) that $\rho_{1}=v_{0}+kv$; combining this and (40) one finds
that:

(41) $v_{i}=v_{i-1}+v\quad
i=1,\dots,k-1\qquad\text{and}\quad\rho_{1}=v_{k-1}+v$

This implies that the fan $\widetilde{\Sigma}$ is obtained from the fan
$\Sigma^{\prime}_{0}$ with rays $\rho_{2},\rho_{3},\rho_{4},v_{0},v$ and
maximal cones

$\langle\rho_{2},\rho_{3},\rho_{4}\rangle\quad\langle\rho_{2},v_{0},v\rangle\quad\langle\rho_{2},\rho_{4},v_{0}\rangle\quad\langle\rho_{3},v_{0},v\rangle\quad\langle\rho_{3},\rho_{4},v_{0}\rangle\quad\langle\rho_{2},\rho_{3},v\rangle$

by the sequence of $k$ star subdivisions:

$\begin{split}&\Sigma^{\prime}_{i}=(\Sigma^{\prime}_{i-1})^{\ast}(v_{i-1}+v)\quad
i=1,\dots,k\\\ \end{split}$

where $\widetilde{\Sigma}={\Sigma}^{\prime}_{k}$. For $i=1,\dots,k$ the toric
variety $G^{\prime}_{i}=G_{\Sigma^{\prime}_{i}}$ is the blow up of the toric
variety $G^{\prime}_{i-1}$ along $D_{v_{i-1}}\cap D_{v}$.

Let $Y^{\prime}$ be the proper transform of $\widetilde{X}$ in
$G^{\prime}_{0}$. The morphism $\widetilde{G}\to G^{\prime}_{0}$ contracts the
divisors $D_{\rho_{1}},D_{v_{k-1}},\dots,D_{v_{1}}$ in $\widetilde{G}$, thus
contracts the curves $C_{i,k},C_{i,k-1},\dots,C_{i,1}$, $i=1,2,3,4$, in
$\widetilde{X}$. For $i=1,2,3,4$ the proper transform of $C_{i,0}$ in
$Y^{\prime}$ is a $(-2)$-curve, and the proper transform of $F$ in
$Y^{\prime}$ is a $(-1)$-curve (see the right of Figure 6).

Now, note that the vector $v$ can be written in terms of
$\rho_{2},\rho_{3},v_{0}$ as:

(42) $v=\rho_{2}+\rho_{3}+4v_{0}$

and that $\rho_{2},\rho_{3},\rho_{4},v_{0}$ span the lattice $N$ and satisfy:

(43) $2v_{0}+\rho_{2}+\rho_{3}+\rho_{4}=0$

This implies that $G^{\prime}_{0}$ is the weighted blow-up of
$\mathbb{P}(2,1,1,1)$ at the smooth point $q=(0:0:0:1)$ with weights
$(4,1,1)$.

Let $X^{\prime}$ be the proper transform of $Y^{\prime}$ in
$\mathbb{P}(2,1,1,1)$. The morphism $G_{0}^{\prime}\to\mathbb{P}(2,1,1,1)$
contracts the divisor $D_{v}\subset G_{0}^{\prime}$ to the point $q$, thus
contracts the proper transform of $F$ in $Y^{\prime}$ to $q$. The proper
transforms $c_{i,0}\subset X^{\prime}$ of the curves
$C_{i,0}\subset\widetilde{X}$, $i=1,2,3,4$, are $(-1)$-curves meeting at the
point $q$ (see the bottom right of Figure 6).

$p_{3}$$p_{4}$$p_{1}$$p_{2}$$p$$\widetilde{X}$$F$$C_{3,k-1}$$C_{4,k-1}$$C_{1,k-1}$$C_{2,k-1}$$C_{2,k}$$C_{3,k}$$C_{1,k}$$C_{4,k}$$C_{1,2}$$C_{3,2}$$C_{4,2}$$C_{1,1}$$C_{2,1}$$C_{3,1}$$C_{4,1}$$C_{3,0}$$C_{1,0}$$C_{4,0}$$C_{2,0}$$C_{2,2}$$c_{3,0}$$c_{4,0}$$c_{1,0}$$c_{2,0}$$q$$c_{3}$$c_{4}$$c_{1}$$c_{2}$$Y$$Y^{\prime}$$X$$X^{\prime}$
Figure 6. A picture of the surfaces $X,Y,\widetilde{X}$, $Y^{\prime}$, and
$X^{\prime}$. The surface $Y$ is the weighted blow-up of $X$ at $p$ with
weights $1/(4k+1)(1,1)$; the surface $Y^{\prime}$ is the blow-up of
$X^{\prime}$ at the smooth point $q$. The curves $F$,
$C_{i,0},\dots,C_{i,k-1}$, $i=1,2,3,4$, are the exceptional curves of the
minimal resolution $\widetilde{X}\to X$, while the curves $F$,
$C_{i,1},\dots,C_{i,k}$, $i=1,2,3,4$, are the exceptional curves of the
morphism $\widetilde{X}\to X^{\prime}$. We use the same color for a curve and
its proper transforms under the maps considered.

It follows from (34) and (43) that the surface $X^{\prime}$ is the smooth del
Pezzo surface of degree two cut out of $\mathbb{P}(2,1,1,1)$ by the section of
$\mathcal{O}(4)$ given by the polynomial $h^{(0)}$ in (37). Indeed, by (43),
$\Delta$ and the fan of $\mathbb{P}(2,1,1,1)$ share the cone
$\sigma=\langle\rho_{2},\rho_{3},\rho_{4}\rangle$. Writing
$U_{\sigma}\simeq\mathbb{C}^{3}/\mathbb{Z}_{2}(1,1,1)$ for the affine toric
variety corresponding to $\sigma$, one has that $X\cap U_{\sigma}\simeq
X^{\prime}\cap U_{\sigma}$, and, by (34), $X\cap U_{\sigma}$ is defined by the
equation:

$(a+h_{4}(y_{1},y_{2})+bz^{2}+h_{2}(y_{1},y_{2})+h_{1}(y_{1},y_{2})z=0)\subset
U_{\sigma}$

where now $y_{1},y_{2},z$ denote affine coordinates on $\mathbb{C}^{3}$. It
follows that $X^{\prime}$ is cut out of $\mathbb{P}(2,1,1,1)$ by $h^{(0)}$,
thus is a smooth del Pezzo surface of degree two.

The union of the four curves $c_{i,0}$ is given by
$\left(x=h_{4}(y_{1},y_{2})=0\right)\subset\mathbb{P}(2,1,1,1)$. The morphism
$\tau\colon\widetilde{X}\to X^{\prime}$ in the statement of the Theorem is the
restriction of the birational morphism $\widetilde{G}\to\mathbb{P}(2,1,1,1)$
constructed above. ∎

An immediate consequence of Theorem 4.4 is:

######  Corollary 4.5.

For $k>0$ integer, let $X=X^{(k)}$ be a generic log del Pezzo surface (1) and
let $\widetilde{X}^{(k)}$ be its minimal resolution. Then
$\widetilde{X}^{(k)}$ is the blow-up of $\mathbb{P}^{2}$ in $8+4k$ points.
Moreover, for all $k>0$, $X^{(k)}$ is a rational surface of Picard rank
$\rho(X^{(k)})=8$.

Theorem 4.4 highlights a relation between the surfaces of the series (1) as
$k$ varies. Fix a generic choice of $a,b\in\mathbb{C}\setminus\\{0\\}$, and of
polynomials $h_{4},h_{2},h_{1}$. This defines a sequence of polynomials
$\\{h^{(k)}\\}_{k>0}$, and with it sequences $\\{X^{(k)}\\}$ and
$\\{\widetilde{X}^{(k)}\\}$ of log del Pezzo surfaces and their minimal
resolutions. By Theorem 4.4, every surface $\widetilde{X}^{(k)}$ admits a
birational morphism $\tau^{(k)}\colon\widetilde{X}^{(k)}\to X^{\prime}$.

######  Corollary 4.6.

For $0<k_{1}<k_{2}$, the map $\tau^{(k_{2})}$ factors through the map
$\tau^{(k_{1})}$, as illustrated in Figure 7.

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{X}^{(k)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{(k)}}$$\scriptstyle{f^{(k)}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{X}^{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{(2)}}$$\scriptstyle{f^{(2)}}$$\textstyle{\widetilde{X}^{(1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{(1)}}$$\textstyle{X^{\prime}=X^{(0)}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{X}^{(k)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{X}^{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{X}^{(1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Figure 7. The morphims $f^{(k)}\colon\widetilde{X}^{(k)}\to X^{(k)}$ and
$\tau^{(k)}\colon\widetilde{X}^{(k)}\to X^{\prime}$. For all $k>0$, the
surface $\widetilde{X}^{(k+1)}$ is the blow-up at four points of the surface
$\widetilde{X}^{(k)}$. We use dashed arrows to denote rational maps and solid
arrows to denote morphisms.

######  Remark 4.7.

Any del Pezzo surface $X^{\prime}$ as in Theorem 4.4 has a generalized Eckardt
point, that is, a point where four $(-1)$-curves of $X^{\prime}$ meet.

By [Kuw05, Theorem 5.1] a del Pezzo surface $S$ of degree two admits a
generalized Eckardt point if and only if, after a suitable change of
coordinates, it can be defined by an equation of the form

(44)
$\left(x^{2}+z^{4}+g_{2}(y_{1},y_{2})z^{2}+g_{4}(y_{1},y_{2})=0\right)\subset\mathbb{P}(2,1,1,1)$

where $x,y_{1},y_{2},z$ are weighted homogeneous coordinates on
$\mathbb{P}(2,1,1,1)$ and $g_{i}(y_{1},y_{2})$ is a polynomial of degree $i$
in the two variables $y_{1},y_{2}$.

It is easy to see that, after a change of coordinates, any equation of the
form (44) can be rewritten as $(h^{(0)}=0)$, where $h^{(0)}$ is a polynomial
as in (37). Indeed, for instance, by sending $x\mapsto
x+\mathtt{i}z^{2}+\frac{\mathtt{i}}{2}g_{2}(y_{1},y_{2})$, the polynomial in
(44) takes the form:

$x^{2}-\frac{g_{2}(y_{1},y_{2})^{2}}{4}+g_{4}(y_{1},y_{2})+x\left(2\mathtt{i}z^{2}+\mathtt{i}g_{2}(y_{1},y_{2})\right)$

Thus the surfaces $X^{\prime}$ appearing in Theorem 4.4 are all del Pezzo
surfaces of degree two with a generalized Eckardt point.

######  Remark 4.8.

Under the assignment $X\mapsto X^{\prime}$ constructed in Theorem 4.4, to each
surface $X$ corresponds bijectively a del Pezzo surface $X^{\prime}$ with a
generalized Eckardt point. Surjectivity is Remark 4.7. For injectivity we may
argue as follows. Suppose two log del Pezzo hypersurfaces, $X_{1}$ and
$X_{2}$, get mapped to isomorphic del Pezzo surfaces of degree two
$X_{1}^{\prime}$ and $X_{2}^{\prime}$ embedded in $\mathbb{P}(2,1,1,1)$. For
$i=1,2$, write defining polynomials $h_{i}^{(0)}$ for $X_{i}^{\prime}$. Since
the surfaces are isomorphic, the $h_{i}^{(0)}$ are related by a change of
coordinates $\psi$ of $\mathbb{P}(2,1,1,1)$. Then, $\psi$ must be a
composition of transformations

(45) $\begin{split}x&\mapsto rx,\\\ y_{j}&\mapsto
p_{1,j}y_{1}+q_{2,j}y_{2},\qquad j=1,2\\\ z&\mapsto
sz+t_{1}y_{1}+t_{2}y_{2}\end{split}$

(these are the projective transformation preserving the form (37)). It is a
straightforward check that the same sequence of transformations, where $z$ is
instead mapped to $sz+t_{1}x^{k}y_{1}+t_{2}x^{k}y_{2}$, relates the
polynomials $h_{1}^{(k)}$ and $h_{2}^{(k)}$ defining $X_{1}$ and $X_{2}$.

###  4.3. Moduli spaces of Johnson–Kollár surfaces

In this Section we use the construction of Theorem 4.4 to obtain a statement
about moduli spaces. As above, we fix $k>0$ integer. We start with the
following remark.

######  Remark 4.9.

The construction of Theorem 4.4 can be carried out in families. More
precisely, let $\mathscr{X}\to B$ be a flat family of Johnson-Kollàr
hypersurfaces with base $B$. In other words,
$\mathscr{X}\subset\mathbb{P}_{B}\coloneqq B\times\mathbb{P}$, and, if
$\mathcal{O}(1)$ be the tautological sheaf on $\mathbb{P}_{B}$, the family
$\mathscr{X}$ is the zero locus of a general section of $\mathcal{O}(8k+4)$.
Equivalently, it is determined by a polynomial $h^{(k)}$ of the form (34),
where $a,b\in\mathcal{O}_{B}(B)$ and $h_{i}\in
H^{0}(B\times\mathbb{P},\mathcal{O}(i(2k+1)))$ for $i=1,2,4$.

In this setting, the same argument as in the proof of Theorem 4.4 gives a
birational correspondence

(46) ${\quad
B\times\widetilde{G}}$${\mathbb{P}_{B}}$${B\times\mathbb{P}(2,1,1,1)}$

The correspondence, restricted to $\mathscr{X}$, maps it to a family
$\mathscr{X}^{\prime}\to B$ of smooth del Pezzo surfaces with a generalized
Eckardt point. On each fiber $\mathscr{X}_{b}$ it coincides with $\tau\circ
f^{-1}\colon\mathscr{X}_{b}\dashrightarrow\mathscr{X}^{\prime}_{b}$. In other
words, $\mathscr{X}^{\prime}$ is the zero locus of the polynomial $h^{(0)}$,
where now $h_{i}\in H^{0}(B\times\mathbb{P}(2,1,1,1),\mathcal{O}(i))$ for
$i=1,2,4$.

Now let $\mathscr{M}=\mathscr{M}_{k}$ be the moduli stack of quasismooth and
wellformed hypersurfaces of degree $8k+4$ in the weighted projective $3$-space
$\mathbb{P}=\mathbb{P}(2,2k+1,2k+1,4k+1)$, i.e. the open substack of the
quotient stack $[|\mathcal{O}_{\mathbb{P}}(8k+4)|/\mathrm{Aut}(\mathbb{P})]$
parametrizing quasismooth and wellformed hypersurfaces. The moduli stack
$\mathscr{M}$ admits a coarse moduli space $M=M_{k}$ which is an algebraic
space as a consequence of [KM97] and [Bun21, Theorem 3.13]121212 [Bun21] only
refers to quasismooth hypersurfaces. However, we observed in Footnote 11 that
any quasismooth surface of the form (1) is wellformed.. Similarly, one defines
the moduli stack $\mathscr{P}$ of smooth del Pezzo surfaces of degree two,
whose coarse moduli space $P$ is a quasiprojective variety by [Bun21, Example
5.22]. We denote by $\mathscr{P}_{E}$ the closed substack of $\mathscr{P}$
parametrizing surfaces with a generalized Eckardt point, and likewise define
$P_{E}\subset P$. The moduli space $P_{E}$ is a stratum of dimension $4$ of
the moduli space $P$. Surfaces in this stratum are called _of type XII_ in
[Dol12, Table 8.9].

######  Proposition 4.10.

The moduli space $M$ is isomorphic to $P_{E}$.

###### Proof.

We begin by constructing a morphism $M\to P_{E}$, using the universal property
of moduli spaces and Yoneda’s Lemma.

Let $[\mathscr{X}\to B]\in\mathscr{M}(B)$ be a flat family of log del Pezzo
surfaces with base $B$. The construction of Remark 4.9 gives rise to a family
$[\mathscr{X}^{\prime}\to B]\in\mathscr{P}_{E}(B)$. In turn, this induces a
natural transformation from $\mathscr{M}$ to the functor of points
$h_{P_{E}}$, and hence a natural transformation $h_{M}\to h_{P_{E}}$ by the
universal property of $M$. By Yoneda’s Lemma, this is equivalent to a morphism
$\phi\colon M\to P_{E}$.

Conversely, consider a family $[\mathscr{X}^{\prime}\to
B]\in\mathscr{P}_{E}(B)$, written as the zero locus of a polynomial $h^{(0)}$
as in Remark 4.7. Then, the inverse of the birational correspondence (46) maps
the family $\mathscr{X}^{\prime}$ to one in $\mathscr{M}(B)$, defined by the
polynomial $h^{(k)}$. Arguing as in Remark 4.8, we see that the induced map
$P_{E}\to M$ inverts $\phi$. ∎

###  4.4. Classification of generalized Eckardt points

Suppose that $S$ is a del Pezzo surface of degree two with a generalized
Eckardt point $q$. Denote by $m_{l}$, $l=1,2,3,4$, the four $(-1)$-curves
meeting at the point $q$. In this Section we classify all the possible
configurations $n_{l}=\pi(m_{l})$, where $\pi\colon S\to\mathbb{P}^{2}$ is the
blow-down of an exceptional set. Keeping this notation:

######  Lemma 4.11.

Let $\pi\colon S\to\mathbb{P}^{2}$ be a blowup of $\mathbb{P}^{2}$ at seven
points $x_{i}$ in general position (equivalently, a choice of an exceptional
set for $S$). Up to reindexing, we have the following possibilities for the
curves $n_{l}$:

  * •

$n_{1}$ and $n_{2}$ are lines, and $n_{3}$ and $n_{4}$ are conics;

  * •

$n_{1}$ is a point, and the remaining are conics, or $n_{1}$ is a cubic and
the remaining are lines;

  * •

$n_{1}$ is a point, and the remaining are a line, a conic, and a cubic.

###### Proof.

Observe first of all that $n_{k}$ is a point for at most one $k$, since the
$m_{k}$ are not disjoint.

If $n_{1}=x_{1}$ is a point, then at most one of the other $n_{k}$ is a line:
two lines in $\mathbb{P}^{2}$ containing $x_{1}$ are disjoint in the blowup,
unless they coincide.

Suppose $n_{1}=x_{1}$ and $n_{2}$ is a line (containing $x_{2}$). At most one
between $n_{3},n_{4}$ is a conic: otherwise, $n_{3}$ and $n_{4}$ pass through
$x_{1}$ with the same tangent direction of $n_{2}$, and therefore they
intersect in $x_{1}$ (with multiplicity 2). Each conic must contain four
points among ($x_{3},...,x_{7}$), but if they share three they coincide, a
contradiction.

If $n_{1},n_{2},n_{3}$ are lines, $n_{4}$ cannot be a line: each line contains
$\pi(q)$ and two of the $x_{i}$, and no point $x_{i}$ can be common to
distinct lines. Similarly, $n_{4}$ is not a conic: if it was, it would have to
share $\pi(q)$ and two of the $x_{i}$ with one among $n_{1},n_{2},n_{3}$, say
$n_{1}$. But then $n_{1}\subset n_{4}$, a contradiction.

At this point, we may argue dually on the curves
$m_{k}^{\prime}=-K_{S}-m_{k}$. In fact, they also intersect at a generalized
Eckardt point $q^{\prime}$: the bitangents corresponding to the $m_{k}$ (and
hence to the $m_{k}^{\prime}$) all meet at the point
$\sigma(q)\in\mathbb{P}^{2}\setminus Q$ (see Section 2.2.1 for notation), and
$q^{\prime}$ is the other point in the preimage of $\sigma(q)$. Since the
image of $m_{k}$ determines that of $m_{k}^{\prime}$ and conversely, we can
exclude all the configurations dual to those excluded above. For example, no
more than one $n_{k}$ is a cubic, otherwise more than one $n_{k}^{\prime}$
would be a point. The only possibilities left are those in the statement:
Example 4.12 shows that these do occur. ∎

######  Example 4.12 (Construction of generalized Eckardt points).

We present three constructions of seven points
$x_{1},...,x_{7}\in\mathbb{P}^{2}$ in general position. The blowup
$\operatorname{Bl}_{\\{x_{i}\\}}\mathbb{P}^{2}$ is a del Pezzo surface of
degree two with a generalized Eckardt point (which is the preimage of $q$ in
the first example, and lies in the preimage of $x_{1}$ in the other cases). In
particular, these constructions are examples of the configurations listed in
Lemma 4.11. Observe moreover that these examples are not in general
degenerations of one another: the families of examples all have dimension 4,
as shown by a quick parameter count.

  1. (a)

Let $t_{1},t_{2}$ be two lines in $\mathbb{P}^{2}$ intersecting at a point
$q$. Pick general points $x_{1},x_{2}\in t_{1}$, $x_{3},x_{4}\in t_{2}$. Let
$s_{1}$ be a generic conic through $q,x_{1},x_{3}$ and $s_{2}$ a generic conic
through $q,x_{2},x_{4}$. Then $s_{1}\cap s_{2}=\\{q,x_{5},x_{6},x_{7}\\}$
(Figure 10).

  2. (b)

Let $x_{1}\in\mathbb{P}^{2}$, and fix a conic $s_{1}$ through $x_{1}$. Define
two general conics $s_{2}$ and $s_{3}$ which pass through $x_{1}$ with the
same tangent direction of $s_{1}$. We have $s_{1}\cap
s_{2}=\\{x_{1},x_{2},x_{3}\\}$, $s_{1}\cap s_{3}=\\{x_{1},x_{4},x_{5}\\}$, and
$s_{2}\cap s_{3}=\\{x_{1},x_{6},x_{7}\\}$. For generic choices, the points
$x_{i}$, $i=1,...,7$ are in general position, and the blowup
$\operatorname{Bl}_{\\{x_{i}\\}}\mathbb{P}^{2}$ is a del Pezzo surface of
degree two with a generalized Eckardt point at the intersection of the strict
transforms of the $s_{k}$ with the exceptional divisor of $x_{1}$ (Figure 10).

  3. (c)

Fix a point $x_{1}\in\mathbb{P}^{2}$, a line $t$ through it and a conic $s$
tangent to $t$ at $x_{1}$. Now pick generic points $x_{2}\in t$, and
$x_{3},...,x_{6}\in s$. Then, pick a nodal cubic in the 2-dimensional linear
system of cubics containing $x_{2},...,x_{6}$ and passing through $x_{1}$ with
tangent direction $t$ (Figure 10).

$q$ Figure 8.

Figure 9.

Figure 10.

In fact, the configurations of Example 4.12 are the only possibility for
generalized Eckardt points. Keeping the same notation as Lemma 4.11, we have:

######  Proposition 4.13.

The map $\pi:S\to\mathbb{P}^{2}$ is one of the blowups described in Example
4.12.

###### Proof.

By Lemma 4.11, there are only three possibilities for the $n_{l}$ (after
possibly considering the dual curves $n_{l}^{\prime}$). We only show one case,
the others being similar.

Suppose $n_{1},n_{2}$ are lines and $n_{3},n_{4}$ are conics. Since $n_{1}$
and $n_{2}$ intersect at $\pi(q)$, they cannot share the $x_{i}$’s they
contain, so we may assume $x_{1},x_{2}\in n_{1}$ and $x_{3},x_{4}\in n_{2}$.
The conic $n_{3}$ must contain 5 of the $x_{i}$ and pass through $q$, moreover
it intersects $n_{1}$ and $n_{2}$ at two points ($n_{3}$ is smooth, otherwise
the $x_{i}$ are not in general position). The only possibility is that $n_{3}$
contains, say, $x_{1}$ and $x_{3}$, whereas $x_{2},x_{4}\in n_{4}$. Then,
$n_{3}$ and $n_{4}$ must meet at $q$ and at $x_{5},x_{6},x_{7}$. This is the
situation of Example 4.12 (a). ∎

Later it will be useful to know that it is always possible to choose $\pi$ as
in Example 4.12 (a). This is Corollary 4.16 below, which follows from more
general statements about cliques on $S$.

In graph theory, a clique is a complete subgraph of an undirected graph. Given
a degree two del Pezzo surface $S$, by a clique on $S$ we mean a set of four
$(-1)$-curves on $S$ whose dual graph is a clique in the dual graph of all
$(-1)$-curves. The $(-1)$-curves meeting at a generalized Eckardt point form a
clique, but the converse does not hold.

######  Remark 4.14.

Lemma 4.11 holds if $\\{m_{1},...,m_{4}\\}$ is only a clique on $S$, with a
nearly identical proof.

We split the following Lemma in two for ease of reference, but part (ii)
clearly implies part (i).

######  Lemma 4.15.

Let $S$ be a del Pezzo surface of degree two and let $\\{m_{1},\dots,m_{4}\\}$
be a clique on $S$. Then:

  1. (i)

There exists an exceptional set $\mathcal{L}$ on $S$ such that
$\mathcal{L}\cap\\{m_{1},\dots,m_{4}\\}=\varnothing$.

  2. (ii)

Let $\\{m_{1}^{\prime},\dots,m_{4}^{\prime}\\}$ be the dual clique. There
exists an exceptional set $\mathcal{L}$ on $S$ such that
$\mathcal{L}\cap\\{m_{1},\dots,m_{4},m_{1}^{\prime},\dots,m_{4}^{\prime}\\}=\varnothing$.

###### Proof.

  1. (i)

The set of $(-1)$-curves on $S$ is in bijection with the set of exceptional
vectors in $I^{1,7}$ (see Section 2.2.1). Identifying the two sets, one checks
with a SageMath computation that on $S$ there are $576$ exceptional sets and
$630$ cliques, and that every clique admits a disjoint exceptional set.
disjoint from itself and its dual.

  2. (ii)

An analogous computation shows that some of the exceptional sets found in (i)
are also disjoint from the dual clique. ∎

######  Corollary 4.16.

Suppose that $S$ is a del Pezzo surface of degree two with a generalized
Eckardt point $q$. Then $S$ can always be realized as the blow-up of
$\mathbb{P}^{2}$ described in Example 4.12 (a).

###### Proof.

An immediate consequence of Proposition 4.13 and Lemma 4.15 (ii). ∎

##  5\. The derived category of $\mathcal{X}$

We construct full exceptional collections for $D(\widetilde{X})$ in Section
5.1; this directly implies the existence of full exceptional collections for
$D(\mathcal{X})$. In Section 5.2, after some additional choices, we compute
the morphism spaces between objects in the above collections. The only non-
vanishing spaces are $\operatorname{Hom}$ and $\operatorname{Ext}^{1}$. In
Section 5.3, we observe that this is sufficient to produce tilting complexes
for $D(\widetilde{X})$ and $D(\mathcal{X})$.

###  5.1. Full exceptional collections for $\widetilde{X}$

Several choices are involved in the construction of full exceptional
collections for $\widetilde{X}$. Our guiding principle is to get exceptional
collections whose Hom-spaces are as simple as possible, and we will make a
choice explicit when it is made.

We start by letting $\pi\colon X^{\prime}\to\mathbb{P}^{2}$ be a blow-down map
which does not contract any of the lines $c_{i,0}$: such a $\pi$ always exists
by Lemma 4.15 (i). Let $\tau\colon\widetilde{X}\to X^{\prime}$ be the morphism
in Theorem 4.4 and write $\alpha=\tau\circ\pi$. For $i=1,2,3,4$ and for $1\leq
l\leq k$, write $b_{i,l}=\mathcal{O}_{\sum_{j=l}^{k}C_{ij}}$ and
$\mathcal{B}_{i}=\left(b_{i,1},\dots,b_{i,k}\right)$. Let $D\coloneqq
F+\sum_{i=1}^{4}\sum_{j=1}^{k}jC_{ij}$.

######  Theorem 5.1.

Let $(A_{0},A_{1},A_{2})$ be a full exceptional collection on $\mathbb{P}^{2}$
and let $L_{1},\dots,L_{7}$ be the strict transforms in $\widetilde{X}$ of the
exceptional curves of $\pi\colon X^{\prime}\to\mathbb{P}^{2}$. Then

(47)
$\sigma_{\widetilde{X}}=\left(\alpha^{\ast}(A_{1}),\alpha^{\ast}(A_{1}),\alpha^{\ast}(A_{2}),\mathcal{O}_{L_{1}},\dots,\mathcal{O}_{L_{7}},\\\
\mathcal{O}_{D},\mathcal{B}_{1},\mathcal{B}_{2},\mathcal{B}_{3},\mathcal{B}_{4}\right)$

is a full exceptional collection on $\widetilde{X}$.

###### Proof.

Let $l_{1},\dots,l_{7}$ be the exceptional curves of $\pi\colon
X^{\prime}\to\mathbb{P}^{2}$. Then, because of our choice of the map $\pi$, we
have that for all $i,j$ $c_{i,0}\neq l_{j}$, thus none of the $l_{j}$ contains
the point $q$. This implies that
$\tau^{\ast}{\mathcal{O}_{l_{i}}}=\mathcal{O}_{L_{i}}$. Then the statement
follows immediately by an iterative application of Orlov’s blow-up formula
[Orl92] (as stated in [Kuz14]). ∎

Let $p,p_{1},\dots,p_{4}$ be the five singular points on $X$, as in Section
4.1. Let $G=\frac{1}{4k+1}(1,1)$ and $H=\frac{1}{2k+1}(1,k)$. For
$j\in\mathbb{Z}/_{4k+1}$ let $\rho_{j}$ be the irreducible representation of
$G$ of weight $j$ and write $e_{j}=\mathcal{O}_{p}\otimes\rho_{j}$. For
$j\in\mathbb{Z}/_{2k+1}$ let $\eta_{j}$, be the irreducible representation of
$H$ of weight $j$ and write $e_{i,j}=\mathcal{O}_{p_{i}}\otimes\eta_{j}$.
Then:

######  Corollary 5.2.

The derived category $D(\mathcal{X})$ admits a full exceptional collection of
the form

(48)
$\sigma_{\mathcal{X}}=\left(\mathcal{E}_{p},\mathcal{E}_{p_{1}},\mathcal{E}_{p_{2}},\mathcal{E}_{p_{3}},\mathcal{E}_{p_{4}},\Phi(\sigma_{\widetilde{X}})\right),$

where $\mathcal{E}_{p}=\left(e_{2},\dots,e_{4k}\right)$,
$\mathcal{E}_{p_{i}}=\left(e_{i,k+1},\dots,e_{i,2k}\right)$, $i=1,2,3,4$.
Here, $\sigma_{\widetilde{X}}$ is the full exceptional collection in (47), and
$\Phi$ is the fully faithful functor from Section 2.3.3.

###### Proof.

An immediate consequence of the global $\operatorname{GL}(2,\mathbb{C})$ McKay
correspondence 2.3.3, Theorem 5.1, Example 2.7 and Example 2.8. ∎

###  5.2. Hom-spaces

There are various standard full exceptional collections $(A_{0},A_{1},A_{2})$
on $\mathbb{P}^{2}$. We choose
$(A_{0},A_{1},A_{2})=(\mathcal{O},\mathcal{T}(-1),\mathcal{O}(1))$, where
$\mathcal{T}$ is the tangent bundle on $\mathbb{P}^{2}$, following [AKO06].
Moreover, we choose the map $\pi\colon X^{\prime}\to\mathbb{P}^{2}$ to be a
blow-up morphism as described in Example 4.12 (a): this is always possible by
Corollary 4.16. This choice implies that the images
$\alpha(C_{i,0})\subset\mathbb{P}^{2}$ of the curves $C_{i,0}$ are two lines
and two conics. Without loss of generality, we assume that $\alpha(C_{i,0})$
is a line if $i=1,2$, is a conic if $i=3,4$. Moreover, setting
$x_{j}=\alpha(L_{j})$, we also assume that $x_{1},x_{2}\in\alpha(C_{1,0})$,
$x_{3},x_{4}\in\alpha(C_{2,0})$, $x_{1},x_{3}\in\alpha(C_{3,0})$, and
$x_{2},x_{4}\in\alpha(C_{4,0})$ (see Figure 10).

With these choices, we describe below the spaces of morphisms of the
exceptional collections $\sigma_{\widetilde{X}}$ 131313 The dimension of the
Hom spaces of $\sigma_{\widetilde{X}}$ are computed in Section 5.2.2 for
arbitrary vector bundles $A_{0},A_{1},A_{2}$. For
$\sigma^{\prime}_{\mathcal{X}}$ they depend on our choices of
$A_{1},A_{2},A_{3}$ and $\pi$. and

(49)
$\sigma^{\prime}_{\mathcal{X}}=\left(\mathcal{E}^{\prime}_{p},\mathcal{E}_{p_{1}},\mathcal{E}_{p_{2}},\mathcal{E}_{p_{3}},\mathcal{E}_{p_{4}},\Phi(\sigma_{\widetilde{X}})\right)$

where $\mathcal{E}^{\prime}_{p}=(e_{2}[-4k+1],\dots,e_{4k-1}[-2],e_{4k}[-1])$.

More precisely, in Lemma 5.6 and Lemma 5.7 we describe the spaces of morphisms
of $\sigma_{\widetilde{X}}$. Since $\Phi$ is a fully faithful functor, the
same two lemmas also describe all morphisms between objects of
$\sigma^{\prime}_{\mathcal{X}}$ of the form $\Phi(E)$, where
$E\in\sigma_{\widetilde{X}}$. Lemma 5.8 and Lemma 5.9 complete the description
of the spaces of morphisms of $\sigma^{\prime}_{\mathcal{X}}$. We omit the
computations of the morphisms between objects in $\mathcal{E}^{\prime}_{p}$
and $\mathcal{E}_{p_{i}}$ since they follow directly from the McKay quivers in
Example 2.7 and Example 2.8.

Our computations show that the collections $\sigma_{\widetilde{X}}$ and
$\sigma^{\prime}_{\mathcal{X}}$ are not strong. However, they have non-trivial
morphisms only in degree 0 and 1, which implies the existence of a tilting
complex in $D(\widetilde{X})$ and $D(\mathcal{X})$ (see Section 5.3).

In fact, one can change the exceptional collection $\sigma_{\widetilde{X}}$ to
another one with simpler spaces of morphisms. The exceptional collection for
$\widetilde{X}$ obtained by the left mutation of $\sigma_{\widetilde{X}}$ in
the pair $(\alpha^{\ast}\mathcal{O}(1),\mathcal{O}_{D})$ 141414More precisely,
one left-mutates $\mathcal{O}_{D}$ through
$\alpha^{\ast}\mathcal{O}(1),\mathcal{O}_{L_{1}},\dots,\mathcal{O}_{L_{7}}$.
However, the left mutation satisfies
$L_{\mathcal{O}_{L_{i}}}(\mathcal{O}_{D})=\mathcal{O}_{D}$ since
$\mathcal{O}_{D}$ and $\mathcal{O}_{L_{i}}$ are orthogonal., that is,

(50)
$L\sigma_{\widetilde{X}}=\left(\mathcal{O}_{\widetilde{X}},\alpha^{\ast}\mathcal{T}(-1),M,\alpha^{\ast}\mathcal{O}(1),\mathcal{O}_{L_{1}},\dots,\mathcal{O}_{L_{7}},\mathcal{B}_{1},\mathcal{B}_{2},\mathcal{B}_{3},\mathcal{B}_{4}\right)$

has only non-trivial morphisms in degree $0$, except for the morphisms from
$\operatorname{Ext}^{\ast}(b_{i,j},b_{i,l})=\mathbb{C}\oplus\mathbb{C}[-1]$
when $j<l$. Here, $M$ denotes the left mutation
$M=L_{\alpha^{\ast}\mathcal{O}(1)}\mathcal{O}_{D}$, defined by the short exact
sequence

(51) $0\to M\to\alpha^{\ast}\mathcal{O}(1)\xrightarrow{r}\mathcal{O}_{D}\to
0.$

Moreover, the corresponding exceptional collection for $\mathcal{X}$

(52)
$L\sigma^{\prime}_{\mathcal{X}}=\left(\mathcal{E}^{\prime}_{p},\mathcal{E}_{p_{1}},\mathcal{E}_{p_{2}},\mathcal{E}_{p_{3}},\mathcal{E}_{p_{4}},\Phi(L\sigma_{\widetilde{X}})\right)$

has only non-trivial morphism in degree $0$, except for the morphisms from
$\Phi(b_{i,j})$ to $\Phi(b_{i,l})$ when $j<l$ (Lemma 5.10), and the morphisms
starting from $e_{i,j}$, which have degree $0$ or $1$ (Lemma 5.11).

We collect the data of the dimensions of the $\operatorname{Hom}$-spaces
between objects of $L\sigma_{\widetilde{X}}$ and
$L\sigma^{\prime}_{\mathcal{X}}$ in Table 1, Table 2 and Table 3. In each
table, the entry in position $(i,j)$ is either the integer
$\dim\operatorname{Hom}(E_{i},E_{j})$, or a pair of integers
$(\dim\operatorname{Hom}(E_{i},E_{j}),\dim\operatorname{Ext}^{1}(E_{i},E_{j}))$.

######  Remark 5.3.

If $k=1$, the exceptional collection $L\sigma_{\widetilde{X}}$ is in fact
strong. Moreover, by shifting by $-1$ any element of $\mathcal{E}_{p_{i}}$,
$i=1,2,3,4$, one obtains a full strong exceptional collection also for the
stack $\mathcal{X}$. The dimensions of the $\operatorname{Hom}$-spaces of this
collection are summarised in Table 4.

It may be interesting to find out whether, for all $k>0$ integer, one could
obtain a full strong exceptional collection for $\mathcal{X}$ through a
sequence of mutations and shifts of $L\sigma^{\prime}_{\mathcal{X}}$.

####  5.2.1. Basic facts

We recall some well-known facts about sheaves of pure dimension 1 on surfaces
which we use repeatedly in the proofs of the lemmas. The first one is a
Riemann-Roch computation. Here, by a curve on S we mean a connected effective
1-cycle.

######  Lemma 5.4.

Let $S$ be a smooth projective surface, and let $V_{1},V_{2}$ be sheaves of
pure dimension one on $S$. For $i=1,2$, let $D_{i}$ be curves on $S$ such that
$\mathrm{ch}_{{1}}(V_{i})=D_{i}\in H^{*}(S)$. Then

$\chi(V_{1},V_{2})=\sum_{i\geq
0}(-1)^{i}\dim\operatorname{Ext}^{i}(V_{1},V_{2})=-D_{1}\cdot D_{2}.$

###### Proof.

Since $V_{1}$ and $V_{2}$ are torsion, their Chern characters satisfy
$\mathrm{ch}_{{0}}(V_{i})=0$. Then the Hirzebruch–Riemann–Roch theorem shows
the statement. ∎

The next Lemma follows directly from the definition of pure sheaf and simple
homological algebra:

######  Lemma 5.5.

Let $V$, $W$ be sheaves of pure dimension one on $S$, supported on curves $C$
and $D$, respectively. Then:

  1. (i)

if $C$ and $D$ have no common irreducible component, then
$\operatorname{Hom}(V,W)=0$;

  2. (ii)

every map $V\to W$ factors through the restriction map $V\to V_{|C\cap D}$;

  3. (iii)

Let $D^{\prime}=\overline{D\setminus(C\cap D)}$. Every map $V\to W$ factors
through the kernel of the restriction map $W\to W_{|D^{\prime}}$.

For future reference, we also list the intersection numbers of
$K_{\widetilde{X}}$ with the curves appearing in Figure 6:

(53) $\begin{split}F\cdot K_{\widetilde{X}}=4k-1\qquad C_{i,0}\cdot
K_{\widetilde{X}}=1\qquad C_{i,j}\cdot K_{\widetilde{X}}=0\qquad C_{i,k}\cdot
K_{\widetilde{X}}=-1\qquad\end{split}$

for $i=1,...,4$ and $j=1,...,k-1$.

####  5.2.2. Morphisms of the collections $\sigma_{\widetilde{X}}$ and
$\sigma^{\prime}_{\mathcal{X}}$

######  Lemma 5.6.

Let $V$ be a vector bundle on $\mathbb{P}^{2}$. Then
$\operatorname{Ext}^{\ast}(\alpha^{\ast}V,E)=\mathbb{C}^{\mathrm{rank}(V)}$,
where $E$ is any object of the collection
$(\mathcal{O}_{L_{1}},\dots,\mathcal{O}_{L_{7}},\mathcal{O}_{D},\mathcal{B}_{1},\mathcal{B}_{2},\mathcal{B}_{3},\mathcal{B}_{4})$.
Moreover, $\mathcal{O}_{L_{i}}$ is mutually orthogonal to
$\mathcal{O}_{L_{j}}$ for all $i\neq j$, and to any object of the collection
$(\mathcal{O}_{D},\mathcal{B}_{1},\mathcal{B}_{2},\mathcal{B}_{3},\mathcal{B}_{4})$.

###### Proof.

To prove the first part of the lemma note that, by adjunction,

$\operatorname{Ext}^{\ast}(\alpha^{\ast}V,E)=\operatorname{Ext}^{\ast}(V,\operatorname{\mathbf{R}\\!}\alpha_{\ast}E)$

and $\operatorname{\mathbf{R}\\!}\alpha_{\ast}E$ is the structure sheaf of a
point. The second part of the lemma follows from the fact that the objects
involved have disjoint supports. ∎

######  Lemma 5.7.

Let $i=1,2,3,4$. Then:

  1. 1.

for $j=1,\dots,k$,
$\operatorname{Ext}^{\ast}(\mathcal{O}_{D},b_{i,j})=\mathbb{C}\oplus\mathbb{C}[-1]$;

  2. 2.

$\mathcal{B}_{i}$ and $\mathcal{B}_{j}$, $j\neq i$, are mutually orthogonal;

  3. 3.

for $1\leq j<l\leq k$,
$\operatorname{Ext}^{\ast}(b_{i,j},b_{i,l})=\mathbb{C}\oplus\mathbb{C}[-1]$ .

###### Proof.

To prove (1) and (3) use Lemma 5.4 and Lemma 5.5. We show for example the
proof of (1) for $j=k$. It follows from Lemma 5.5 (ii) that there is only one
map $\mathcal{O}_{D}\to b_{i,k}$. Moreover, by Lemma 5.4

$\chi(\mathcal{O}_{D},b_{i,k})=-D\cdot C_{i,k}=-(F+(k-1)\cdot C_{i,k-1}+k\cdot
C_{i,k})\cdot C_{i,k}=0$

On the other hand,
$\operatorname{Ext}^{2}(\mathcal{O}_{D},b_{i,k})=\operatorname{Ext}^{2}(\mathcal{O}_{D},\mathcal{O}_{C_{i,k}})\simeq\operatorname{Hom}(\mathcal{O}_{C_{i,k}}(1),\mathcal{O}_{D})^{*}$
by Serre duality and (53), and the latter is trivial by Lemma 5.5 (iii).
Assertion (2) is immediate since $\mathcal{B}_{i}$ and $\mathcal{B}_{j}$ have
disjoint supports. ∎

To carry out the rest of the computations, we will combine our result on the
left adjoint functor $\Psi$ to $\Phi$ with the configuration of curves
described in Theorem 4.4. In particular, the local description from Examples
3.5 and 3.6 gives

(54) $\displaystyle\Psi(e_{i})\simeq\begin{cases}0&\mbox{if
}i\in{0,\dots,4k-2}\\\ \mathcal{O}_{F}(-4k-1)[1]&\mbox{ if }i=4k-1\\\
\mathcal{O}_{F}(-4k)&\mbox{ if }i=4k\end{cases}$

and, writing $C_{i}=\sum_{j=0}^{k-1}C_{i,j}$,

(55) $\displaystyle\Psi(e_{ij})\simeq\begin{cases}0&\mbox{if
}j\in{0,\dots,k-1}\\\ \mathcal{O}_{C_{i}}(C_{i})[1]&\mbox{ if }j=k\\\
\mathcal{O}_{C_{i,2k-j}}(-1)&\mbox{ if }j\in\\{k+1,\dots,2k-1\\}\\\
\mathcal{O}_{C_{i,0}}(-2)&\mbox{ if }j=2k\end{cases}$

######  Lemma 5.8.

We have that:

  1. 1.

for $2\leq l<4k-1$ , $\operatorname{Ext}^{*}(e_{l},\Phi(E))=0$ for all $E\in
D^{b}(\operatorname{Coh}(\widetilde{X}))$;

  2. 2.

for $V$ a vector bundle on $\mathbb{P}^{2}$,
$\operatorname{Ext}^{\ast}(e_{4k-1},\Phi(\alpha^{\ast}(V)))=\mathbb{C}^{\mathrm{rank(V)}}[-2]$;

  3. 3.

$\operatorname{Ext}^{\ast}(e_{4k-1},\Phi(\mathcal{O}_{D}))=\mathbb{C}^{2}[-1]\oplus\mathbb{C}[-2]$;

  4. 4.

for $i=1,2,3,4$, $j=1,\dots,k$,
$\operatorname{Ext}^{\ast}(e_{4k-1},\Phi(b_{i,j}))=\mathbb{C}[-1]$;

  5. 5.

for $V$ a vector bundle on $\mathbb{P}^{2}$,
$\operatorname{Ext}^{\ast}(e_{4k},\Phi(\alpha^{\ast}(V)))=0$;

  6. 6.

$\operatorname{Ext}^{\ast}(e_{4k},\Phi(\mathcal{O}_{D}))=\mathbb{C}$;

  7. 7.

for $=1,2,3,4$, $j=1,\dots,k$,
$\operatorname{Ext}^{\ast}(e_{4k},\Phi(b_{i,j}))=\mathbb{C}[-1]$;

  8. 8.

$\mathcal{E}_{p}$ and
$(\Phi(\mathcal{O}_{L_{1}}),\dots,\Phi(\mathcal{O}_{L_{7}}))$ are mutually
orthogonal.

###### Proof.

In all the statements above, we use that $\Psi$ is left adjoint to $\Phi$, and
conclude with techniques similar to the previous lemmas. We show (3) as an
illustration:

$\operatorname{Ext}^{\ast}(e_{4k-1},\Phi(\mathcal{O}_{D}))=\operatorname{Ext}^{\ast}_{\widetilde{X}}(\mathcal{O}_{F}(-4k-1)[1],\mathcal{O}_{D})$

by (54). The restriction of $\mathcal{O}_{D}$ to $\overline{D\setminus F}$ has
kernel $\mathcal{O}_{F}(-4k)$, so we have

$\operatorname{Hom}_{\widetilde{X}}(\mathcal{O}_{F}(-4k-1),\mathcal{O}_{D})=\operatorname{Hom}_{\widetilde{X}}(\mathcal{O}_{F}(-4k-1),\mathcal{O}_{F}(-4k))=\mathbb{C}^{2}$

by Lemma 5.5 (iii). On the other hand,

$\operatorname{Ext}^{2}_{\widetilde{X}}(\mathcal{O}_{F}(-4k-1),\mathcal{O}_{D})\simeq\operatorname{Hom}(\mathcal{O}_{D},\mathcal{O}_{F}(-2))^{*}=\operatorname{Hom}(\mathcal{O}_{F},\mathcal{O}_{F}(-2))^{*}=0$

by Serre duality, (53), and Lemma 5.5 (ii). Since
$\chi(\mathcal{O}_{F}(-4k-1),\mathcal{O}_{D})=-F\cdot D=1$ (using Lemma 5.4),
we show (3).

∎

######  Lemma 5.9.

Let $i=1,2,3,4$. Then:

  1. 1.

for $V$ a vector bundle on $\mathbb{P}^{2}$,
$\operatorname{Ext}^{\ast}(e_{i,j},\Phi(\alpha^{\ast}(V)))=0$ for all $j\neq
2k$;

  2. 2.

$e_{i,j}$ and $\Phi(\mathcal{O}_{L_{h}})$ are mutually orthogonal for all
$j\neq 2k$;

  3. 3.

$\operatorname{Ext}^{\ast}(e_{i,2k},\Phi(\mathcal{O}_{\widetilde{X}}))=0$;

  4. 4.

$\operatorname{Ext}^{\ast}(e_{i,2k},\Phi(\alpha^{\ast}(\mathcal{T}(-1))))=\operatorname{Ext}^{\ast}(e_{i,2k},\Phi(\alpha^{\ast}(\mathcal{O}(1))))=\begin{cases}\mathbb{C}[-1]&\mbox{
if \ }i=1,2\\\ \mathbb{C}^{2}[-1]&\mbox{ if \ }i=3,4\\\ \end{cases}$

  5. 5.

$\operatorname{Ext}^{\ast}(e_{i,2k},\Phi(O_{L_{h}}))=\mathbb{C}^{a_{i,j}}[-1]$,
where $a_{i,j}$ is the entry $(i,j)$ of the matrix:

$A=\begin{pmatrix}1&1&0&0&0&0&0\\\ 0&0&1&1&0&0&0\\\ 1&0&1&0&1&1&1\\\
0&1&0&1&1&1&1\end{pmatrix}$

  6. 6.

$\operatorname{Ext}^{\ast}(e_{i,j},\Phi(\mathcal{O}_{D}))=\begin{cases}0&\mbox{
if }j<2k\\\ \mathbb{C}[-1]&\mbox{ if }j=2k\\\ \end{cases}$

  7. 7.

$\operatorname{Ext}^{\ast}(e_{i,j},\Phi(b_{i,l}))=\begin{cases}\mathbb{C}&\mbox{
if }l=2k-j\\\ \mathbb{C}[-1]&\mbox{ if }l=2k-j+1\\\ 0&\mbox{ otherwise
}\end{cases}$

  8. 8.

$\mathcal{E}_{i}$ and $\Phi(\mathcal{B}_{j})$, $j\neq i$, are mutually
orthogonal.

###### Proof.

Again, in all the statements above, we use that $\Psi$ is left adjoint to
$\Phi$ and we conclude using ideas similar to those of the previous lemmas. As
an example, we show (4).

By (55), Serre duality, (53), and adjunction,

$\begin{split}\operatorname{Ext}^{\ast}(e_{i,2k},\Phi(\alpha^{\ast}(\mathcal{T}(-1))))&=\operatorname{Ext}^{\ast}(\mathcal{O}_{C_{i,0}}(-2),\alpha^{\ast}(\mathcal{T}(-1)))\\\
&=\operatorname{Ext}^{2-\ast}(\alpha^{\ast}(\mathcal{T}(-1)),\mathcal{O}_{C_{i,0}}(-1))^{*}\\\
&=\operatorname{Ext}^{2-\ast}(\mathcal{T}(-1),\mathcal{O}_{\alpha(C_{i,0})}(-1))^{*}\end{split}$

Now, by the long exact sequence for the Hom functor applied to the Euler
sequence, one finds that

$\operatorname{Ext}^{\ast}(\mathcal{T}(-1),\mathcal{O}_{\alpha(C_{i,0})}(-1))=\operatorname{Ext}^{\ast-1}(\mathcal{O}(-1),\mathcal{O}_{\alpha(C_{i,0})}(-1))$

since $\mathrm{H}^{\ast}(\mathbb{P}^{2},\mathcal{O}_{\alpha(C_{i,0})}(-1))=0$
because $C_{i,0}$ is rational. One concludes by observing that

$\operatorname{Ext}^{\ast}(\mathcal{O}(-1),\mathcal{O}_{\alpha(C_{i,0})}(-1))=\begin{cases}\operatorname{Ext}^{\ast}(\mathcal{O},\alpha(C_{i,0}))=\mathbb{C}&\mbox{
if }\alpha(C_{i,0})\mbox{ is a line,}\\\
\operatorname{Ext}^{\ast}(\mathcal{O},\alpha(C_{i,0})(1))=\mathbb{C}^{2}&\mbox{
if }\alpha(C_{i,0})\mbox{ is a conic.}\end{cases}$

Analogously,

$\operatorname{Ext}^{\ast}(e_{i,2k},\Phi(\alpha^{\ast}(\mathcal{O}(1))))=\operatorname{Ext}^{\ast}(\mathcal{O}_{C_{i,0}}(-2),\alpha^{\ast}(\mathcal{O}(1)))=\operatorname{Ext}^{2-\ast}(\mathcal{O}(1),\mathcal{O}_{\alpha(C_{i,0})}(-1))^{*}$

Hence,
$\operatorname{Ext}^{\ast}(e_{i,2k},\Phi(\alpha^{\ast}(\mathcal{O}(1))))$ is
$\mathbb{C}[-1]$ if $\alpha(C_{i,0})$ is a line, $\mathbb{C}^{2}[-1]$ if
$\alpha(C_{i,0})$ is a conic. ∎

####  5.2.3. Morphisms of the mutated collections

######  Lemma 5.10.

Let $M$ be as in (51). Then:

  1. 1.

$\operatorname{Ext}^{\ast}(\mathcal{O}_{\widetilde{X}},M)=\mathbb{C}^{2}$

  2. 2.

$\operatorname{Ext}^{\ast}(\alpha^{\ast}(\mathcal{T}(-1)),M)=\mathbb{C}$

  3. 3.

$\operatorname{Ext}^{\ast}(M,\alpha^{\ast}(\mathcal{O}(1)))=\mathbb{C}$

  4. 4.

for $i=1,\dots,7$,
$\operatorname{Ext}^{\ast}(M,\mathcal{O}_{L_{i}})=\mathbb{C}$

  5. 5.

for $i=1,2,3,4$, $j=1,\dots,k$,
$\operatorname{Ext}^{\ast}(M,b_{i,j})=\mathbb{C}$

###### Proof.

All the computations follow from the long exact sequence of the
$\operatorname{Hom}$ functor applied to (51), combined with Lemmas 5.6 and
5.7. For example, composition with $r$ gives a surjection

$\operatorname{Hom}(\alpha^{*}(\mathcal{T}(-1)),\alpha^{*}(\mathcal{O}(1)))\twoheadrightarrow\operatorname{Hom}(\alpha^{*}(\mathcal{T}(-1)),\mathcal{O}_{D}),$

which immediately implies (2). ∎

######  Lemma 5.11.

Let $M$ be as in (51). Then:

  1. 1.

$\operatorname{Ext}^{\ast}(e_{l},\Phi(M))=\begin{cases}0&\mbox{if }\ 2\leq
l<4k-1\\\ \mathbb{C}^{2}[-2]&\mbox{if }\ l=4k-1\\\ \mathbb{C}[-1]&\mbox{if }\
l=4k\end{cases}$

  2. 2.

for $i=1,2,3,4$ and for $k<j<2k$,
$\operatorname{Ext}^{\ast}(e_{i,j},\Phi(M))=0$;

  3. 3.

$\operatorname{Ext}^{\ast}(e_{i,2k},\Phi(M))=\begin{cases}0&\mbox{if }\
i=1,2\\\ \mathbb{C}[-1]&\mbox{if }\ i=3,4\\\ \end{cases}$

###### Proof.

All the computations follow from the long exact sequence of the
$\operatorname{Hom}$ functor applied to (51), together with Lemma 5.8 and
Lemma 5.9. For example, assertions (1) and (6) in Lemma 5.9 immediately imply
(2). ∎

###  5.3. Exceptional collections and tilting

Let $\mathcal{D}$ be a triangulated category. We say that $T\in\mathcal{D}$
generates $\mathcal{D}$ if, for every $E\in\mathcal{D}$, we have that
$\operatorname{Hom}(T,E)=0$ implies $E\simeq 0$, and that $T$ classically
generates $\mathcal{D}$ if the smallest full triangulated subcategory
containing $T$, closed under isomorphisms and direct summands, is
$\mathcal{D}$ itself. An object $T\in\mathcal{D}$ is partial tilting if
$\operatorname{Hom}(T,T[l])=0$ for all $l\neq 0$. We say that $T$ is a tilting
object if, moreover, it generates $\mathcal{D}$.

Generally speaking, tilting objects produce equivalences between $\mathcal{D}$
and derived categories of modules over their endomorphism algebra. More
precisely, suppose $\mathcal{D}=D^{b}(\operatorname{Coh}(Z))$ where $Z$ is a
smooth Deligne-Mumford stack with finite stabilizers and projective coarse
moduli space. Then we have:

######  Proposition 5.12.

Let $T$ be a tilting complex in $\mathcal{D}$ and denote by
$\Lambda=\operatorname{Hom}(T,T)$ its endomorphism algebra. Then, the functor
$\operatorname{\mathbf{R}\\!}\operatorname{Hom}_{\mathcal{O}_{Z}}(T,-)$
defines an equivalence of triangulated categories

$\mathcal{D}\xrightarrow{\sim}D^{b}(\mathrm{mod-}\Lambda)$

with inverse $-\otimes^{\operatorname{\mathbf{L}\\!}}_{\Lambda}T$.

###### Proof.

For schemes, this is [HVdB07, Theorem 7.6]. If $\mathcal{X}$ is a smooth DM
stack, $D(\operatorname{QCoh}(Z))$ is compactly generated (this is shown for
the more general class of perfect stacks in [BZFN10, Prop. 3.9]151515Smooth DM
stacks with finite stabilizers have affine diagonal, and hence are perfect.
Our setting is analogous to that of [AU15, Lemma 4.1], which follows a similar
argument.). Then, $T$ generates $D(\operatorname{QCoh}(Z))$ if and only if it
classically generates $\mathcal{D}$ [BvdB03, Theorem 2.1.2], and the standard
proof (see e.g. [Bae88, §3]) suffices to conclude. ∎

If $(E_{1},...,E_{n})$ is a strong full exceptional collection in
$\mathcal{D}$, then $\mathcal{E}=\oplus_{i=1}^{n}E_{i}$ is a tilting complex.
If the collection is not strong,
$\Lambda_{\mathcal{E}}=\operatorname{Hom}(\mathcal{E},\mathcal{E})$ is a
$\mathrm{dg}$-algebra, and one may still obtain an equivalence, analogous to
Prop. 5.12, between $\mathcal{D}$ and the derived category of
$\mathrm{dg}$-modules over $\Lambda_{\mathcal{E}}$. Alternatively, if the
exceptional collection only admits maps in degrees $0$ and $1$, we obtain an
explicit tilting complex from the so called universal extension trick, which
we briefly recall here:

######  Lemma 5.13.

Assume that $(E_{1},...,E_{n})$ is a collection of objects of $\mathcal{D}$
such that:

  1. (i)

each $E_{i}$ is partial tilting;

  2. (ii)

the sum $\oplus_{i=1}^{n}E_{i}$ generates
$D^{b}(\operatorname{Coh}(\mathcal{X}))$;

  3. (iii)

if $i<j$, then $\operatorname{Hom}(E_{i},E_{j}[l])=0$ for all $l\neq 0,1$;

  4. (iv)

if $i<j$, then $\operatorname{Hom}(E_{j},E_{i}[l])=0$ for all
$l\in\mathbb{Z}$.

Then $\mathcal{D}$ admits a tilting complex.

###### Proof.

We direct the reader to [Har21, Lemma 2.4] for the details of the proof, we
only recall the explicit construction here.

Proceed by induction on $n$: the case $n=1$ is clear. Assume now that $n>1$.
Let $r$ denote the dimension of $\operatorname{Ext}^{1}(E_{1},E_{2})$, then
the universal extension map produces a triangle

(56) $E_{2}\to F\to E_{1}^{\oplus r}.$

Put $E_{2}^{\prime}=E_{2}\oplus F$ and $E_{i}^{\prime}=E_{i}$ for $i=3,...,n$.
Then, $\\{E_{2}^{\prime},E_{3}^{\prime},...,E_{n}^{\prime}\\}$ still satisfies
the assumptions. This claim is proven as in [Har21]. The only significant
difference is that we do not assume the $E_{i}$ to be sheaves, and therefore
need more care to control negative Ext groups. ∎

######  Corollary 5.14.

Suppose $(E_{1},...,E_{n})$ is a full exceptional collection of $\mathcal{D}$,
such that $\operatorname{Hom}(E_{i},E_{j}[l])=0$ for all $l\neq 0,1$. Then
$\mathcal{D}$ admits a tilting complex.

###### Proof.

This is immediate from Lemma 5.13. ∎

Applying this to the case at hand:

######  Proposition 5.15.

The exceptional collections $L\sigma_{\widetilde{X}}$ and
$L\sigma^{\prime}_{\mathcal{X}}$ from Section 5.2 give rise to tilting
complexes in $D(\widetilde{X})$ and $D(\mathcal{X})$.

| $\mathcal{O}_{\widetilde{X}}$ | $\alpha^{\ast}\mathcal{T}(-1)$ | $M$ | $\alpha^{\ast}\mathcal{O}(1)$ | $\mathcal{O}_{L_{t}}$ | $b_{i,j}$ | $b_{i,l}$
---|---|---|---|---|---|---|---
$\mathcal{O}_{\widetilde{X}}$ | $1$ | $3$ | $2$ | $3$ | $1$ | $1$ | $1$
$\alpha^{\ast}\mathcal{T}(-1)$ |  | $1$ | $1$ | $3$ | $2$ | $2$ | $2$
$M$ |  |  | $1$ | $1$ | $1$ | $1$ | $1$
$\alpha^{\ast}\mathcal{O}(1)$ |  |  |  | $1$ | $1$ | $1$ | $1$
$\mathcal{O}_{L_{t}}$ |  |  |  |  | $1$ | $0$ | $0$
$b_{i,j}$ |  |  |  |  |  | $1$ | $(1,1)$
$b_{i,l}$ |  |  |  |  |  |  | $1$
Table 1. The Hom-spaces of the collection $L\sigma_{\widetilde{X}}$. | $e_{4k-j}[-j-1]$ | $\Phi(\mathcal{O}_{\widetilde{X}})$ | $\Phi\alpha^{\ast}\mathcal{T}(-1)$ | $\Phi M$ | $\Phi\alpha^{\ast}\mathcal{O}(1)$ | $\Phi\mathcal{O}_{L_{t}}$ | $\Phi b_{i,1}$
---|---|---|---|---|---|---|---
$\vdots$ |  McKay quiver (with shifts) Example 2.7 |  |  |  |  |  |
$e_{4k-2}[-3]$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{4k-1}[-2]$ | $1$ | $2$ | $2$ | $1$ | $0$ | $1$
$e_{4k}[-1]$ | $0$ | $0$ | $1$ | $0$ | $0$ | $1$
Table 2. The Hom-spaces between objects in $\mathcal{E}^{\prime}_{p}$ and from $\mathcal{E}^{\prime}_{p}$ to $\Phi(L\sigma_{\widetilde{X}})$. The shifts of the objects in $\mathcal{E}_{p}^{\prime}$ make the arrows from the McKay quiver into maps of degree $0$. | $e_{i,2k-j}$ | $\Phi(\mathcal{O}_{\widetilde{X}})$ | $\Phi\alpha^{\ast}\mathcal{T}(-1)$ | $\Phi M$ | $\Phi\alpha^{\ast}\mathcal{O}(1)$ | $\Phi\mathcal{O}_{L_{t}}$ | $\Phi b_{i,2k-j}$ | $\Phi b_{i,2k-j+1}$
---|---|---|---|---|---|---|---|---
$e_{i,k+1}$ |  McKay quiver Example 2.8 | $0$ | $0$ | $0$ | $0$ | $0$ | $\cdots$ | $\cdots$
$\vdots$ |  |  |  |  |  |  |
$e_{i,j}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $(1,0)$ | $(0,1)$
$\vdots$ |  |  |  |  |  |  |
$e_{i,2k-1}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\cdots$ | $\cdots$
$e_{1,2k}$ | $0$ | $(0,1)$ | $0$ | $(0,1)$ | $(0,a_{1,t})$ | $\cdots$ | $\cdots$
$e_{3,2k}$ | $0$ | $(0,2)$ | $(0,1)$ | $(0,2)$ | $(0,a_{3,t})$ | $\cdots$ | $\cdots$
Table 3. The Hom-spaces from between objects in $\mathcal{E}_{p_{i}}$ and from $\mathcal{E}_{p_{i}}$ to $\Phi(L\sigma_{\widetilde{X}})$. The arrows of the McKay quiver involved in the first column are morphisms of degree $1$. Thus every $e_{i,j}$ admits both $\operatorname{Hom}$’s and $\operatorname{Ext}^{1}$’s to other objects of the collection. | $e_{2}[-3]$ | $e_{3}[-2]$ | $e_{4}[-1]$ | $e_{i,2}[-1]$ | $\Phi\alpha^{*}\mathcal{O}_{\widetilde{X}}$ | $\Phi\alpha^{*}\mathcal{T}(-1)$ | $\Phi M$ | $\Phi\alpha^{*}\mathcal{O}(1)$ | $\Phi\mathcal{O}_{L_{t}}$ | $\Phi\mathcal{O}_{C_{i,1}}$
---|---|---|---|---|---|---|---|---|---|---
$e_{2}[-3]$ | $1$ | $2$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{3}[-2]$ |  | $1$ | $2$ | $O_{2\times 4}$ | $1$ | $2$ | $2$ | $1$ | $O_{2\times 7}$ | $1$
$e_{4}[-1]$ |  |  | $1$ | $0$ | $0$ | $1$ | $0$ | $1$
$e_{1,2}[-1]$ |  |  |  | $\mathrm{Id}_{4}$ | $0$ | $1$ | $0$ | $1$ | $A$ | $\mathrm{Id}_{4}$
$e_{2,2}[-1]$ |  |  |  | $0$ | $1$ | $0$ | $1$
$e_{3,2}[-1]$ |  |  |  | $0$ | $2$ | $1$ | $2$
$e_{4,2}[-1]$ |  |  |  | $0$ | $2$ | $1$ | $2$
Table 4. The strong exceptional collection for $\mathcal{X}$ when $k=1$.

## References

  * [ACC+16] Mohammad Akhtar, Tom Coates, Alessio Corti, Liana Heuberger, Alexander Kasprzyk, Alessandro Oneto, Andrea Petracci, Thomas Prince, and Ketil Tveiten.  Mirror symmetry and the classification of orbifold del Pezzo surfaces.  Proc. Amer. Math. Soc., 144(2):513–527, 2016.
  * [AKO06] Denis Auroux, Ludmil Katzarkov, and Dmitri Orlov.  Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves.  Invent. Math., 166(3):537–582, 2006.
  * [AKO08] Denis Auroux, Ludmil Katzarkov, and Dmitri Orlov.  Mirror symmetry for weighted projective planes and their noncommutative deformations.  Ann. of Math. (2), 167(3):867–943, 2008.
  * [Art66] Michael Artin.  On isolated rational singularities of surfaces.  Amer. J. Math., 88:129–136, 1966.
  * [AU15] Tarig Abdelgadir and Kazushi Ueda.  Weighted projective lines as fine moduli spaces of quiver representations.  Comm. Algebra, 43(2):636–649, 2015.
  * [Aus86] Maurice Auslander.  Rational singularities and almost split sequences.  Trans. Amer. Math. Soc., 293(2):511–531, 1986.
  * [Bae88] Dagmar Baer.  Tilting sheaves in representation theory of algebras.  Manuscripta Math., 60(3):323–347, 1988.
  * [BM02] Tom Bridgeland and Antony Maciocia.  Fourier-Mukai transforms for $K3$ and elliptic fibrations.  J. Algebraic Geom., 11(4):629–657, 2002.
  * [Bri02] Tom Bridgeland.  Flops and derived categories.  Invent. Math., 147(3):613–632, 2002.
  * [Bun21] Dominic Bunnett.  On the moduli of hypersurfaces in toric orbifolds.  arXiv e-prints, page arXiv:1906.00272, February 2021.
  * [BvdB03] A. Bondal and M. van den Bergh.  Generators and representability of functors in commutative and noncommutative geometry.  Mosc. Math. J., 3(1):1–36, 258, 2003.
  * [BZFN10] David Ben-Zvi, John Francis, and David Nadler.  Integral transforms and Drinfeld centers in derived algebraic geometry.  J. Amer. Math. Soc., 23(4):909–966, 2010.
  * [CCG+13] Tom Coates, Alessio Corti, Sergey Galkin, Vasily Golyshev, and Alexander Kasprzyk.  Mirror symmetry and Fano manifolds.  In European Congress of Mathematics, pages 285–300. Eur. Math. Soc., Zürich, 2013.
  * [CG21] Alessio Corti and Giulia Gugiatti.  Hyperelliptic integrals and mirrors of the Johnson-Kollár del Pezzo surfaces.  Trans. Amer. Math. Soc., 374(12):8603–8637, 2021.
  * [CLS11] David A Cox, John B Little, and Henry K Schenck.  Toric varieties, volume 124.  American Mathematical Soc., 2011.
  * [Dem80] Michel Demazure.  Surfaces de del Pezzo II, III, IV, V.  In Séminaire sur les singularités des surfaces, Lecture notes in Mathematics, volume 777, pages 23–69. Springer, Berlin, 1980.
  * [Dol12] Igor V. Dolgachev.  Classical algebraic geometry.  Cambridge University Press, Cambridge, 2012.  A modern view.
  * [Ela07] A. D. Elagin.  Exceptional sets on some del Pezzo surfaces with one log-terminal singularity.  Mat. Zametki, 82(1):36–51, 2007.
  * [Esn85] Hélène Esnault.  Reflexive modules on quotient surface singularities.  J. Reine Angew. Math., 362:63–71, 1985.
  * [FMN10] Barbara Fantechi, Etienne Mann, and Fabio Nironi.  Smooth toric Deligne-Mumford stacks.  J. Reine Angew. Math., 648:201–244, 2010.
  * [Giv98] Alexander Givental.  A mirror theorem for toric complete intersections.  In Topological field theory, primitive forms and related topics (Kyoto, 1996), volume 160 of Progr. Math., pages 141–175. Birkhäuser Boston, Boston, MA, 1998.
  * [Gol07] Vasily Golyshev.  Classification problems and mirror duality.  In Surveys in geometry and number theory: reports on contemporary Russian mathematics, volume 338 of London Math. Soc. Lecture Note Ser., pages 88–121. Cambridge Univ. Press, Cambridge, 2007.
  * [Gol08] Vasily Golyshev.  Spectra and strains.  arXiv e-prints, page arXiv:0801.0432, January 2008.
  * [GPS15] Sheel Ganatra, Timothy Perutz, and Nick Sheridan.  Mirror symmetry: from categories to curve counts.  arXiv e-prints, page arXiv:1510.03839, October 2015.
  * [GS18] Mark Gross and Bernd Siebert.  Intrinsic mirror symmetry and punctured Gromov–Witten invariants.  In Algebraic geometry: Salt Lake City 2015, volume 97 of Proc. Sympos. Pure Math., pages 199–230. Amer. Math. Soc., Providence, RI, 2018.
  * [Hac04] Paul Hacking.  Compact moduli of plane curves.  Duke Math. J., 124(2):213–257, 2004.
  * [Har21] Wahei Hara.  On the Abuaf-Ueda flop via non-commutative crepant resolutions.  SIGMA Symmetry Integrability Geom. Methods Appl., 17:Paper No. 044, 22, 2021.
  * [HV00] Kentaro Hori and Cumrun Vafa.  Mirror Symmetry.  arXiv e-prints, February 2000.
  * [HVdB07] Lutz Hille and Michel Van den Bergh.  Fourier-Mukai transforms.  In Handbook of tilting theory, volume 332 of London Math. Soc. Lecture Note Ser., pages 147–177. Cambridge Univ. Press, Cambridge, 2007\.
  * [IF00] A. R. Iano-Fletcher.  Working with weighted complete intersections.  In Explicit birational geometry of 3-folds, volume 281 of London Math. Soc. Lecture Note Ser., pages 101–173. Cambridge Univ. Press, Cambridge, 2000.
  * [Ish02] Akira Ishii.  On the McKay correspondence for a finite small subgroup of ${\rm GL}(2,\mathbb{C})$.  J. Reine Angew. Math., 549:221–233, 2002.
  * [IU15] Akira Ishii and Kazushi Ueda.  The special McKay correspondence and exceptional collections.  Tohoku Math. J. (2), 67(4):585–609, 2015.
  * [JK01] Jennifer M. Johnson and János Kollár.  Kähler–Einstein metrics on log del Pezzo surfaces in weighted projective 3-spaces.  Ann. Inst. Fourier (Grenoble), 51(1):69–79, 2001.
  * [Kaw06] Yujiro Kawamata.  Derived categories of toric varieties.  Michigan Math. J., 54(3):517–535, 2006.
  * [KM92] János Kollár and Shigefumi Mori.  Classification of three-dimensional flips.  J. Amer. Math. Soc., 5(3):533–703, 1992.
  * [KM97] Sean Keel and Shigefumi Mori.  Quotients by groupoids.  Annals of Mathematics, 145(1):193–213, 1997.
  * [KM98] János Kollár and Shigefumi Mori.  Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics.  Cambridge University Press, Cambridge, 1998.  With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original.
  * [Kol96] János Kollár.  Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics].  Springer-Verlag, Berlin, 1996.
  * [Kon95] Maxim Kontsevich.  Homological algebra of mirror symmetry.  In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 120–139. Birkhäuser, Basel, 1995.
  * [KSB88] J. Kollár and N. I. Shepherd-Barron.  Threefolds and deformations of surface singularities.  Invent. Math., 91(2):299–338, 1988.
  * [KSC04] János Kollár, Karen E. Smith, and Alessio Corti.  Rational and nearly rational varieties, volume 92 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2004.
  * [Kuw05] Masato Kuwata.  Twenty-eight double tangent lines of a plane quartic curve with an involution and the Mordell-Weil lattices.  Comment. Math. Univ. St. Pauli, 54(1):17–32, 2005.