# Error terms for the motives of discriminant complements and a Cayley-
Bacharach theorem

Ishan Banerjee

###### Abstract

In this paper we prove under some simplifying hypotheses questions of Picoco
and Levinson-Ullery on Cayley-Bacharach sets. Our results imply that, under
suitable hypotheses Cayley-Bacharach sets lie on curves of low degree. We then
use these results to estimate error terms to the normalized motive of the
space of smooth degree $d$ hypersurfaces in $\mathbb{P}^{n}$ as $d$ grows to
infinity. The error term can be expressed in terms of a certain ‘sum over
points’ on plane cubic curves and the associated Hodge structure can be
expressed in terms of the cohomology of the moduli space of elliptic curves.
We also prove convergence of the motive of degree $d$ hypersurfaces in
$\mathbb{P}^{n}$ as $n$ grows to infinity as well as other results on
discriminant complements of high dimensional varieties.

##  1 Introduction

Let $X$ be a smooth projective variety over a field $\mathbb{F}$. Let
$\mathcal{L}$ be an ample line bundle on $X$. Let

$\Sigma(X,\mathcal{L}):=\\{f\in H^{0}(X,\mathcal{L})|\exists p\in
X,f(p)=0,df(p)=0\\}.$

We call $\Sigma(X,\mathcal{L})$ the discriminant variety associated to $X$ and
$\mathcal{L}$, it consists of those sections of $\mathcal{L}$ which define a
singular hypersurface in $X$. Let

$U(X,\mathcal{L})=H^{0}(X,\mathcal{L})\setminus\Sigma(X,\mathcal{L}).$

Theorems of Vakil-Wood and Poonen mply that after some suitable normalization,
the motive of $U(X,\mathcal{L}^{j})$ converges (in an appropriate sense) as
$j\to\infty$ (see [16] and [14]).

In this paper we study the _rate of convergence_ of the motive of
$U(X,\mathcal{L}^{j})$. Our main contributions in this regard are:

  1. 1.

An improvement of known rates of convergence for all $X$ and $\mathcal{L}$
(Theorem 1.6). Interestingly enough, our results suggest that the rate of
convergence grows faster with the increase in the dimension of $X$, as in the
upper bounds we establish on the ‘error term’ decrease as $\dim X\to\infty$.

  2. 2.

In the special case of $\mathbb{P}^{n}$, we obtain a leading error term (see
Theorem 1.9 ) for the convergence of $U(\mathbb{P}^{n},\mathcal{O}(d))$ as
$d\to\infty$.

  3. 3.

We establish upper bounds on the difference between the motive of
$U(X,\mathcal{O}(d))$ and a special value of the motivic zeta function of $X$
for all smooth projective varieties $X$ (see Theorem 1.7). Our bound depends
only on the dimension of $X$. Our bounds improve as the dimension of $X$
increases. As a consequence we establish (for $d\geq 3$ ) that after a
suitable normalisation, the motive of $U(\mathbb{P}^{n},\mathcal{O}(d))$
converges to a certain limit of motivic zeta functions as $n\to\infty$ (see
Theorem 1.7). It is important to note here that we do not assume that
$d\to\infty$ or take some kind of limit in which the line bundle $\mathcal{L}$
becomes more and more ample.

In order to establish the above results, we prove several theorems about
Cayley Bacharach sets in $\mathbb{P}^{n}$ which are of independent interest
(the definition of Cayley-Bacharach sets will be recalled in Section 2). Our
results regarding Cayley-Bacharach sets are as follows:

  1. 1.

We show that under certain hypotheses, Cayley-Bacharach subsets of
$\mathbb{P}^{n}$ are forced to lie on curves of low degree (see Theorem 1.13
). This is partial progress on a question of Picoco(see Question 1.1 of [13] )

  2. 2.

We establish Conjecture 1.2 of [11] under certain additional hypotheses
(Theorem 1.15 ). The conjecture states that a subset $Z$ of $\mathbb{P}^{n}$
that is Cayley-Bacharach for $\mathcal{O}(r)$, that satisfies
$|Z|\leq(d+1)r+1$ lies on a plane configuration of dimension $d$.

###  1.1 Grothendieck ring of varieties

In this section, will briefly discuss the Grothendieck ring of varieties. We
largely follow section 1 of [16] where the Grothendieck ring of varieties is
discussed in more detail.

The Grothendieck ring of varieties $\mathcal{M}$ over the field $\mathbb{F}$
(also denoted $K_{0}(Var_{\mathbb{F}})$) is defined as follows, in terms of
generators and relations.

  1. 1.

Given a finite type $\mathbb{F}$ scheme $X$, we have an element
$[X]\in\mathcal{M}$. Isomorphic schemes define the same element of
$\mathcal{M}$.

  2. 2.

The collection of all $[X]$ generate $\mathcal{M}$ as an abelian group.

  3. 3.

Given any finite type scheme $X$ and a closed subscheme $Y$ with complement
$U=X\setminus Y$, we have the relation

$[X]=[U]+[Y].$

  4. 4.

Given $[X],[Y]\in\mathcal{M}$ we define $[X][Y]=[X\times Y]$. This gives
$\mathcal{M}$ the structure of a ring.

Let $\mathbb{L}=[\mathbb{A}^{1}].$ Let
$\mathcal{M}_{\mathbb{L}}=\mathcal{M}[\mathbb{L}^{-1}]$. There is an
increasing filtration on $\mathcal{M}$, with varieties of dimension
$\leq\ell$, generating the $\ell$th piece of the filtration. We will call this
filtration the dimensional filtration on $\mathcal{M}$. The filtration extends
to a filtration on $\mathcal{M}_{\mathbb{L}}$. We denote the completion of
$\mathcal{M}_{\mathbb{L}}$ with respect to this filtration by
$\hat{\mathcal{M}_{\mathbb{L}}}$.

######  Remark 1.

In this paper we will often have formulae where an element $x\in\mathcal{M}$
is multiplied by a vector space over the base field $\mathbb{F}$. These
formulae are to be understood in the following sense- we identify a vector
space $V$ with $\mathbb{L}^{\dim V}$. Any expression of the form $xV$ where
$x\in\mathcal{M}$ is simply shorthand for $x\mathbb{L}^{\dim V}$. Similarly an
expression of the form $xV^{-1}$ is understood to mean $x\mathbb{L}^{-\dim
V}$.

Let us now assume $\mathbb{F}=\mathbb{C}$. Let $\mathcal{M}^{Hdg}$ denote the
Grothendieck group of mixed Hodge structures . Let
$\mathbb{L}^{Hdg}=[H^{2}_{c}(\mathbb{A}^{1},\mathbb{Q})]$. We define
$\mathcal{M}_{\mathbb{L}}^{Hdg}=\mathcal{M}^{Hdg}[(\mathbb{L}^{Hdg})^{-1}].$
There is an increasing filtration on $\mathcal{M}^{Hdg}$ where the $\ell$th
piece of the filtration is generated by Hodge structures of weight $\leq\ell$.
The filtration extends to a filtration on $\mathcal{M}_{\mathbb{L}}^{Hdg}$. We
denote the completion of $\mathcal{M}_{\mathbb{L}}^{Hdg}$ with respect to this
filtration $\hat{\mathcal{M}_{\mathbb{L}}}^{Hdg}$

There is a specialisation map $\phi:\mathcal{M}\to\mathcal{M}^{Hdg}$ defined
on smooth proper generators by
$[X]\mapsto\sum_{k}(-1)^{k}H^{k}_{c}(X,\mathbb{Q}).$ The map extends to a map
$\mathcal{M}_{\mathbb{L}}\to\mathcal{M}_{\mathbb{L}}^{Hdg}$ and a map
$\hat{\mathcal{M}_{\mathbb{L}}}\to\hat{\mathcal{M}_{\mathbb{L}}}^{Hdg}.$

There is a further specialisation of the map
$\hat{\mathcal{M}_{\mathbb{L}}}\to\hat{\mathcal{M}_{\mathbb{L}}}^{Hdg}$ to a
map

$e:\hat{\mathcal{M}_{\mathbb{L}}}\to\mathbb{Z}[x,y][[x^{-1},y^{-1}]]$

defined on generators by

$e(X)=\sum_{(p,q)\in\mathbb{N}^{2}}\sum_{k\in\mathbb{N}}(-1)^{k}h^{p,q}(H^{k}_{c}(X,\mathbb{Q}))x^{p}y^{q}.$

Here $h^{p,q}$ denotes the $(p,q)^{th}$ Hodge number of the mixed Hodge
structure $H^{k}_{c}(X,\mathbb{Q})$. We will call the polynomial $e(X)$ the
Serre polynomial of $X$.

#### Motivic zeta function

Let $X$ be a quasi-projective variety over $\mathbb{F}$. For $n>0$ we define
$\textrm{Sym}^{n}X$ to be the $n^{th}$ symmetric power of $X.$ For $n=0$, we
set $\textrm{Sym}^{0}X$ to be $\textrm{Spec}\mathbb{F}$.

We define the Kapranov motivic zeta function of $X$,
$Z_{X}(t)\in\mathcal{M}_{\mathbb{L}}[[t]]$ by

$Z_{X}(t)=\sum_{k=0}^{\infty}[\textrm{Sym}^{k}X]t^{k}.$

We define

$\zeta_{X}(s)=Z_{X}(\mathbb{L}^{-s}).$

For $s>\dim X$, $\zeta_{x}(s)\in\hat{\mathcal{M}}_{\mathcal{L}}.$

####  1.1.1 A note on brackets

In this paper we will deal with many large formulae in the Grothendieck ring.
As a result we have decided to omit the square brackets in our formulae when
referring to the class of a variety, i.e. we will refer to $[X]$ as $X$. This
is because keeping the brackets in the notation would lead to a huge number of
brackets and be quite difficult to parse.

###  1.2 Meta problems for convergence

In this subsection we describe several meta problems about convergence of a
sequence in the completion of a filtered ring. Theorems 1.6, 1.9, and 1.7 will
be instances of these metaproblems. This subsection is in a strict sense not
essential to the rest of the paper and the main results of the paper will
still be completely understandable without this subsection. However we include
this section for motivation and to place some of our results in a wider
context.

Let $R$ be a ring with an increasing bi-infinite filtration $F^{l}$. Assume
that
$\cap_{\ell\in\mathbb{Z}}F^{\ell}R=0,\cup_{\ell\in\mathbb{Z}}F^{\ell}R=R$. Let
$\hat{R}$ be the completion of $R$ with respect to the filtration.

Now we define a list of meta problems regarding convergence in $\hat{R}$ in
increasing order of difficulty.

######  Problem 1.1.

Let $((x_{n}))$ be a sequence in $\hat{R}$. Show that $\lim_{n\to\infty}x_{n}$
exists.

######  Problem 1.2.

Let $((x_{n}))$ be a sequence in $\hat{R}$. Show that $\lim_{n\to\infty}x_{n}$
exists. Give an explicit description of the limit.

######  Problem 1.3.

Let $((x_{n}))$ be a sequence in $\hat{R}$. Show that
$\lim_{n\to\infty}x_{n}=L$. Find $\ell_{n}\in\mathbb{Z},\ell_{n}\to\infty$
such that $x_{n}-L\in F^{-\ell_{n}}\hat{R}$.

######  Problem 1.4.

Let $((x_{n}))$ be a sequence in $\hat{R}$. Show that
$\lim_{n\to\infty}x_{n}=L$. Find $\ell_{n}\in\mathbb{Z},\ell_{n}\to\infty$
such that $x_{n}-L\in F^{-\ell_{n}}\hat{R}$ and for $n\gg 0$ $x_{n}-L\not\in
F^{-\ell_{n}+1}\hat{R}$.

One could just as well propose a variant of Problem 4 where we require
$x_{n}-L\not\in F^{-\ell_{n}+C}\hat{R}$ where $C\in\mathbb{R}$ is some
constant not depending on $n$.

######  Problem 1.5.

Let $((x_{n}))$ be a sequence in $\hat{R}$ such that
$\lim_{n\to\infty}x_{n}=L$ Construct a sequence $y_{n}$ such that
$x_{n}-L=y_{n}+z_{n}$, where $z_{n}$ is much smaller than $y_{n}$.

Of course Problem 5 as stated is quite imprecise and one would need to decide
what it means for $z_{n}$ to be much smaller than $y_{n}$. For example, if the
$y_{n}$ we construct are all units one could ask that,
$\lim_{n\to\infty}\frac{z_{n}}{y_{n}}=0$.

###  1.3 Statement of results on convergence

We will adopt the following notation:

  1. 1.

Let $X$ be a scheme over a field $\mathbb{F}$. For a closed subscheme
$Y\subseteq X$ we denote the ideal sheaf of $Y$ by $I_{Y}$.

  2. 2.

Let $\mathcal{F}$ be a coherent sheaf on $X$. We define
$h^{i}(X,\mathcal{F}):=\dim_{\mathbb{F}}H^{i}(X,\mathcal{F}).$

  3. 3.

We define $N_{X}^{*}$ to be the conormal sheaf of $Y$ in $X$ it is the
pullback of the sheaf $I_{Y}$ to $Y$.

  4. 4.

Given $X$ and $Y$ as above we define $N_{X}(Y)$ to be the subscheme of $X$
defined by the sheaf $I^{2}_{Y}$, i.e. the first infinitesmal neighbourhood of
$Y$.

Given $x,y\in\hat{\mathcal{M}}_{\mathbb{L}}$, we will say $x=y$ up to
dimension $r$ if $x-y$ lies in the $F^{r}\hat{\mathcal{M}}_{\mathbb{L}}$,
where $F^{r}$ refers to the dimension filtration on
$\hat{\mathcal{M}}_{\mathbb{L}}$.

Let $X$ be a smooth variety over a field $\mathbb{F}$, $\mathcal{L}$ an ample
line bundle bundle on $X$. Then Proposition 3.7 of [16] states that

$\lim_{d\to\infty}\frac{U(X,\mathcal{L}^{d})}{H^{0}(X,\mathcal{L}^{d})}=\zeta_{X}^{-1}(\dim
X+1).$

This can be seen as as an instance of Problem 1. However the proof of
Proposition 3.7 in [16] actually gives more. Let $m(d)$ be the largest integer
such that for all zero dimensional reduced subschemes $Z\subseteq X$ of length
$\leq m(d)$, the map $H^{0}(X,\mathcal{L}^{d})\to
H^{0}(N_{X}(Z),\mathcal{L}^{d})$ is surjective. Then the proof of Proposition
3.7 in [16] actually implies that

$\frac{U(X,\mathcal{L}^{d})}{H^{0}(X,\mathcal{L}^{d})}-\zeta_{X}^{-1}(n+1)$

lies in $F^{-m(d)}\hat{\mathcal{M}_{\mathcal{L}}}$. We improve this result to
the following:

######  Theorem 1.6.

Let $X,\mathcal{L},m(d)$ be as above. Then there exists $C>0$ such that for
all $d\gg 0$,

$\frac{U(X,\mathcal{L}^{d})}{H^{0}(X,\mathcal{L}^{d})}=\zeta_{X}^{-1}(n+1)$

up to dimension $-m(d)(\dim X)+C$.

######  Remark 2.

The constant $C$ in the above theorem depends on $X$ and $\mathcal{L}$, it is
related to the dimensions of the spaces of curves in $X$ such that the line
bundle $\mathcal{L}$ restricts to a low degree. However, it is fairly
straightforward to adapt the proof to obtain a constant $C^{\prime}$,
satisfying $C<C^{\prime}$ where $C^{\prime}$ depends only on the dimension of
$X$.

We note that the improvement is greater for varieties of high dimension. This
motivates us to study whether there are bounds on the difference

$\frac{U(X,\mathcal{L}^{d})}{H^{0}(X,\mathcal{L}^{d})}-\zeta_{X}^{-1}(\dim
X+1)$

that improve with the dimension of $X$ and are valid for small values of $d$.

######  Theorem 1.7.

Let $X$ be a smooth projective variety, let $\mathcal{L}$ be a _very_ ample
line bundle. Then for any $d\geq 3$,

$\frac{U(X,\mathcal{L}^{d})}{H^{0}(X,\mathcal{L}^{d})}=\zeta_{X}^{-1}(\dim
X+1),$

up to dimension $\sqrt{\dim X}/3$.

As a corollary we have:

######  Corollary 1.8.

Let $X_{n}$ be a sequence of smooth projective varieties such that $\dim
X_{n}\to\infty$ and $\lim_{n\to\infty}\zeta_{X}^{-1}(\dim X_{n}+1)$ exists.
Let $\mathcal{L}_{n}$ be a very ample line bundle on $X_{n}$. Let $d_{n}\geq
3$ be a sequence of positive integers. Then

$\lim_{n\to\infty}\frac{U(X_{n},\mathcal{L}_{n}^{d_{n}})}{H^{0}(X_{n},\mathcal{L}_{n}^{d_{n}})}=\lim_{n\to\infty}\zeta_{X_{n}}^{-1}(\dim
X_{n}+1).$

In particular

$\lim_{n\to\infty}\frac{U(\mathbb{P}^{n},\mathcal{O}(d_{n}))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d_{n}))}=\prod_{k=1}^{\infty}(1-\mathbb{L}^{-k}).$

Finally we identify the leading error term for the sequence
$\frac{U(\mathbb{P}^{n},\mathcal{O}(d))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d))}$,
where $n$ is fixed and $d$ goes to infinity, providing a complete answer to
Problem 5 in this instance.

######  Theorem 1.9.

Let $n\geq 2$. Then for $d\gg n$, there exists $\epsilon>0$.

$\frac{U(\mathbb{P}^{n},\mathcal{O}(d))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d))}-Z_{\mathbb{P}^{n}}^{-1}(n+1)=Y(d)+Z(d)$

where:

  1. 1.

$\dim Z(d)\leq-(\frac{3n}{2}+\epsilon)d$.

  2. 2.

For $d\equiv 1\mod{4}$ , the virtual Hodge structure of $Y(d)$ has highest
weight term $E_{k}(\mathbb{L}^{Hdg})^{n^{2}-k(n+1)+1}$ in $K_{0}(MHS),$ where
$E_{d}$ is the highest weight term of the virtual Hodge structure
corresponding to $H^{1}(\mathcal{M}_{1,1},H_{k})$. Here $\mathcal{M}_{1,1}$ is
the moduli space of elliptic curves, $H_{k}$ is the Hodge bundle corresponding
to $k^{th}$ tensor power of the relative canonical bundle and
$k=\frac{3d+1}{2}$.

$Y(d)$ is explicitly defined in Section 6.

######  Remark 3.

$Y(d)$ has an explicit description in terms of configurations of points on
smooth plane cubic curves a precise description is given in Section 6 (defined
just before Proposition 6.14) and its virtual Hodge structure may be readily
computed from that description using the results of [7].

###  1.4  Motivation for our results on convergence

Our motivation for the results proven above comes from two different sources,
homological stability and number theory. We will first discuss some motivation
related to homological stability.

####  1.4.1 Homological stability

It was established by [15] that the spaces $U(\mathbb{P}^{n},\mathcal{O}(d))$
satisfy homological stability. Later on Aumonier in [1] and Das-Howe in [3]
established homological stability for the sequence of spaces
$U(X,\mathcal{L}^{d}$. However, it was known (and this is even mentioned in
[15]) that the range of homological stability given in [15] is not optimal. It
was of interest therefore to see where and how homological stability fails and
explain the first class in
$H^{*}(U(\mathbb{P}^{n},\mathcal{O}(d)),\mathbb{Q})$ that is not stable.

Related to the above is the desire to establish ‘secondary homological
stability’-like results for the family of spaces
$U(\mathbb{P}^{n},\mathcal{O}(d)).$ For instance the paper [5] establishes
that.

It would be interesting to establish some analogue of these results in the
case of $U(\mathbb{P}^{n},\mathcal{O}(d)),$ however there are several
difficulties in trying to do so, first of all there are no maps between
different $U(\mathbb{P}^{n},\mathcal{O}(d))$ for different values of $d$ and
secondly the nature of the spaces $U(\mathbb{P}^{n},\mathcal{O}(d))$ is very
different from the kind of spaces people have established ‘secondary
stability’ for; secondary stability results are known in the setting of
configuration spaces on manifolds and other more topological and less
algebraic geometric spaces as compared to discriminant complements. The
techniques used to establish such results are much more topological than the
ones in this paper.

In the end we were unable to establish such a ‘secondary stability’ theorem
for the homology of the spaces $U(\mathbb{P}^{n},\mathcal{O}(d))$ and this
paper does not really deal with results at the level of homology.

One may in fact combine the techniques of this paper with those of [15] to
conclude that $H^{k}(U(\mathbb{P}^{n},\mathcal{O}(d),\mathcal{O}(d))\cong
H^{k}(GL_{n+1}(\mathbb{C}),\mathbb{Q}),$ for $k\leq\frac{nd}{2}$. However we
do not include this result as we do not believe this is optimal either and we
do not how to obtain the optimal answer. In some sense our failure to get such
a result is a consequence of some of the differences between cohomology and
the Grothendieck ring, in the Grothendieck ring setting, if a variety $X$
admits a stratification with understandable pieces, we understand the class of
$[X]$ in the Grothendieck ring. However if a variety $X$ has a stratification
with understandable pieces, it is still not clear what the cohomology of $X$
is, the best we can do is obtain a spectral sequence whose $E_{1}$ page is
related to the cohomology of the strata. However it is often not clear in
these situations what the differentials in this spectral sequence are.

As a result we were unable to adapt some of the arguments involved in proving
Theorem 1.9 to a cohomological setting. It still may very well be possible to
establish a cohomological analogue of Theorem 1.9 but it would require
somewhat different techniques than we have used/ are aware of.

However it is possible to establish that for fixed $d\geq 3$,
$H^{k}(U(\mathbb{P}^{n},\mathcal{O}(d)))\cong H^{k}(GL_{n+1}(\mathbb{C}))$ for
$k\leq f(n)$ where $f(n)$ is some function of $n$ that grows to $\infty$. This
is a consequence of combining the techniques involved in proving Theorem 1.7
with those of [15]. //

We believe it should be possible to get a cohomological analogue of Theorem
1.7 as well.

While we have not established any secondary stability results for the sequence
of spaces $U(\mathbb{P}^{n},\mathcal{O}(d))$, we have established an analogue
of secondary stability in the setting of $K_{0}(Var_{\mathbb{F}});$ by the
results of [16], we have

$\lim_{d\to\infty}\frac{U(\mathbb{P}^{n},\mathcal{O}(d))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d))}\to\frac{1}{Z_{\mathbb{P}^{n}}(n+1)}$

Theorem 1.9 refines this result to give us an estimate of the difference
between the limit and the elements of the sequence. Furthermore the proof of
Theorem 1.9 identifies this leading error term as arising from the failure of
the proof of [16] to give an equality between
$\frac{1}{Z_{\mathbb{P}^{n}}(n+1)}$ and
$\frac{U(\mathbb{P}^{n},\mathcal{O}(d))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d))}$;
the proof of Theorem 1.13 in [16] crucially involves the fact that if $Z$ is a
collection of points in $X$ whose length is small compared to $d$, then the
vanishing of $1$-jets at the points of $Z$ impose linearly independent
conditions. This fails when the length of $Z$ is comparable to $d$ and our
error term $Y(d)$ is related to (and in fact defined in terms) of the variety
parametrizing collections of points where this linear independence just begins
to fail.

####  1.4.2 Arithmetic statistics

A second source of motivation comes from number theory, in particular from
arithmetic statistics. In many situations in arithmetic statistics, one is
interested in establishing asymptotic expressions for the number of some
quantity of arithmetic interest. For instance as a prototype, one has Roberts
conjecture (established in [2]) which says that the number of degree $3$
extensions of $\mathbb{Q}$ of discriminant $\leq X$ is
$aX+bX^{\frac{5}{6}}+o(X^{\frac{5}{6}})$.

We view Theorem 1.9 as a Theorem of this kind, we have essentially established
that the class of the discriminant complement $U(X,\mathcal{L})$ is

$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))Z_{\mathbb{P}^{n}}^{-1}(n+1)+H^{0}(\mathbb{P}^{n},\mathcal{O}(d))Y(d)+\textrm{
lower order terms},$

where lower order is to be understood in terms of the dimension filtration.

A natural question to ask here is if Theorem 1.9 and 1.7 have consequences
with regard to point counting over finite fields. More precisely, one may ask
if one has a result of the following form: Let $\mathbb{F}=\mathbb{F}_{q}$.
Let $X,\mathcal{L},d$ be as in Theorem 1.7. Is there a bound on

$\frac{\\#U(X,\mathcal{L}^{d})}{\\#3H^{0}(X,\mathcal{L}^{d})}-\\#Z_{X}^{-1}(-n-1)$

analogous to Theorem 1.9? Does

$\lim_{n\\\
to\infty}\\#\frac{U(\mathbb{P}^{n},\mathcal{O}(d))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d))}$

exist and equal

$\prod_{i=1}^{\infty}(1-\frac{1}{q^{i}})?$

Does one have

$\\#\frac{(U(\mathbb{P}^{n},\mathcal{O}(d)))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d))}=\\#Z_{\mathbb{P}^{n}}(-n-1)+\\#Y(d)+\textrm{
lower order terms}?$

We believe that the answer to all the above questions is yes and we are
currently working on solving the above questions. The reason why point
counting formulae like the above do not immediately follow from Theorems 1.7
and 1.9 is that the functor $\\#$ is not continuous with respect to the
dimension filtration, varieties of small dimension can have an arbitrary
number of points. In addition to this some of the methods we use are not well
adapted to the setting of point counts, e.g. the sum over ordered partitions
constructions that we use involve a very large number of terms that may cause
problems when added up. However we believe that we may combine our methods
with that of [14] to obtain the above results.

###  1.5 Cayley-Bacharach sets

In this section, we will define what Cayley-Bacharach sets are and prove a few
basic propositions about them. This may appear to be quite unrelated to the
material of the previous section, however it is not as the results of the
previous section rely on our results on Cayley-Bacharach sets.

In this section, we assume $\mathbb{F}$ to be an algebraically closed field.
Let $X$ be a projective variety over $\mathbb{F}$. Let $\mathcal{L}$ be a line
bundle on $X$.

######  Definition 1.10.

A zero dimensional reduced subscheme $Z\subseteq X$ is said to be Cayley-
Bacharach for $\mathcal{L}$ if given $f\in H^{0}(X,\mathcal{L})$, if $f$
vanishes on all but one point of $Z$ then $f$ vanishes on all of $Z$.

We will now reformulate the above definition. Given any $Z^{\prime}\subseteq
Z\subseteq X$ we have restriction maps $H^{0}(X,\mathcal{L})\to
H^{0}(Z,\mathcal{L})\to H^{0}(Z^{\prime},\mathcal{L})$. We hence have a map

$\textrm{Im}(H^{0}(X,\mathcal{L})\to
H^{0}(Z,\mathcal{L}))\to\textrm{Im}(H^{0}(X,\mathcal{L})\to
H^{0}(Z^{\prime},\mathcal{L})).$

Given a zero dimensional scheme $Z$, we will use $|Z|$ to denote the length of
$Z$.

######  Proposition 1.11.

A zero dimensional reduced subscheme $Z\subseteq X$ is Cayley-Bacharach for
$\mathcal{L}$ iff for all $Z^{\prime}\subseteq Z$, $|Z\setminus
Z^{\prime}|=1$, the restriction map

$\textrm{Im}(H^{0}(X,\mathcal{L})\to
H^{0}(Z,\mathcal{L}))\to\textrm{Im}(H^{0}(X,\mathcal{L})\to
H^{0}(Z^{\prime},\mathcal{L}))$ (1)

is an isomorphism.

###### Proof.

If $Z^{\prime}\subseteq Z$, then the homomorphism in (1) is always surjective.
For it to be to an isomorphism is equivalent therefore to it having zero
kernel. But if $f$ is in the kernel of (1), it is an element of
$H^{0}(X,\mathcal{L})$ that restricts to $0$ on $Z^{\prime}$. If $Z$ is
Cayley-Bacharach this implies that $f$ rstricts to $0$ on $Z$ and hence the
kernel of the (1) is trivial. Conversely if the kernel of (1) is trivial, then
any $f$ restricting to zero on $Z^{\prime}$ restricts to $0$ on $Z$, and hence
if the kernel of (1) is trivial for all choices of $Z^{\prime}$, then $Z$ is
Cayley-Bacharach for $\mathcal{L}$. ∎

###  1.6 Results on Cayley-Bacharach sets

In[13], Picoco asks the following question.

######  Question 1.12.

Let $e,d>0$. Suppose $Z\subseteq\mathbb{P}^{n}$ is a reduced set of points and
that $|Z|<e(d-e+3)-1$. Then is it true that if $Z$ is Cayley-Bacharach for
$\mathcal{O}(d)$, it lies on a curve of degree $<e$?

While we do not know if Picoco’s question has an affirmative answer we have
made the following partial result.

######  Theorem 1.13.

Let $d\gg e\geq 1$. Let $Z$ be a Cayley Bacharach subset of $\mathbb{P}^{n}$
for $\mathcal{O}(d)$. Then there exists a function $f$, depending on $n$ but
not on $d$ such that if

$|Z|<ed-f(e),$

$Z$ lies on a curve of degree $\leq e$.

In [11](Conjecture 1.2 in the paper) Levinson and Ullery made the following
conjecture

######  Conjecture 1.14.

Let $d\geq 1$. Suppose $Z\subseteq\mathbb{P}^{n}$ is a Cayley-Bacharach set
for $\mathcal{O}(d)$. Suppose $|Z|\leq(e+1)d+1$ Then $Z$ is contained in a
union of positive dimensional linear subspaces $L_{1},\dots L_{k}$ of
$\mathbb{P}^{n}$ such that $\sum_{i=1}^{k}\dim L_{i}\leq e$.

######  Theorem 1.15.

Let $d\gg e$. Then Conjecture 1.14 is true.

One can replace $d\gg e$ in 1.15 with $d$ greater than a certain quartic
polynomial in $e$, by going through the proof. The proof of Theorem 1.15
(assuming Theorem 1.13 ) is fairly straightforward and is essentially a
consequence of the fact that an integral curve of degree $e$ lies in an $e$
dimensional linear subspace of projective space.

###  1.7 Acknowledgements

I would like to thank Eduard Looijenga and Madhav Nori for helpful and
clarifying discussions. I’d like to thank Jake Levinson, Brooke Ullery and
David Stapleton for discussing some of their ideas with me. I’d like to thank
Ronno Das, and Aaron Landesmann for suggestion on how to improve the paper.
I’d like to thank Jordan Ellenberg for a question that led to Theorem 1.7.
Finally, I would like to thank Joshua Mundinger for extensive comments on
several drafts of this paper.

##  2 Cayley-Bacharach subsets lie on curves

In this section we will prove Theorems 1.13 and 1.15 . Our strategy to prove
Theorem 1.13 is to induct on the dimension of the ambient space. In the case
of $\mathbb{P}^{2}$ (the base case of our induction), we have the following
theorem of Lopez and Pirola (this is a subcase of Lemma 2.5 of [12]).

######  Proposition 2.1.

Let $Z\subseteq\mathbb{P}^{2}$ be a Cayley Bacharach set for $\mathcal{O}(d)$.
Let $e\geq 1$. If $|Z|\leq e(d-e+3)-1$, then $Z$ lies on a curve of degree
$\leq e-1$.

Our induction strategy will involve linearly projecting onto smaller
dimensional projective spaces. We will need the following propositions in our
proof, which deal with projections and Cayley Bacharach sets.

######  Proposition 2.2.

Let $n\geq 2$. Let $Z\subseteq\mathbb{P}^{n}$ be a Cayley-Bacharach set for
$\mathcal{O}(d)$. Let $P\in\mathbb{P}^{n}\setminus Z$. Let
$H\cong\mathbb{P}^{n-1}$ be a hyperplane not containing $P$. Let
$\pi:\mathbb{P}^{n}\setminus\\{P\\}\to H$ be the projection map. Suppose that
the scheme theoretic image of $Z$, $\pi(Z)$ is reduced. Then $\pi(Z)$ is a
Cayley-Bacharach set for $\mathcal{O}(d)$.

###### Proof.

We note that given a linear projection $\pi$, defined by projecting from a
point $P$ and a zero dimensional reduced subscheme $Z$, $\pi(Z)$; the
conditions that $P$ does not lie on a line joining any two of the points in
$Z$ and that $\pi(Z)$ is reduced are equivalent.

Without loss of generality, we may assume $P=[0:0:\dots:0:1]$ and $H$ is the
hyperplane defined by $X_{n+1}=0.$ Then the projection map $\pi$ is defined by
$[x_{1}:x_{2}:\dots:x_{n}:x_{n+1}]\mapsto[x_{1}:x_{2}:\dots:x_{n}].$ Let
$Z=\\{P_{1},P_{2},\dots,P_{m}\\}.$ Let
$\pi(Z)=\\{\pi(P_{1}),\pi(P_{2}),\dots,\pi(P_{m})\\}$. Let $f\in
H^{0}(\mathbb{P}^{n-1},\mathcal{O}(d))$ be a section vanishing at
$\\{\pi(P_{1}),\dots,\pi(P_{m-1})\\}.$ Such an $f$ is given by a degree $d$
homogenous polynomial $p$ in the variables $X_{1}\dots,X_{n}$. However the $p$
can be thought of as a polynomial in the variables $X_{1}\dots,X_{n},X_{n+1}$
as well, with no dependence on $X_{n+1}$. We observe that for any
$Q\in\mathbb{P}^{n}-\\{[0:\dots:0:1]\\}$, $p(Q)=0$ iff $p(\pi(Q))=0$. Hence
$p(P_{i})=0$ for $i\in\\{1,\dots m-1\\}.$ But since $Z$ is Cayley-Bacharach
for $\mathcal{O}(d)$ this implies $p$ vanishes on $P_{m}$ as well. But this
implies $p$ vanishes on $\pi(P_{m})$. Hence $\pi(Z)$ is Cayley-Bacharach for
$\mathcal{O}(d)$. ∎

######  Proposition 2.3.

Let $n\geq 3$. Let $P_{1},P_{2}$ be two distinct points in $\mathbb{P}^{n}$.
Let $H$ be a hyperplane in $\mathbb{P}^{n}$ such that $P_{1},P_{2}\not\in H$.
Let $C_{1},C_{2}$ be two curves in $H$ such that $\textrm{deg}(C_{i})\leq
e_{i}$. Let $S_{i}$ be the cone over $C_{i}$ with apex $P_{i}$. Then
$S_{1}\cap S_{2}=C\cup\Gamma$, where $C$ is a curve and $\Gamma$ is zero
dimensional. Furthermore $\deg(C)+|\Gamma|\leq e_{1}e_{2}$.

###### Proof.

We begin by noting that a cone over an irreducible curve is irreducible and
that cones with different apexes are distinct subvarieties. This immediately
implies that $S_{1}$ and $S_{2}$ have no irreducible components in common and
$S_{1}\cap S_{2}$ is a lower dimensional subvariety of $\mathbb{P}^{n}$ and
hence is of the form $C\cup\Gamma$, where $C$ is a curve and $\Gamma$ is zero
dimensional.

We note that $\deg(S_{i})=\deg(C_{i})$. We now apply the generalised Bezout’s
theorem of [4] (bottom of page 223), which states that the total degree of
$S_{1}\cap S_{2}$ (i.e. $\deg C+|\Gamma|$) is less than or equal to
$\deg(S_{1})\deg(S_{2})\leq e_{1}e_{2}$. ∎

######  Proposition 2.4.

Let $d\gg e>0$. Let $Z\subseteq\mathbb{P}^{n}$ be a Cayley-Bacharach set for
$\mathcal{O}(d)$. Let $C\subseteq\mathbb{P}^{n}$ be a reduced curve. Let
$\Gamma\subseteq\mathbb{P}^{n}$ be a finite set of points disjoint from $C$.
Suppose $\deg(C)+|\Gamma|\leq e^{2}$. If $Z\subseteq C\cup\Gamma$, then
$Z\subseteq C$.

###### Proof.

For a fixed value of $\deg(C)+|\Gamma|$, there exists $d_{0}$ such that for
$d>d_{0}$, the map $H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(C\cup\Gamma,\mathcal{O}(d))$ is surjective. Suppose $P\in Z\cap\Gamma$,
and that $P\not\in C$. Then there exists a section $f\in
H^{0}(C\cup\Gamma,\mathcal{O}(d))$ that is nonzero at $P$ and vanishes on
$C\cup\Gamma\setminus\\{P\\}$. Since the map
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(C\cup\Gamma,\mathcal{O}(d))$ is
surjective, we can find a polynomial $p\in
H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$ that restricts to $f$. Such a $p$
vanishes on $Z\setminus\\{P\\}$ and doesn’t vanish on $P$. This contradicts
the Cayley-Bacharach property of $Z$. Hence no such $p$ exists, i.e. $Z$ is
contained in $C$. ∎

######  Proposition 2.5.

Let $d\gg e>0$. Let $C\subseteq\mathbb{P}^{n}$ be a reduced curve of degree
$e$. Let $Z\subseteq C$ be a Cayley-Bacharach set for $\mathcal{O}(d)$. We
assume that $Z$ is not contained in any curve of degree $<e$. Then there is a
positive real valued increasing function $f$ not depending on $Z$ or $d$ such
that $|Z|\geq ed-f(e)$.

###### Proof.

The proof of this proposition is essentially Riemann-Roch for curves. However,
we first need to do some work to convert the given curve $C$ into a smooth
curve to which we can apply the Riemann Roch theorem.

We first note that a Cayley-Bacharach set must contain at least $d$ points
(see for instance [11]). Furthermore there is a function $\phi_{1}$, such that
any curve $C_{0}$ of degree $e$ satisfies
$|\textrm{Sing}(C_{0})|\leq\phi_{1}(e)$ . Hence for $d\gg e$, $Z$ is not
contained in $\textrm{Sing}(C)$. Let $C_{1},\dots,C_{k}$ be the irreducible
components of $C$. We note that $k\leq e$.

We can similarly conclude that for any irreducible component $C_{i}$ of $C$,
the intersection $Z\cap(C_{i}\setminus\textrm{Sing}(C))$ is nonempty. If this
were not the case, $Z$ would be a Cayley-Bacharach set contained in the union
of a curve $C^{\prime}=\cup_{j\neq i}C_{j}$ and a finite set,
$\textrm{Sing}C_{i}$. But for $d\gg e$,we may apply the argument of
Proposition 2.4 to conclude $Z\subseteq C^{\prime}$ which contradicts the fact
that $Z$ is not contained in a curve of degree $<e$.

Let $\pi:\tilde{C}\to C$ be a resolution of singularities. Let
$\tilde{Z}=\pi^{-1}(Z)\cup\pi^{-1}(\textrm{Sing}C)$. Note that there is some
function $\phi_{2}$ such that $|\tilde{Z}|-|Z|<\phi_{2}(e)$. Let $P\in
Z\setminus\textrm{Sing}(C)$. Let $\tilde{P}=\pi^{-1}(P)$. Suppose there exists
$\tilde{f}\in H^{0}(\tilde{C},\mathcal{O}(d))$ vanishing at
$\tilde{Z}-\tilde{P}$, such that $\tilde{f}(\tilde{P})\neq 0$. Then
$\tilde{f}$ is the pullback of a section $f\in H^{0}(C,\mathcal{O}(d))$
vanishing at $Z\setminus\\{P\\}$ and not vanishing at $P$. For $d\gg e$,
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$ surjects onto $H^{0}(C,\mathcal{O}(d))$
and hence $f$ is the restriction of some element of
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$. This contradicts the Cayley-Bacharach
property of $Z$. Hence no $f$ and consequently no such $\tilde{f}$ can exist.
We may now apply the Riemann Roch theorem to the curve $\tilde{C}$.

Let $\tilde{C}_{1},\dots,\tilde{C}_{k}$ be the corresponding components of
$\tilde{C}$.

Let $\tilde{P}\in\pi^{-1}((Z\cap C_{i})\setminus\textrm{Sing}(C))$. Let
$\pi(\tilde{P})=P$. We note that any $\tilde{f}\in
H^{0}(\tilde{C}_{i},\mathcal{O}(d))$ that vanishes on
$\pi^{-1}(\textrm{Sing}(C))\cup\pi^{-1}(Z)\setminus\\{P\\}$ is automatically
the pullback of a section $f\in H^{0}(C,\mathcal{O}(d))$ vanishing at
$Z\setminus\\{P\\}$ and hence $\tilde{f}$ is forced to vanish at $P$. Let
$Y_{i}=\pi^{-1}(\textrm{Sing}(C))\cup\pi^{-1}(Z)$. We have established that
the restriction map $H^{0}(\tilde{C}_{i},\mathcal{O}(d))\to
H^{0}(Y_{i},\mathcal{O}(d))$ is not surjective. Let $|Y_{i}|$ denote the line
bundle on $\tilde{C}_{i}$ associated with the divisor $Y_{i}$. Thus,

$H^{1}(\tilde{C}_{i},\mathcal{O}(d)\otimes|Y_{i}|^{\vee})\neq 0.$

By Riemann-Roch this implies that

$d\deg C_{i}-|\pi^{-1}(Z\cap C_{i})|-|\pi^{-1}(\textrm{Sing}(C)\cap
C_{i})|\leq\textrm{genus(C)}.$ (2)

If we sum (2) over all $i$ we obtain $d\deg(C)-\sum_{i}(|\pi^{-1}(Z\cap
C_{i})|-|\pi^{-1}(\textrm{Sing}(C)\cap
C_{i})|)\leq\sum_{i}\textrm{genus}(C_{i}).$ Hence

$\sum_{i}(|\pi^{-1}(Z\cap C_{i})|\geq
d\deg(C)-\sum_{i}\textrm{genus}(C_{i})-\sum|\pi^{-1}(\textrm{Sing}(C)\cap
C_{i})|).$ (3)

We note that $|\pi^{-1}(Z\cap C_{i})|-|Z\cap C_{i}|\leq\phi_{2}(e)$, with
$\phi_{2}$ as defined above. Hence, $|Z\cap C_{i}|\geq d(\deg
C_{i})-\phi_{2}(e)-\phi_{1}(e)$. Furthermore $\sum_{i}|Z\cap C_{i}|-|Z|\leq
k\textrm{Sing}(C)\leq e\phi_{1}(e).$

Combining this with 3 we have

$|Z|\geq
d\deg(C)-\sum_{i}(\textrm{genus}(C_{i}))-\sum_{i}|\pi^{-1}(\textrm{Sing}(C)\cap
C_{i})|)-e\phi_{2}(e)$ $\geq
d\deg(C)-\sum_{i}(\textrm{genus}(C_{i}))-2e\phi_{2}(e).$

Now we note that the genus of each individual $C_{i}$ is bounded in terms of
$e$ and $n$ by the Castelnuovo bound. Let us call this bound $\phi_{3}(e)$.
This gives us

$|Z|\geq d\deg(C)-e(\phi_{3}(e)+2\phi_{2}(e)).$

We let $f(e)=\max_{j\leq e}(1,j(\phi_{3}(j)+2\phi_{2}(j)))$. This establishes
the result. ∎

We now prove Theorem 1.13.

###### Proof of Theorem 1.13.

We will prove the theorem by induction on $n$. For $n=2$, the Theorem follows
from Proposition 2.1. Let us assume the Theorem for Cayley-Bacharach sets in
$\mathbb{P}^{n}$ for some $n\geq 2$. Let $Z\subseteq{\mathbb{P}^{n+1}}$ be a
Cayley Bacharach set for $\mathcal{O}(d)$, with $|Z|<ed-f(e)$. Let
$P_{1},P_{2}$ be two distinct points disjoint from $Z$ such that no line
passing through two of the points of $Z$ contains $P_{1}$ or $P_{2}$. Let $H$
be a hyperplane in $\mathbb{P}^{n+1}$ not containing $P_{1}$ or $P_{2}$. We
project $Z$ linearly from $P_{i}$ onto $H$ to obtain a set $Z_{i}$. The set
$Z_{i}$ is Cayley-Bacharach by Proposition 2.2. By the induction hypothesis
$Z_{i}$ lies on a curve $C_{i}$ of degree $e_{i}\leq e$. Therefore $Z$ lies on
the intersection of $S_{1}$ and $S_{2}$, where $S_{i}$ is the cone over
$C_{i}$ with apex $P_{i}$. By Proposition 2.3 $S_{1}\cap S_{2}=C\cup\Gamma$
where $\deg(C)+|\Gamma|\leq e_{1}e_{2}$. By Proposition 2.4, $Z\subseteq C$.
Let us replace $C$ by the curve of minimal degree containing $Z$. By
Proposition 2.5, $|Z|\geq\deg(C)d-f(\deg(C))$. But our assumption implies that
$|Z|\leq ed-f(e)$. Since $d\gg e$, this implies $\deg(C)\leq e$.

∎

We now move on to the proof of Theorem 1.15.

###### Proof of Theorem 1.15.

Let $Z$ be the given Cayley-Bacharach set. By Theorem 1.13, for $d\gg e$ and
$(e+1)d+1\leq ed-f(e)$ $Z$ is contained in a curve $C$ of degree at most
$e+1$. We remind the reader that an integral curve of degree $e$ is contained
in an $e$ dimensional linear subspace and an integral curve of degree $e$ that
is not contained in an $e-1$ dimensional subspace is a rational normal curve.

As a result the only curves of degree $e+1$ that are not contained in a union
of linear subspaces whose dimensions sum to $\leq e$ are unions of rational
normal curves. Hence for our purposes we may assume $Z$ is a Cayley- Bacharach
set for $\mathcal{O}(d)$ that is contained in a union of $k$ rational normal
curves $C_{i}$ with $\deg(C_{i})=r_{i}$ and $\sum r_{i}=e+1$, as the result
follows in all other cases.

. Suppose two rational normal curves $C_{i}$ and $C_{j}$ intersect at $k$
points. Then by the properties of rational normal curves one may conclude that
their union $C_{i}\cup C_{j}$ is contained in a subspace of dimension $\deg
C_{i}+\deg C_{j}+1-k$. So if $k>1$ for any pair $(i,j)$ we may replace
$P_{i},P_{j}$ by the span of $P_{i}$ and $P_{j}$ to obtain a new sequence of
linear subspaces whose dimensions sum to $\leq e$.

Hence we may assume $|C_{i}\cap C_{j}|\leq 1$ as the result follows in all
other cases.

Furthermore we claim that the intersection graph of the $C_{i}$ is a forest.
This is established as follows:

If

$|C_{i}\cap C_{j}|=|C_{j}\cap C_{k}|=|C_{i}\cap C_{k}|=1$

, then $\dim\textrm{span}P_{i}\cup P_{j}\cup P_{k}<\dim P_{i}+\dim P_{j}+\dim
P_{k}$ unless $|C_{i}\cap C_{j}\cap C_{k}|\neq 0$. A similar argument implies
that there are no cycles in the intersection graph, and hence it is a forest.

Let $d(C_{i})$ denote the degree of the vertex $C_{i}$ in the intersection
graph. We claim that $|Z\cap C_{i}|+d(C_{i})\geq r_{i}d+2$. For suppose that
$|Z\cap C_{i}|+d(C_{i})<r_{i}d+2$. Let $S$ be the set of points in $C_{i}$
that are also in some $C_{j}$ for $j\neq i$. Let $C_{i}^{\circ}=C_{i}\setminus
S$, i.e. the interior of $C_{i}$ in $C$. Let $p\in Z\cap C_{i}^{\circ}$. We
note that $|Z\cap C_{i}^{\circ}\setminus{p}|+|S|\leq|Z\cap
C_{i}|-1+d(C_{i})\leq r_{i}d$, by assumption. However, given any set of $\leq
r_{i}d$ points not containing $p$ in $C_{i}$, there is a section of $f\in
H^{0}(C_{i},\mathcal{O}(d))$ vanishing at those points and not at $p$ (Here we
use the fact that $C_{i}$ is isomorphic to $\mathbb{P}^{1}$). We may extend
$f$ from $C_{i}$ to $C$, by defining it to be zero outside $C_{i}$. Then $f$
vanishes on $Z\setminus\\{p\\}$ but not at $p$. This contradicts the Cayley-
Bacharach property of $Z$. Hence $|Z\cap C_{i}^{\circ}|+d(C_{i})\geq
r_{i}d+2$.

By summing this inequality over $i$, we obtain

$|Z|+\sum_{i}(d(C_{i}))\geq\sum_{i}|Z\cap
C_{i}^{\circ}|+\sum_{i}d(C_{i})\geq\sum_{i}r_{i}d+2k=(e+1)d+2k.$

By the above inequality, to establish that $|Z|\geq d(e+1)$ suffices to
establish that $2k-\sum_{i}d(C_{i})\geq 1$. We note that $\sum_{i}d(C_{i})$ is
twice the number of edges in the intersection graph. As a result,
$2k-\sum_{i}d(C_{i})$ is twice the Euler characteristic of the intersection
graph. Since the intersection graph is a forest, the Euler characteristic is
always $\geq 1$.

∎

##  3 Preparatory lemmas

The purpose of this section is to state and prove Lemma 3.1 which will be used
later on in the paper to prove our theorems on convergence of the motives of
discriminant complements. It is essentially a corollary of Theorem 1.13 .

######  Lemma 3.1.

Let $d\gg e>0$. Let $f$ be as in Theorem 1.13 Let $Z\subseteq\mathbb{P}^{n}$
be a reduced zero dimensional subscheme. Suppose $2|Z|\leq ed-f(e)$. Suppose
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))\neq 0$. Then there is a function $f$ such
that if $2|Z|\leq ed-f(e)$, then there is a subset $Z^{\prime}\subseteq Z$ and
a curve $C$ of degree $\leq e$ such that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime}}(d))=h^{1}(N_{\mathbb{P}^{n}}(C),I^{2}_{Z^{\prime}}(d))$.
Furthermore the curve of minimal degree satisfying the above conditions is
unique.

Before embarking on the proof of lemma, we will need a few propositions.

######  Proposition 3.2.

Assume that $\mathbb{F}$ is algebraically closed. Let
$Z\subseteq\mathbb{P}^{n}$ be a reduced zero dimensional subscheme. Suppose
that the restriction map $H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(Z,\mathcal{O}(d))$ is not surjective. Then $Z$ has a subset $Z^{\prime}$
that is Cayley-Bacharach for $\mathcal{O}(d)$.

###### Proof.

Let $Z^{\prime}\subseteq Z$ be such that

$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(Z^{\prime},\mathcal{O}(d))$

is not surjective and for any proper subset $Z^{\prime\prime}\subseteq
Z^{\prime}$, the restriction map

$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(Z^{\prime\prime},\mathcal{O}(d))$

is surjective. Clearly, such a $Z^{\prime}$ exists as the restriction map is
surjective for singleton sets. Let us now take any subset
$Z^{\prime\prime}\subseteq Z^{\prime}$ such that $|Z^{\prime}\setminus
Z^{\prime\prime}|=1$.

Since $H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(Z^{\prime},\mathcal{O}(d))$ is not surjective,

$\dim Im(H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(Z^{\prime},\mathcal{O}(d)))<\dim
H^{0}(Z^{\prime},\mathcal{O}(d)))=|Z^{\prime}|.$

However $Im(H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(Z^{\prime},\mathcal{O}(d)))$ surjects onto

$Im(H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(Z^{\prime\prime},\mathcal{O}(d)))=H^{0}(Z^{\prime\prime},\mathcal{O}(d))$

which is of dimension $|Z^{\prime\prime}|=|Z^{\prime}|-1$. However, since the
dimension of the source is at most $|Z^{\prime}|-1$, the restrcition map is
forced to be an isomorphism, which establishes that $Z^{\prime}$ is Cayley
Bacharach.

∎

######  Proposition 3.3.

Assume that $\mathbb{F}$ is algebraically closed. Let
$Z\subseteq\mathbb{P}^{n}$ be a reduced zero dimensional subscheme. Suppose
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))\neq 0$. Then $Z$ has a subset $Z^{\prime}$
that is Cayley-Bacharach for $\mathcal{O}(\lceil(d/2)\rceil)$.

###### Proof.

We observe that Theorem 1.1 of [6] of implies that if
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))\neq 0$, then
$h^{1}(\mathbb{P}^{n},I_{Z}(\lceil(d/2)\rceil)\neq 0$. Associate to the short
exact sequence

$0\to
I_{Z}(\lceil(d/2)\rceil)\to\mathcal{O}(\lceil(d/2)\rceil)\to\mathcal{O}_{Z}(\lceil(d/2)\rceil)\to
0$

, i we have a long exact sequence in cohomology, part of which is as follows:

$H^{0}(\mathbb{P}^{n},\mathcal{O}(\lceil(d/2)\rceil))\to
H^{0}(Z,\mathcal{O}(\lceil(d/2)\rceil))\to
H^{1}(\mathbb{P}^{n},I_{Z}(\lceil(d/2)\rceil))\to
H^{1}(\mathbb{P}^{n},\mathcal{O}(\lceil(d/2)\rceil)))$

But $H^{1}(\mathbb{P}^{n},\mathcal{O}(\lceil(d/2)\rceil)))\neq 0,$. As a
result the fact that $h^{1}(\mathbb{P}^{n},I_{Z}(\lceil(d/2)\rceil)\neq
0$implies that the map $H^{0}(\mathbb{P}^{n}\mathcal{O}(\lceil(d/2)\rceil))\to
H^{0}(Z\mathcal{O}(\lceil(d/2)\rceil))$ can not be surjective.

We then apply Proposition 3.2 to conclude that $Z$ has a subset $Z^{\prime}$
that is Cayley-Bacharach for $\mathcal{O}(\lceil(d/2)\rceil)$. ∎

######  Proposition 3.4.

Let $e,n>0$. Let $Z\subseteq\mathbb{P}^{n}$ be a reduced set of points. Let
$C$ be a curve of degree $e$. Let $d\gg e,n$. Suppose
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))\neq 0$. Let $Z^{\prime}=Z\cap C$. Suppose

$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))>h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime}}(d)).$

Let $Z^{\prime\prime}=Z\setminus Z^{\prime}$. Then
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d-2e))\to
H^{0}(N(Z^{\prime\prime}),\mathcal{O}(d-2e))$ is not surjective.

###### Proof.

Let $D\subseteq\mathbb{P}^{n}$ be a divisor containing $C$ not containing any
point of $Z^{\prime\prime}$. Let $|D|$ denote the degree of $D$. Suppose for
the sake of contradiction that $H^{0}(\mathbb{P}^{n},\mathcal{O}(d-2|D|)$
surjects onto $H^{0}(N(Z^{\prime\prime}),\mathcal{O}(d-2|D|)$. Let $f\in
H^{0}(\mathbb{P}^{n},|D|)$ be a section whose zero locus is $D$. We have a
subspace

$f^{2}\otimes H^{0}(\mathbb{P}^{n},\mathcal{O}(d-2|D|)\subseteq
H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$

that surjects onto $H^{0}(Z^{\prime\prime},\mathcal{O}(d))$ and maps to $0$ in
$H^{0}(N(Z^{\prime}),\mathcal{O}(d))$. We note that
$H^{0}(N(Z),\mathcal{O}(d))\cong H^{0}(N(Z^{\prime}),\mathcal{O}(d))\oplus
H^{0}(N(Z^{\prime\prime}),\mathcal{O}(d)).$ Some elementary linear algebra
then implies that the codimension of the image of
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$ in $H^{0}(N(Z),\mathcal{O}(d))$ is
equal to the codimension of the image of
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$ in
$H^{0}(N(Z^{\prime}),\mathcal{O}(d))$, since the map surjects onto the other
factor. This contradicts the assumption that

$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))>h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime}}(d)).$

∎

We now proceed to prove Lemma 3.1

###### Proof of Lemma 3.1.

Let us first prove the lemma under the assumption that our base field
$\mathbb{F}$ is algebraically closed. We begin by applying Proposition 3.2 to
conclude that $Z$ contains a subset $Z_{0}$ that is Cayley Bacharach for
$\mathcal{O}(\lceil d/2\rceil)$. We then apply Theorem 1.13 to conclude that
there is a curve $C$ that contains $Z_{0}$. Let $Z^{\prime}=Z\cap C$. If
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime}}(d))$
we are done and $C$ is our required curve. If not, we proceed as follows:

  1. 1.

By Proposition 3.4 $Z^{\prime\prime}=Z\setminus Z^{\prime}$ must satisfy
$h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime\prime}}(d-2e)>0$. Hence it must contain
a Cayley Bacharach subset $Z^{\prime\prime\prime}$ for $\mathcal{O}(\lceil
d/2\rceil-e)$, which is contained on a curve $C^{\prime}$.

  2. 2.

We replace our original curve $C$ by $C\cup C^{\prime}$ and replace
$Z^{\prime}$ by $Z\cap(C\cup C^{\prime})$.

  3. 3.

We repeat steps 1 and 2 until
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime}}(d))$.
Since $|Z^{\prime}|$ increases every time we perform step $2$ and
$|Z^{\prime}|<|Z|$ this process terminates.

We then have our required curve $C$ and subset $Z^{\prime}$.

If our base field $\mathbb{F}$ is not algebraically closed, we may pass to the
algebraic closure $\bar{\mathbb{F}}$ and then find a curve $C$ defined over
$\bar{\mathbb{F}}$ containing $Z$ and of degree $\leq e$. We will prove that
$C$ is defined over $\mathbb{F}$ which will complete the proof of the lemma.
Let $g\in Gal(\bar{\mathbb{F}},\mathbb{F})$. We claim that $g(C)=C$. If
$g(C)\neq C$, there is some irreducible component $C_{i}\subseteq C$ such that
$g(C_{i})\neq C_{i}$. But $Z\cap C_{i}\subseteq|g(C_{i})\cap
C_{i}|\leq\deg(C_{i})^{2}\leq e^{2}.$ However, $Z\cap C_{i}\geq e^{2}$ (see
proof of Proposition 2.4). Hence for $d\gg e$, $g(C_{i})=C_{i}$ and hence $C$
is defined over $\mathbb{F}$. This completes the proof of existence.

Let us now turn to proving uniqueness. It suffices to prove uniqueness over an
algebraically closed field. Suppose that $C_{1},C_{2}$ are curves of the same
degree, with $\deg(C_{i})\leq e\ll d$, such that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z_{i}}(d))=l$
where $Z_{i}=Z\cap C_{i}$ and no curve of strictly smaller degree satisfies
the conditions of the lemma. Let $W=Z_{1}\cap Z_{2}$. Suppose
$h^{1}(\mathbb{P}^{n},I^{2}_{W}(d))<h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=l$. We
will show that this implies that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))>h^{1}(\mathbb{P}^{n},I^{2}_{Z_{i}}(d))$
which contradicts our assumption. Let $Z_{i}^{\prime}=Z_{i}\setminus W$. Let
$U\subseteq H^{0}(Z_{1}^{\prime}\cup Z_{2}^{\prime}\cup W,\mathcal{O}(d))$ be
the image of $H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$ which is of codimension
$l$. Let $\pi_{i}:H^{0}(Z,\mathcal{O}(d))\to H^{0}(Z_{i},\mathcal{O}(d))$ be
the restriction map. Let $\pi:H^{0}(Z,\mathcal{O}(d))\to
H^{0}(W,\mathcal{O}(d))$ be the restriction map. By our assumption,
$\pi_{i}(U)$ is of codimension $l$ in $H^{0}(Z_{i},\mathcal{O}(d))$. Therefore
$U\subseteq\pi_{i}^{-1}(\pi_{i})(U)$ and both are of codimension $l$ in
$H^{0}(Z,\mathcal{O}(d))$. Hence we have

$U=\pi_{1}^{-1}(\pi_{1}(U))=\pi_{2}^{-1}(\pi_{2}(U)).$

The only way the above equation can hold is if $U=\pi^{-1}(\pi(U))$ and as a
result

$\pi(U)$ is also of codimension $l$, which contradicts
$h^{1}(\mathbb{P}^{n},I^{2}_{Z_{0}}(d))<h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=l$.

Hence
$h^{1}(\mathbb{P}^{n},I^{2}_{Z_{0}}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=l>0$.
However in this case we note that $Z_{0}\subseteq C_{1}\cap C_{2}$. Let
$C_{1}\cap C_{2}=C\cup\Gamma$ where $C$ is a reduced curve and $\Gamma$ is a
finite set of points. Either $h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap
C}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$, in which case $C$ is a curve of
smaller degree than $C_{1}$ or $C_{2}$ satisfying the conditions of the lemma.
Otherwise $h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap
C}(d))<h^{1}(\mathbb{P}^{n},I^{2}_{Z_{0}}(d))$ However, this implies that
$Z_{0}\setminus(Z\cap C)\subseteq\Gamma$ contains a Cayley Bacharach set for
$\mathcal{O}(d-2\deg C)$ by the combination of Proposition 3.4 and Proposition
3.3. But $|\Gamma|\leq e^{2}$ and a Cayley Bacharach set for
$\mathcal{O}(d-2\deg C)$ must contain at least $d-2e$ points. Since $d\gg e$,
we are done.

∎

##  4 Proof of Theorem 1.6

In this section we prove Theorem 1.6.

We will begin by reviewing some material in section 2 of [16].

######  Definition 4.1.

Let $S$ be a nonempty finite set. An ordered partition $\lambda$ of the set
$S$ is a tuple $(A_{1},\dots,A_{m})$, where the $A_{i}$ are nonempty disjoint
subsets of $S$ such that $A_{1}\cup\dots\cup A_{m}=S$. The empty set
$\varnothing$ has a single ordered partition called the empty partition and
also denoted $\varnothing$. We will denote the ordered partition of
$\\{1,\dots,n\\}$ with only one set by $[n]$.

We can associate two numbers to an ordered partition $\lambda$. If
$\lambda=(A_{1},A_{2},\dots.A_{m})$ is an ordered partition of a nonempty
finite set $S$, we define $|\lambda|=|S|$ and $||\lambda||=m$. For the empty
partition $\varnothing$, we define $||\varnothing||=|\varnothing|=0$.

In what follows we will have to consider sums over ordered partitions of a
certain length. By $\sum_{|\lambda|=k}f(\lambda)$ we mean that we sum the
values of $f(\lambda)$ as $\lambda$ ranges over all ordered partitions of
$\\{1,\dots,k\\}$.

######  Definition 4.2.

. Let $X$ be a quasiprojective variety over $\mathbb{F}$. Let $S$ be a
nonempty finite set. Let $\lambda=(A_{1},A_{2},\dots,A_{m})$ be an ordered
partition of $S$. We define
$\textrm{Sym}^{\lambda}(X):=\prod_{i=1}^{m}\textrm{Sym}^{|A_{i}|}(X)$.

There is a quotient map $\pi:X^{|S|}\to\textrm{Sym}^{\lambda}(X)$ We define
the $\lambda$ configuration space of $X$,
$w_{\lambda}(X)\subseteq\textrm{Sym}^{\lambda}(X)$ to be
$\pi(X^{|S|}\setminus\Delta)$, where $\Delta$ is the big diagonal.

We define
$\textrm{Sym}^{\varnothing}(X)=w_{\varnothing}(X)=\textrm{Spec}\textrm{
}\mathbb{F}$.

If $\lambda$ is an ordered partition and $X$ is a quasiprojective variety, any
$Z\in w_{\lambda}(X)$ defines a reduced zero dimensional subscheme of $X$
which we will denote $\textrm{supp}(Z)$. For any subvariety $Y\subseteq X$, we
say $Z\subseteq Y$ if $\textrm{supp}(Z)\subseteq Y$.

There is a forgetful covering map $w_{\lambda}(X)\to w_{|\lambda|}(X)$ defined
by $Z\mapsto\textrm{supp}(Z).$

######  Definition 4.3.

Let $X$ be a quasi projective variety over $\mathbb{F}$. Let $\mathcal{L}$ be
a line bundle on $X$. We define

$W_{\geq\lambda}(X)=\\{(f,Z)\in H^{0}(X,\mathcal{L})\times
w_{\lambda}(X)|Z\subseteq\textrm{Sing}(f)\\}.$

For $N\geq 0$, we define

$W_{\lambda,\geq N}(X):=\\{(f,Z)\in W_{\geq\lambda}(X)||\textrm{Sing}(f)|\geq
N\\}.$

We define

$W_{\lambda}(X):=\\{(f,Z)\in
W_{\geq\lambda}(X)||\lambda|=|\textrm{Sing}(f)|\\}.$

Our notation does not involve the line bundle $\mathcal{L}$. We do not believe
this will cause confusion as the line bundle $\mathcal{L}$ will be fixed
throughout. We note that $W_{\varnothing}(X)=U(X,\mathcal{L})$ the
discriminant complement.

######  Definition 4.4.

Let $S_{1}\subseteq S_{2}$ be two nonempty finite sets. Let
$\lambda_{i}=(A_{1}^{i},A_{2}^{i},\dots,A^{i}_{m_{i}})$ be ordered partitions
of $S_{i}$. We say $\lambda_{1}$ is a subpartition of $\lambda_{2}$, denoted
$\lambda_{1}\subseteq\lambda_{2}$, if there is an increasing function
$f:\\{1,\dots,m_{1}\\}\to\\{1,\dots,m_{2}\\}$ such that $A_{j}^{1}\subseteq
A_{f(j)}^{2}$ for all $j\in\\{1,\dots,m_{1}\\}$. If this is the case we may
form an ordered partition of $S_{3}=S_{1}\setminus S_{2}$ as follows. We
define $B_{i}=A_{i}^{2}-A_{j}^{1}$ if $i=f(j)$ for $j\in\\{1,2\dots,m_{1}\\}$,
otherwise we define $B_{i}=A_{i}$. We define $\lambda_{3}$ to be the partition
corresponding to the tuple $(B_{1},\dots B_{m_{2}})$ after deleting empty
sets.

In what follows we will need to sum over all subpartitions of a given
partition $\lambda$. By $\sum_{\alpha\subseteq\lambda}f(\alpha)$ we mean the
sum of $f(\alpha)$ where $\alpha$ ranges over all subpartitions of $\lambda$.

######  Definition 4.5.

We assume $X$ is a quasi projective variety and $\mathcal{L}$ is a very ample
line bundle on it. We define the _forced positive dimensional singularities_
subset of $w_{\lambda}(X)$ denoted $w_{\lambda}^{p}(X)$ by

$w_{\lambda}^{p}(X)=\\{Z\in w_{\lambda}(X)|\textrm{if }f\in
H^{0}(X,\mathcal{L}),Z\subseteq\textrm{Sing}(f),\textrm{ then
}\dim\textrm{Sing}(f)\geq 1\\}.$

Let $w_{\lambda}^{n}(X)=w_{\lambda}(X)\setminus w_{\lambda}^{p}(X).$

Let $W_{\geq\lambda}^{p}(X)=\\{(f,Z)\in W_{\geq\lambda}(X)|Z\in
w_{\lambda}^{p}(X)\\}.$ Let
$W_{\geq\lambda}^{n}(X)=W_{\geq\lambda}(X)\setminus W_{\geq\lambda}^{p}(X).$
Let $W_{\lambda,\geq N}^{p}(X)=\\{(f,Z)\in W_{\lambda,\geq N}(X)|Z\in
w_{\lambda}^{p}(X)\\}.$ Let $W_{\lambda,\geq N}^{n}(X)=W_{\lambda,\geq
N}(X)\setminus W_{\lambda,\geq N}^{p}(X).$

######  Proposition 4.6.

Let $X,\lambda$ be as above. The set $w_{\lambda}^{p}(X)$ is a constructible
subset of $w_{\lambda}(X).$

###### Proof.

Let $j$ be a positive integer. Let

$w_{\lambda,j}(X)=\\{(Z,f_{1},\dots,f_{j})|Z\in w_{\lambda}(X),f_{i}\in
H^{0}(X,\mathcal{L}),Z\subseteq\textrm{Sing}(f_{i})\\}.$

There is a natural forgetful map $w_{\lambda,j}(X)\to w_{\lambda}(X)$. Let

$w_{\lambda,j}^{p}(X)=\\{(Z,f_{1},\dots f_{j})|f_{i}\textrm{form a basis for
}H^{0}(X,I^{2}_{Z}\otimes\mathcal{L})\\}.$

Clearly $w_{\lambda}^{p}(X)$ is the union of the images of
$w_{\lambda,j}^{p}(X)$. Hence it suffices to establish that the
$w_{\lambda,j}^{p}(X)$ are constructible.

Let $E_{\lambda,j}(X)=\\{(P,Z,f_{1},\dots,f_{j})\in X\times
w_{\lambda^{j}}(X)|P\in\cap\textrm{Sing}(f_{i})\\}.$ There is an obvious
forgetful map $\pi:E_{\lambda_{j}}(X)\to w_{\lambda,j}(X)$. Consider a
flattening stratification of $S_{1},\dots S_{k}$ $w_{\lambda,j}$ with respect
to $\pi.$ We note that $(Z,f_{1},\dots,f_{j})\in w_{\lambda,j}^{p}(X)$ iff the
$f_{i}$ form a basis of $H^{0}(X,I^{2}_{Z}\otimes\mathcal{L})$ and if
$\pi^{-1}((Z,f_{1},\dots,f_{j}))$ is positive dimensional. However the
dimension of $\pi^{-1}((Z,f_{1},\dots,f_{j}))$ is constant on each stratum.
Hence $w_{\lambda,j}^{p}(X)$ is just the union of a collection of strata,
intersected with the set of $(Z,f_{1},\dots,f_{j})$ where the $f_{i}$ form a
basis of $H^{0}(X,I^{2}_{Z}\otimes\mathcal{L})$. However the latter is clearly
a constructible set as is each stratum. Hence $w_{\lambda,j}^{p}(X)$ is the
union of an intersection of constructible sets and is hence constructible. ∎

######  Remark 4.

The p and n in the notation $w_{\lambda}^{p},w_{\lambda^{n}}$ is intended to
mean positive singularities and not positive dimensional singularities. These
variants of the $w_{\lambda}$ are largely used in the final section of this
paper and are necessary because we will often want to disregard elements of
$w_{\lambda}^{p}$ for various technical reasons. The necessity of using the
$w_{\lambda}^{n}$ is almost exclusive to Proposition 6.16 and the reader is
advised to not pay too much attention to it at a first reading.

######  Proposition 4.7.

Let $N$ be a positive integer. The following equations are true, where both
sides are treated as part of $\mathcal{M}_{\mathbb{L}}$

$W_{\varnothing}=\sum_{|\lambda|\leq
N}(-1)^{||\lambda||}W_{\geq\lambda}-\sum_{|\lambda|\leq
N}(-1)^{||\lambda||}W_{\lambda,\geq N+1}.$

$W_{\varnothing}=\sum_{|\lambda|\leq
N}(-1)^{||\lambda||}W^{n}_{\geq\lambda}-\sum_{|\lambda|\leq
N}(-1)^{||\lambda||}W^{n}_{\lambda,\geq N+1}.$

###### Proof.

The first equation follows from the proof of Proposition 3.7 in [16]. The
second follows immediately from the first and the fact that

$W_{\geq\lambda}-W_{\lambda,\geq
N-|\lambda|+1}=W^{n}_{\geq\lambda}-W^{n}_{\lambda,\geq N+1}.$

∎

######  Proposition 4.8.

Let $Z$ be a constructible subset of $w_{[n]}(X)$. Let
$\pi_{\lambda}:w_{\lambda}(X)\to w_{[|\lambda|]}(X)$ be the covering map. Let
$\phi:\mathcal{M}\to\mathcal{M}_{Hdg}$ denote the specialisation homomorphism.
Then

$\phi(\sum_{|\lambda|=n}(-1)^{||\lambda||}(\pi_{\lambda}^{-1}(Z)))=H^{*}_{c}(Z,\pm\mathbb{Q})$

, where $\pm\mathbb{Q}$ is the restriction of the alternating sheaf on
$w_{[n]}(X)$.

Also, for $k>3$,
$H^{*}_{c}(w_{k}(\mathbb{P}^{1}),\pm\mathbb{Q})=H^{*}_{c}(w_{k}(\mathbb{A}^{1}),\pm\mathbb{Q})=0.$

###### Proof.

Let $S_{\lambda}$ be the local system $(\pi_{\lambda})_{*}\mathbb{Q}$. Then by
[3] $\sum_{|\lambda|=n}(-1)^{||\lambda||}S_{\lambda}=\pm\mathbb{Q}$ as a
virtual representation. Thus,
$\sum_{|\lambda|=n}(-1)^{||\lambda||}H^{*}_{c}(Z,S_{\lambda})=H^{*}_{c}(Z,\pm\mathbb{Q})$.
However
$H^{*}_{c}(Z,S_{\lambda})=H^{*}_{c}(\pi_{\lambda}^{-1}(Z),\mathbb{Q})$. This
completes the proof. The equality
$H^{*}_{c}(w_{k}(\mathbb{P}^{1}),\pm\mathbb{Q})=H^{*}(w_{k}(\mathbb{A}^{1}),\pm\mathbb{Q})=0,$
is established in Lemma 2 of [17]. ∎

We define

$w_{\lambda}^{l}(X)=\\{Z\in
w_{\lambda}(X)|h^{0}(X,I^{2}_{Z}(d))+h^{0}(N(Z),\mathcal{O}(d))=l+h^{0}(X,\mathcal{O}(d))\\}.$

We note that the $w_{\lambda}^{l}(X)$ are constructible, this follows
immediately from the fact that $\\{Z\in
w_{\lambda}(X)|h^{0}(X,I^{2}_{Z}(d))+h^{0}(N(Z),\mathcal{O}(d))\geq
l+h^{0}(X,\mathcal{O}(d))\\}$ is closed, which is a consequence of
semicontinuity of cohomology.

######  Proposition 4.9.

Let $X$ be a smooth projective variety of dimension $n$. Let $\lambda$ be an
ordered partition. Then,

$[W_{\geq\lambda}(X)]=H^{0}(X,\mathcal{L})\sum_{l=0}^{\infty}[w_{\lambda}^{l}(X)]\mathbb{L}^{l-|\lambda|(n+1)}.$

###### Proof.

Let $\pi:W_{\lambda}(X)\to w_{\lambda}(X)$ be the projection map. We note that
$w_{\lambda}(X)$ is the disjoint union of $w_{\lambda}^{l}(X)$. Therefore
$W_{\lambda}(X)$ is the disjoint union of $\pi^{-1}(w_{\lambda}^{l}(X)),$ and
hence $[W_{\lambda}(X)]=\sum_{l=0}^{\infty}[\pi^{-1}(w_{\lambda}^{l}(X))].$

The map $\pi:\pi^{-1}(w_{\lambda}^{l}(X))\to w_{\lambda}^{l}(X)$ is a vector
bundle of dimension $h^{0}(X,\mathcal{L})-|\lambda|(n+1)+l$ by Grauert’s
theorem (see for instance p. 288 Cor. 12.9 [9] ). Hence

$[\pi^{-1}(w_{\lambda}^{l}(X))]=H^{0}(X,\mathcal{L})[w_{\lambda}^{l}(X)]\mathbb{L}^{l-|\lambda|(n+1)}.$

∎

######  Proposition 4.10.

Let $X$ be a smooth projective variety of dimension $n$. Let $C\subseteq X$ be
a reduced curve of degree $e$. Let $Z\subseteq C$ be a reduced set of points.
Let $d\gg\deg C$. Then

$h^{1}(X,I^{2}_{Z}(d))\leq\max(2|Z|-ed,0)+(n-1)\max(|Z|-ed,0)-k.$

###### Proof.

This follows immediately from Corollary 1.1.25 in [10]. ∎

Let $\psi_{e,d}(l)$ denote the minimum $m$ such that
$l\geq\max(2m-ed,0)+(n-1)\max(m-ed,0)-k.$

######  Proposition 4.11.

Let $X$ be a smooth projective variety of dimension $n\geq 2$. Let $l\geq 1$.
Let $d\gg e>0$. Let $k$ be as in Proposition 4.10.

Let $\lambda$ be an ordered partition such that $e\geq|\lambda|/d$. Then

$\dim w_{\lambda}^{l}(X)\leq\max_{e>0}(|\lambda|(n)-(n-1)\psi_{e,d}(l))+K$

where $K=\max_{r\leq e}\dim Chow_{r}(X)$ where $Chow_{r}(X)$ is the Chow
variety of reduced curves of degree $r$ in $X$. Let $r_{0}$ denote the minimal
degree of any curve in $X$.

Let $(nr_{0}+1)d>N>r_{0}nd$ Then, there exists $C$ such that

$\dim W_{\lambda,\geq N}^{l}(X)\leq h^{0}(X,\mathcal{O}(d))-r_{0}nd+K.$

###### Proof.

By Lemma 3.1 any $Z\in w_{\lambda}^{l}(X)$ must be of the form $Z^{\prime}\cup
Z^{\prime\prime}$ where $Z^{\prime}=Z\cap C$ for some curve $C$ of degree
$e^{\prime}\leq e$. Furthermore
$l=h^{1}(X,I^{2}_{Z}(d))=h^{1}(X,I^{2}_{Z^{\prime}}(d))$. By Proposition 4.10,
this implies that $|Z^{\prime}|\geq\psi_{e,d}(l)$.

As a result,

$\dim
w_{\lambda}^{l}(X)\leq\max_{|\lambda|/d>e>0}(|\lambda|(n)-(n-1)\psi_{e,d}(l))+\max_{|\lambda|/d>e>0}\dim
Chow(e).$

∎

We now prove Theorem 1.6, The essential idea is as follows- we use the
techniques of [16] to argue that the difference between
$\frac{U(X,\mathcal{L})}{H^{0}(X,\mathcal{L})}-\zeta^{-1}(\dim X+1)$ is a
certain sum of contributions corresponding to configurations of points
contained in $w_{\lambda}^{l}(X)$ where $l\geq 1$. We then use Proposition
4.11 to bound this contribution.

###### Proof of Theorem 1.6.

By Proposition 4.7

$U(X,\mathcal{L})=W_{\varnothing}(X)=\sum(-1)^{||\lambda||}(W_{\geq\lambda}(X)-W_{\lambda,\geq
N+1}).$

By Proposition 4.11 the terms $W_{\lambda,\geq N+1})=0$ up to dimension
$-m(d)(\dim X)+C$. Also for $||\lambda||\leq N,l\geq 1$, $w_{\lambda}^{l}$
lies in $F$. Hence we may apply Proposition and conclude that
$W_{\geq\lambda}=w_{\lambda}\mathbb{L}^{-|\lambda|(n+1)}$ upto dimension
$-m(d)(\dim X)+C$.

This implies that
$W_{\varnothing}(X)=\sum_{\lambda}(-1)^{||\lambda||}w_{\lambda}(X)\mathbb{L}$,
up to dimension $-m(d)(\dim X)+C$.. But by Proposition 3.7 of [16] the right
hand side of this last equation equal $Z_{X}^{-1}(\dim X+1)$.

∎

##  5 Proof of Theorem 1.7

In this section we will prove Theorem 1.7. The proof of Theorem 1.7 is similar
to that of Theorem 1.6 in that we use Proposition 4.7 and provide bounds on
the dimensions of $w^{l}_{\lambda}(X)$. However our assumptions are different.
This section is strictly speaking independent of Sections 2 and 3 , and we do
not use any results on Cayley-Bacharach sets.

We define the projective hull of a subscheme $Y\subseteq\mathbb{P}^{n}$,
denoted $phull(Y)$ to be the intersection of all projective subspaces of
$\mathbb{P}^{n}$ containing $Y$.

######  Proposition 5.1.

Let $n,m\geq 0$. Let $Z\subseteq\mathbb{P}^{n}$ be a reduced zero dimensional
subscheme. SUppose $|Z|\geq m+2$. Suppose that $phull(Z)$ in $\mathbb{P}^{n}$
is $m$ dimensional. Let $d\geq 3$. Then the kernel of the restriction map
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(N(Z),\mathcal{O}(d))$ is of
codimension $\geq(m+1)(n+1)+n-1$.

###### Proof.

We claim that it suffices to prove the Proposition in the special case when
$Z=\\{P_{0},\dots,P_{m},P\\}$ where the projective hull of the $P_{i}$ is $m$
dimensional, and $P$ is in $phull(\\{P_{0},\dots,P_{m}\\}).$ To see this,
first note that if $Z^{\prime}\subseteq Z$ the kernel of
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(N(Z),\mathcal{O}(d))$ is
contained in the kernel of $H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(N(Z^{\prime}),\mathcal{O}(d)).$ As a result, to establish the
Proposition for $Z$, it suffices to establish it for any subset
$Z^{\prime}\subseteq Z$. We then note that given any $Z$ whose projective hull
is $m$ dimensional, we can find a $Z_{ind}\subseteq Z$ such that
$|Z_{ind}|=m+1$ and $phull(Z_{ind})=phull(Z)$. We now consider any point $P\in
Z\setminus Z_{ind}$ and note that $Z^{\prime}=Z_{ind}\cup\\{P\\}$ is a subset
of $Z$ satisfying the conditions of the claim and that to prove the
Proposition for $Z$ it suffices to prove it for $Z^{\prime}$.

In light of the above paragraph, we now assume $Z=\\{P_{0},\dots,P_{m},P\\},$
where $phull(P_{0},\dots,P_{m})$ is $m$ dimensional and contains $P$. We may
further assume that $P_{i}=[0:0\dots:0:1:0\dots 0],$ here the $1$ is in the
$i+1^{st}$ position and $P=[x_{0}:\dots x_{m}:0\dots:0].$ This is because
there is an element $g\in PGL_{n+1}(\mathbb{F})$ satisfying
$g(P_{i})=[0:0\dots:0:1:0\dots 0].$

We now claim that it suffices to prove the proposition in the case where
$P=[1:x:0\dots,:0]$. to see this, note that we may assume without loss of
generality that if $P=[x_{0}:x_{1}:\dots,x_{m}:0\dots:0],$ $x_{0}\neq 0$ and
$x_{1}\neq 0$. For $t\in\mathbb{A}^{1}$, let $Z_{t}$ be the reduced zero
dimensional subscheme $\\{P_{1},\dots P_{m}.P_{t}\\}$, where
$P_{t}=[x_{0}:x_{1}:tx_{2},\dots tx_{m}:0\dots 0].$ The $Z_{t}$ form a family
of subschemes over $\mathbb{A}^{1},$ with the original subscheme $Z$ being
equal to $Z_{1}$. Furthermore, given $t\neq 0$, there is a $g\in
PGL_{n+1}(\mathbb{F})$ such that $g(Z_{t})=Z_{1}$. This has the consequence
that the dimension of the kernel of the restriction of
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(N(Z_{t}),\mathcal{O}(d))$ is
independent of $t$ as long as $t\neq 0$. The kernel of
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(N(Z_{t}),\mathcal{O}(d))$ is
isomorphic to $H^{0}(\mathbb{P}^{n},I^{2}_{Z_{t}}(d)).$ Semicontinuity now
implies that $h^{0}(\mathbb{P}^{n},I^{2}_{Z_{0}}(d))\geq
h^{0}(\mathbb{P}^{n},I^{2}_{Z_{1}}(d)).$ As a result it suffices to prove the
proposition for $Z_{0}$. However $Z_{0}$ is of the required form.

To complete the proof, we directly compute the bound on the codimension of the
kernel when $Z$ is of the form $\\{P_{0},\dots,P_{m},P\\}$ where
$P_{i}=[0:0\dots:0:1:0\dots 0],$ and $P=[1:x:0\dots:0]$. Let $f$ be a section
of $H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$, we may think of $f$ as a homogenous
polynomial. Then requiring $f$ to vanish to second order at $P_{i}$ implies
that the coefficients of $X_{i}^{d}$ and $X_{j}X_{i}^{d-1}$ are zero. Let
$a_{k,r}$ be the coefficient to $X_{k}X_{1}^{r}X_{0}{d-1-r}$. Requiring that
the partial derivatives of $f$ vanish at $P$ is equivalent to requiring that
for $k\in\\{2,\dots n\\}$,

$\sum_{r=0}^{d}x^{r}a_{k,r}=0.$

We note that the above linear relations are linearly independent and hence the
kernel of $H^{0}(\mathbb{P}^{,}\mathcal{O}(d))\to H^{0}(N(Z),\mathcal{O}(d))$
is of codimension at least $(m+1)(n+1)+n-1$.

∎

The above proof gives us a little more information than the statement of
Proposition 5.1. We record this information in the following Proposition.

######  Proposition 5.2.

Let $m,n\geq 1$. Let $d\geq 3$ Let $P_{i}=[0:\dots
1::0\dots:0]\in\mathbb{P}^{n}$, where the $1$ is in the $i+1^{th}$ position.
Let $P\in phull\\{P_{0},\dots,P_{m}\\}$ Let $V_{i}^{j}$ be any basis of
$T_{P_{i}}(\mathbb{P}^{n})$. Let $V^{j}$ be a basis of $T_{P}\mathbb{P}^{n}$.

Then requiring a polynomial $f\in H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$ to
satisfy

$f(P_{i})=0,\frac{\partial}{\partial
V_{i}^{j}}(f)(P_{i})=0,\frac{\partial}{\partial v^{j}}f(p)=0,$

imposes $(m+1)(n+1)+n-1$ linearly independent conditions on $f$.

###### Proof.

We omit the proof as this is already present in the proof of Proposition 5.1.
∎

######  Proposition 5.3.

Let $n,m\geq 0$. Let $Z\subseteq X\subseteq\mathbb{P}^{n}$ be a reduced set of
points, $m+1<|Z|<\dim X$. Let $d\geq 3$. Let $\dim phull(Z)=m$. Then the
kernel of the restriction map $H^{0}(X,\mathcal{O}(d))\to
H^{0}(N(Z),\mathcal{O}(d))$ is of codimension $\geq(m+1)(\dim X+1)+\dim X-1$.

###### Proof.

This proof is similar to that of Proposition 5.1 . It suffices to prove the
proposition in the case where $Z=\\{P_{0},\dots,P_{m},P\\}$ where
$P_{i}=[0:\dots:0:1:\dots:0]$ where there is a $1$ in the $i+1$th position.
This is by an argument identical to that in Proposition 5.1.

Let $V_{i}^{j}$ be a basis of $T_{P_{i}}X$. Let $v^{j}$ be a basis of
$T_{P}X$.It suffices to prove that requiring a polynomial $f$ to satisfy at
$f(P_{i})=0,$ for all $i$, $f(P_{i})=0$ and that
$\partial_{v_{i}^{j}}(f)(P_{i})$, $\partial_{v_{i}}(f)(P)$ impose linearly
independent conditions on $f$. However, this follows from Proposition 5.2,
since in that proof we show that a larger set of linear equations impose
linearly independent conditions on $f$. ∎

######  Proposition 5.4.

Let $n\geq 0$. Let $X$ be a smooth projective variety of dimension $n$. Let
$\lambda$ be an ordered partition, $|\lambda|\leq\sqrt{n}/3$. Let $l\geq 1$.
Then

$\dim w_{\lambda}^{l}(X)+l-(n+1)(|\lambda|)\leq-\sqrt{n}/{3}$

and

$\dim W_{\lambda,\geq N}^{l}(X)\leq h^{0}(X,\mathcal{O}(d))-\sqrt{n}/{3}.$

In addition we have $\dim W_{\lambda,\geq N}^{0}(X)\leq
h^{0}(X,\mathcal{O}(d))-\sqrt{n}/{3}.$

###### Proof.

Let $l\geq 1$. Let $Z\in w_{\lambda}^{l}(X)$. Then by definition,
$h^{0}(X,I^{2}_{Z}(d))=h^{0}(X,\mathcal{O}(d))-|Z|(\dim X+1)+l$. Let $\dim
phull(Z)=m$. Then by Proposition 5.3, $|Z|(\dim X+1)-l\geq(m+1)(\dim X+1)-l$.
But a simple dimension count shows that the space of $|\lambda|$ points in $X$
contained in an $m$ dimensional subspace is of dimension $\leq(m+1)\dim
X+(|\lambda|-m-1)m$.

This implies that $\dim w_{\lambda}^{l}(X)\leq\max_{m}(m+1)\dim
X+(|\lambda|-m-1)m.$

We will now bound the quantity $(m+1)\dim X+(|\lambda|-m-1)m+l-(\dim
X+1)|\lambda|$. By the above we have

$(m+1)\dim X+(|\lambda|-m-1)m+l-(\dim X+1)|\lambda|$

$\leq(\dim X+1)|\lambda|-l-\dim X+1+(|\lambda|-m-1)m+l-(\dim X+1)|\lambda|$

$=-\dim X+1+(|\lambda|-m-1)m\leq-\dim X+1+(|\lambda|)^{2}\leq-\dim
X+1+\frac{\dim X}{9}$

$\leq-\frac{8}{9}\dim X+1\leq-\frac{\sqrt{\dim X}}{3}.$

As a result

$\dim w_{\lambda}^{l}(X)+l-(n+1)(|\lambda|)\leq-\sqrt{n}/{3}$

. The inequality

$\dim W_{\lambda,\geq N+1}^{l}(X)\leq h^{0}(X,\mathcal{O}(d))-\sqrt{n}/{3}$

follows similarly to the above for $l\geq 1$ and is immediate if $l=0$.

This implies the result. ∎

We note here that the proof of the above Proposition gives us a better bound
on the dimension of $w_{\lambda}^{l}(X)$ than we claimed in the result.
However we are unable to utilise this improved bound. The sticking factor is
that we only have the bound $\dim W_{\lambda,\geq N}^{0}(X)\leq
h^{0}(X,\mathcal{O}(d))-N$, which we believe to be tight.

###### Proof of Theorem 1.7.

Let $N=\sqrt{n}/3$. By Proposition 4.7

$\frac{W_{\varnothing}(X)}{H^{0}(X,\mathcal{O}(d))}=\sum_{|\lambda|\leq
N}(-1)^{||\lambda||}\frac{W_{\geq\lambda}(X)}{H^{0}(X,\mathcal{O}(d))}-\frac{W_{\lambda,\geq
N+1}(X)}{H^{0}(X,\mathcal{O}(d)}.$

By Proposition 4.7 the right hand side equals

$\sum_{|\lambda|\leq N}w_{\lambda}^{0}(X)\mathbb{L}^{-|\lambda|(\dim
X+1)}+\sum_{|\lambda|\leq N}w_{\lambda}^{l}(X)(\mathbb{L}^{-|\lambda|)(\dim
X+1)+l}-\sum_{|\lambda|\leq N}\frac{W_{\lambda,\geq
N+1}(X)}{H^{0}(X,\mathcal{O}(d))}.$

By Proposition 4.7 the first term equals $Z_{X}^{-1}(\dim
X+1)-\sum_{|\lambda|\leq N}w_{\lambda}^{l}(X)\mathbb{L}^{-|\lambda|(\dim
X+1)}-\sum_{|\lambda|>N}(-1)^{||\lambda||}w_{|\lambda}(X)\mathbb{L}^{-|\lambda|(\dim
X+1)}=Z_{X}^{-1}(\dim X+1)$ up to dimension $-N$. By Proposition 5.4 the
latter two terms are of dimension $\leq-N$ which implies the result. ∎

##  6 Proof of Theorem 1.9

In this section we prove Theorem 1.9 . The proof of Theorem 1.9 is similar to
that of Theorems 1.6 and 1.7 , albeit a bit more involved.

In this section we will exclusively work with $X=\mathbb{P}^{n}$. As a result
we will drop the $\mathbb{P}^{n}$ from our notation, i.e. we will refer to
$w_{\lambda}(\mathbb{P}^{n})$ as $w_{\lambda}$,
$W_{\geq\lambda}(\mathbb{P}^{n})=W_{\geq\lambda}$ etc. However if the argument
of $w_{\lambda},$ is not $\mathbb{P}^{n}$ we will explicitly write down what
the space is.

The following basic proposition will be used repeatedly in this section.

######  Proposition 6.1.

Let $C$ be a smooth curve in $\mathbb{P}^{n}$. Then for $d\gg\deg C$,

$h^{0}(N(C),\mathcal{O}(d))=h^{0}(C,\mathcal{O}(d))+h^{0}(C,N_{C}^{*}(d)).$

###### Proof.

Let $I$ denote the ideal sheaf of $C$. Consider the following chain of
inclusions of sheaves on $\mathbb{P}^{n}$-

$0\subseteq I^{2}(d)\subseteq I(d)\subseteq\mathcal{O}(d).$

We will break the above exact sequence into short exact sequences. We have a
short exact sequence

$0\to I^{2}(d)\to I(d)\to I/I^{2}(d)\to 0.$

Under the assumption that $d\gg\deg(C)$, Serre vanishing implies that
$H^{1}(\mathbb{P}^{n},I^{2}(d))=0$. As a result,

$h^{0}(\mathbb{P}^{n},I(d))=h^{0}(\mathbb{P}^{n},I^{2}(d))+h^{0}(\mathbb{P}^{n},I/I^{2}(d)).$

We also have the short exact sequence $0\to
I(d)\to\mathcal{O}(d)\to\mathcal{O}/I(d).$ Since $d\gg\deg(C)$,

$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))=h^{0}(\mathbb{P}^{n},I(d))+h^{0}(\mathbb{P}^{n},\mathcal{O}/I(d)).$

Finally we have the short exact sequence $0\to
I^{2}(d)\to\mathcal{O}(d)\to\mathcal{O}/I^{2}(d).$ Since $d\gg\deg(C)$,

$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))=h^{0}(\mathbb{P}^{n},I^{2}(d))+h^{0}(\mathbb{P}^{n},\mathcal{O}/I^{2}(d)).$

As a result, we have

$h^{0}(\mathbb{P}^{n},\mathcal{O}(d)/I^{2}(d))=h^{0}(\mathbb{P}^{n},\mathcal{O}/I(d))+h^{0}(\mathbb{P}^{n},I/I^{2}(d)).$

But we now note that $\mathcal{O}/I^{2}$ is the push forward of the structure
sheaf of $N(C)$ and $\mathcal{O}/I$ is the pushforward of the structure sheaf
of $C$. As a result

$h^{0}(\mathbb{P}^{n},\mathcal{O}(d)/I^{2}(d))=h^{0}(N(C),\mathcal{O}(d))$

and

$h^{0}(\mathbb{P}^{n},\mathcal{O}(d)/I(d))=h^{0}(C,\mathcal{O}(d)).$

Finally, we note that $I/I^{2}$ is the pushforward of the conormal sheaf of
$C$ and as a result $h^{0}(\mathbb{P}^{n},I/I^{2}(d))=h^{0}(C,N^{*}(d))$. This
gives us the required result.

∎

######  Definition 6.2.

Let $\textrm{Chow}(r)$ be the Chow variety of degree $d$ curves in
$\mathbb{P}^{n}$. Let $M\subseteq\mathbb{P}^{n}$ be a constructible subset of
$Chow(r)$. Let $\lambda$ be an ordered partition. We define
$w_{\lambda}^{M,l}\subseteq w_{\lambda}^{l}$ as consisting of all $Z\in
w_{\lambda}$ such that there exists $C\in M$ satisfying

$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap C}(d))$

and furthermore there is no curve $C^{\prime}$ of degree less than $r$ such
that

$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap
C^{\prime}}(d)).$

######  Proposition 6.3.

Let $d,r,l\geq 0$. Let $\lambda$ be an ordered partition. Let $M$ be a
constructible subset of $Chow(r)$. Assume $d\gg r.$ The subset
$w_{\lambda}^{M,l}$ is constructible and hence defines an element in
$\mathcal{M}$.

###### Proof.

Let

$\tilde{w}_{\lambda}^{M,l}:=\\{(Z,C)\in w_{\lambda}^{M,l}\times
M|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap
C}(d))\\}.$

We note that since $d\gg r$, by Proposition 3.1 for any $Z\in
w_{\lambda}^{M,l}$, there exists a unique $(Z,C)\in\tilde{w}_{\lambda}^{M,l}$
mapping onto it under the projection, thus it suffices to establish that
$\tilde{w}_{\lambda}^{M,l}$ is constructible. Consider $X=w_{\lambda}\times
M\times\mathbb{P}^{n}$. The variety $X$ has two ideal sheaves on it
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ corresponding to the subvarieties
$E_{1}=\\{(Z,C,P)\in\tilde{w}_{\lambda}^{M,l}\times\mathbb{P}^{n}|P\in Z\\}$
and $E_{2}=\tilde{w}_{\lambda}^{M,l}\times\mathbb{P}^{n}|P\in Z\cap C\\}$. Let
$\pi:\tilde{w}_{\lambda}^{M,l}\times\mathbb{P}^{n}\to\tilde{w}_{\lambda}^{M,l}$
denote the projection. Now we may consider a stratification $S_{i}$ of
$\tilde{w}_{\lambda}^{M,l}$ such that the sheaves $\pi_{*}\mathcal{F}_{i}(d)$
are flat on each stratum. The semicontinuity theorem then implies that one may
further refine the stratification $S_{i}$ to a stratification $S_{i}^{\prime}$
on which $h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap C}(d))$ and
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ are constant. We then note that
$\tilde{w}_{\lambda}^{M,l}$ is precisely the subset of $X$ where
$h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap C}(d))$ and
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ are equal to $l$. Thus
$\tilde{w}_{\lambda}^{M,l}$ is a union of strata and is thus constructible. ∎

Let $\pi:W_{\geq\lambda}\to w_{\lambda}$ denote the natural projection. Let
$W_{\geq\lambda}^{M,l}(\mathbb{P}^{n})=\pi^{-1}(w_{\lambda}^{M,l})$. Let
$W_{\lambda}^{M,l}=W_{\geq\lambda}^{M,l}\cap W_{\lambda}$. We define

$\bar{w}_{\lambda}^{M,l}=\\{Z\in w_{\lambda}|\exists C\in
M,h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap C}(d))=l\\}.$

We note that

$w_{\lambda}^{M,l}\subseteq\bar{w}_{\lambda}^{M,l}.$

We start by decomposing the varieties $w_{\lambda}^{l}$ into pieces.

######  Proposition 6.4.

Let $l\geq 1$.

$w_{\lambda}^{l}=\sum_{r=1}^{\infty}w_{\lambda}^{\textrm{Chow}(r),l}.$

###### Proof.

We note that the sum is finite, for $r$ sufficiently large
$w_{\lambda}^{\textrm{Chow}(r),l}$ is empty.

It then suffices to prove that the collection
$w_{\lambda}^{\textrm{Chow}(r),l}$ partitions $w_{\lambda}^{l}$. This follows
from Proposition 3.1. ∎

######  Proposition 6.5.

Let $e\geq 1$. Let $\mathcal{A}$ be a finite collection of constructible
subsets defining a partition of $\textrm{Chow}(e).$ Then
$w_{\lambda}^{\textrm{Chow}(e),l}=\sum_{M\in\mathcal{A}}w_{\lambda}^{M,l}.$

###### Proof.

This follows from the fact that a partition of $\textrm{Chow}(e)$ induces a
partition of $w_{\lambda}^{\textrm{Chow}(e),l}$. ∎

We now prove a dimension bound on several of the $w_{\lambda}^{M,l}$.

######  Proposition 6.6.

Let $e\geq 4$. Let $M\subseteq\textrm{Chow}(e)$ be a constructible set. Let
$n\geq 2,l\geq 1$. Then there exists $\epsilon>0$ such that,

$\dim(W_{\geq\lambda}^{M,l}-W^{M,l}_{\geq\lambda,\geq N+1})\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d.$

Furthermore $\dim
w_{\lambda}^{M,l}\leq|\lambda|(n)-l-(\frac{3n}{2}+\epsilon)d$.

###### Proof.

Let $M\subseteq\textrm{Chow}(e)$. We will first prove that $\dim
w_{\lambda}^{M,l}\leq|\lambda|(n)-l-(\frac{3n}{2}+\epsilon)d$. We note that
$\dim w_{\lambda}^{M,l}\leq\dim\tilde{w}_{\lambda}^{M,l}$. By Proposition 4.10
any $(Z,C)\in w_{\lambda}^{M,l}$ must have $|Z\cap C|\geq\psi_{e,d}(l)$.

Thus

$\dim w_{\lambda}^{M,l}\leq\dim\tilde{w}_{\lambda}^{M,l}\leq\dim
M+(|\lambda|-\psi_{e,d}(l))(n)+\psi_{e,d}(l).$

An inspection of the function $\psi_{e,d}$ then implies that for $e\geq 4$ and
$d\gg 0$, $\dim w_{\lambda}^{M,l}\leq|\lambda|(n)-l-(\frac{3n}{2}+\epsilon)d.$
Since
$W_{\geq\lambda,l}^{M}=w_{\lambda}^{M,l}H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\mathbb{L}^{-|\lambda|(n+1)+l}$,
the result follows.

∎

######  Proposition 6.7.

Let $d_{0}\geq 1$. Let $M\subseteq Chow(d_{0})$ be a constructible set. Then
$w_{\lambda}^{M,l}\subseteq\bar{w}_{\lambda}^{M,l}$. Furthermore
$\bar{w}_{\lambda}^{M,l}\setminus
w_{\lambda}^{M,l}=\cup_{d>d_{0},l^{\prime}>l}w_{\lambda}^{M^{\prime}(d),l^{\prime}},$
where $M^{\prime}(d)$ is the subset of $Chow(d)$ consisting of reducible
curves that contain a curve in $M$.

###### Proof.

This follows from Lemma 3.1. ∎

######  Proposition 6.8.

Let $X$ and $Y$ be algebraic varieties over $\mathbb{F}$. Then
$w_{\lambda}(X\amalg
Y)=\sum_{\alpha\subseteq\lambda}w_{\alpha}(X)w_{\lambda\setminus\alpha}(Y).$

###### Proof.

This follows immediately from the fact that $w_{\lambda}(X\amalg Y)$ is
partitioned into subsets isomorphic to $w_{\alpha}(X)\times
w_{\lambda\setminus\alpha}(Y)$ as $\alpha$ ranges over all subpartitions of
$\lambda$. ∎

######  Proposition 6.9.

Let $X$ be an algebraic variety. Let $Y$ be a subvariety of $X$. Suppose
$[X],[Y]$ are polynomials in $\mathbb{L}$. Then for any $\lambda$,
$w_{\lambda}(X),w_{\lambda}(Y)$ and $w_{\lambda}(X\setminus Y)$ are
polynomials in $\mathbb{L}$.

###### Proof.

For $X=\mathbb{A}^{n}$,
$[\textrm{Sym}^{\lambda}(\mathbb{A}^{n})]=\mathbb{L}^{|\lambda|n}$. Let $X$
and $Y$ be such that $Y\subseteq X$ and $w_{\lambda}(X)$, $w_{\lambda}(Y)$ can
always be expressed as a polynomial in $\mathbb{L}$.

We claim that if $U=X\setminus Y$ then $w_{\lambda}(U)$ can also be expressed
as a polynomial in $\mathbb{L}$ for all $\lambda$. Suppose for the sake of
contradiction that $\lambda_{0}$ is an ordered partition such that
$w_{\lambda_{0}}(U)$ is not a polynomial in $\mathbb{L}$ and furthermore for
all $\lambda^{\prime}\subseteq\lambda_{0}$, $w_{\lambda^{\prime}}(U)$ is a
polynomial in $\mathbb{L}$. Then by Proposition 6.8,
$w_{\lambda_{0}}(U)=w_{\lambda_{0}}(X)-\sum_{\lambda^{\prime}\subsetneq\lambda_{0}}w_{\lambda^{\prime}}(U)w_{\lambda_{0}\setminus\lambda^{\prime}}(Y)$
which is a polynomial in $\mathbb{L}$.

Hence the class of varieties such that $w_{\lambda}(X)$ is always a polynomial
contains $\mathbb{A}^{n}$ and is closed under differences and unions. This
completes the proof of the proposition.

∎

In the next proposition we describe the possible reduced $0$\- dimensional
subschemes $Z$ contained in a curve $C$ of low degree ($\leq 3$) such that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=l>0$.

######  Proposition 6.10.

  1. 1.

Let $T\subseteq\mathbb{P}^{n}$ be a smooth rational normal curve of degree
$r\leq 3$. Let $Z\subseteq T$ be a reduced zero dimensional subscheme. Let
$d\gg r$. Then, $h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ depends only on $|Z|$.
Furthermore,
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-\max(2|Z|-rd,0)|+(n-1)\max(|Z|-d,0)\leq
K$ for some constant $K$.

  2. 2.

Let $d\gg e$. Let $C_{1},C_{2}$ be disjoint reduced curves of degree $\leq e$.
Let $Z=Z_{1}\amalg Z_{2}$, with $Z_{i}\subseteq C_{i}$. Then
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z_{1}}(d))+h^{1}(\mathbb{P}^{n},I^{2}_{Z_{2}}(d))$.

  3. 3.

Let $L_{1},L_{2}$ be two coplanar lines. Let $Z=Z_{1}\amalg Z_{2}$ such that
$Z_{i}\subseteq L_{i}$ and $Z\cap L_{1}\cap L_{2}=\varnothing$. Then
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends on $|Z_{1}|$ and $|Z_{2}|$
and furthermore,
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L_{1}),I^{2}_{Z_{1}}(d))-h^{1}(N(L_{2}),I^{2}_{Z_{2}}(d))|\leq
2(n+1)$.

  4. 4.

Let $L_{1},L_{2}$ be two coplanar lines. Let $\\{p\\}=L_{1}\cap L_{2}$ Let
$Z=Z_{1}\amalg Z_{2}\amalg\\{p\\}$ such that $Z_{i}\subseteq L_{i}$ and
$Z_{i}\cap L_{1}\cap L_{2}=\varnothing$. Then Then
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends on $|Z_{1}|$ and $|Z_{2}|$
and furthermore,

$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L_{1}),I^{2}_{Z_{1}}(d))-h^{1}(N(L_{2}),I^{2}_{Z_{2}}(d))|\leq
4(n+1).$

  5. 5.

Let $L,C$ be a line and a conic intersecting at one point, transversally.Let
$Z=Z_{1}\amalg Z_{2}$ such that $Z_{1}\subseteq L,Z_{2}\subseteq C$ and $Z\cap
L\cap C=\varnothing$. Then $h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends
on $|Z_{1}|$ and $|Z_{2}|$ and furthermore,

$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L),I^{2}_{Z_{1}}(d))-h^{1}(N(C),I^{2}_{Z_{2}}(d))|\leq
2(n+1).$

  6. 6.

Let $L,C$ be a line and a conic intersecting at one point, transversally.Let
$Z=Z_{1}\amalg Z_{2}\amalg\\{p\\}$ such that $Z_{1}\subseteq L,Z_{2}\subseteq
C$ and $Z_{i}\cap L\cap C=\varnothing$. Then
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends on $|Z_{1}|$ and $|Z_{2}|$
and furthermore,
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L),I^{2}_{Z_{1}}(d))-h^{1}(N(C),I^{2}_{Z_{2}}(d))|\leq
4(n+1)$.

  7. 7.

Let $L,C$ be a line and a conic intersecting at one point,
nontransversally.Let $Z=Z_{1}\amalg Z_{2}$ such that $Z_{1}\subseteq
L,Z_{2}\subseteq C$ and $Z\cap L\cap C=\varnothing$.

Then $h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends on $|Z_{1}|$ and
$|Z_{2}|$ and furthermore,

$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L),I^{2}_{Z_{1}}(d))-h^{1}(N(C),I^{2}_{Z_{2}}(d))|\leq
4(n+1).$

  8. 8.

Let $L,C$ be a line and a conic intersecting at one point,
nontransversally.Let $Z=Z_{1}\amalg Z_{2}\amalg\\{p\\}$ such that
$Z_{1}\subseteq L,Z_{2}\subseteq C$ and $Z_{i}\cap L\cap C=\varnothing$. Then
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends on $|Z_{1}|$ and $|Z_{2}|$
and furthermore,
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L),I^{2}_{Z_{1}}(d))-h^{1}(N(C),I^{2}_{Z_{2}}(d))|\leq
8(n+1)$.

  9. 9.

Let $L,C$ be a line and a conic intersecting at two points. Let $Z_{1}=Z\cap
L,Z_{2}=Z\cap C$. Then $h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends on
$|Z_{1}|,|Z_{2}|,|Z_{1}\cap Z_{2}|$ and furthermore,
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L),I^{2}_{Z_{1}}(d))-h^{1}(N(C),I^{2}_{Z_{2}}(d))|\leq
8(n+1)$.

  10. 10.

Let $L_{1},L_{2},L_{3}$ be three lines meeting in the following configuration.
Let $Z\subseteq L_{1}\cup L_{2}\cup L_{3}$. Let $Z_{i}=Z\cap L_{i}$. Then

$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-\sum
h^{1}(N(L_{i}),I^{2}_{Z_{i}}(d))|\leq 12(n+1).$

Furthermore, $h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ depends only on the
configuration of the lines as well as $|Z_{i}|,|Z_{i},\cap Z_{j}|,|Z_{i}\cap
Z_{j}\cap Z_{k}|$.

  11. 11.

Let $E$ be a plane cubic in $\mathbb{P}^{n}$. Let $Z\subseteq\mathbb{P}^{n}$.
Then
$h^{1}(\mathbb{P}^{n},I^{2}_{E}(d))=h^{1}(E,\mathcal{O}(d)-2Z)+h^{1}(E,N^{*}\otimes(\mathcal{O}(d)-Z))$.
Since $E$ is a complete intersection, $N^{*}$ splits and the term
$h^{1}(E,N^{*}\otimes(\mathcal{O}(d)-Z))=0$, if $|Z|<3d-5$. The first term

$h^{1}(\mathcal{O}(d)-2Z)=\begin{cases}0&\text{if}\
2|Z|<3d\text{or}2|Z|=3d\text{and}2Z\neq\mathcal{O}(d)\\\ 1,&\text{if}\
2Z=\mathcal{O}(d)\\\ 2|Z|-3d,&\text{otherwise}\end{cases}$ (4)

###### Proof.

We begin with the proof of (1). By Proposition 6.1,
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(N(T),I^{2}_{Z}(d))=h^{1}(T,\mathcal{O}(d)-2|Z|)+h^{1}(T,N^{*}\otimes(\mathcal{O}(d)-|Z|))$
Here $N^{*}$ is the conormal bundle of $T$ in $\mathbb{P}^{n}$, and $|Z|$ is
the line bundle on $T$ corresponding to $Z$. Since $T$ is isomorphic to
$\mathbb{P}^{1}$, the above two $h^{1}$ terms only depend on $|Z|$ and it is
also easy to see that

$|h^{1}(T,\mathcal{O}(d)-2|Z|)+h^{1}(T,N^{*}\otimes(\mathcal{O}(d)-|Z|))$
$-\max(2|Z|-rd,0)|+(n-1)\max(|Z|-d,0)\leq K$

for some constant $K$.

We move on to the proof of (2). We note that since $d\gg e$, the restriction
map

$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(N(C_{1}\cup
C_{2}),\mathcal{O}(d))$

is surjective (by Serre vanishing). Since $C_{1}$ and $C_{2}$ are disjoint,
$N(C_{1}\cup C_{2})=N(C_{1})\cup N(C_{2})$ and also

$H^{0}(N(C_{1}\cup C_{2}),\mathcal{O}(d))\cong
H^{0}(N(C_{1}),\mathcal{O}(d))\oplus H^{0}(N(C_{2}),\mathcal{O}(d)).$

The map

$H^{0}(N(C_{1}\cup C_{2}),\mathcal{O}(d))\to H^{0}(N(Z),\mathcal{O}(d))$

is then the direct sum of the maps

$H^{0}(N(C_{1}),\mathcal{O}(d))\to H^{0}(N(Z_{1}),\mathcal{O}(d))$

and

$H^{0}(N(C_{2}),\mathcal{O}(d))\to H^{0}(N(Z_{2}),\mathcal{O}(d)).$

We note that $h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ is the codimension of the
image of first map and $h^{1}(\mathbb{P}^{n},I^{2}_{Z_{1}}(d))$ (resp.
$h^{1}(\mathbb{P}^{n},I^{2}_{Z_{2}}(d))$) is the codimension of the second
(resp. third) map. Since codimension is additive over direct sums, we have the
required equality.

We now move on to the proofs of (3) and (4). Since $d\gg 0$,
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(N(L_{1}\cup L_{2}),I^{2}_{Z}(d))$.
We note that we have a map $\pi:N(L_{1})\amalg N(L_{2})\to N(L_{1}\cup
L_{2})$. There is a short exact sequence of sheaves on $N(L_{1}\cup L_{2})$,

$0\to I^{2}_{Z}(d)\to\pi_{*}\pi^{*}I^{2}_{Z}(d)\to\mathcal{F}\to 0,$

where the quotient sheaf $\mathcal{F}$ is supported on $N(p)$, where $p$ is
the intersection point of the two lines. Associated to the above short exact
sequence of sheaves we have a long exact sequence in cohomology, a part of
which is as follows:

$H^{0}(N(L_{1}\cup L_{2}),\mathcal{F})\to H^{1}(N(L_{1}\cup
L_{2}),I^{2}(d))\to H^{1}(N(L_{1}\cup L_{2}),\pi_{*}\pi^{*}I^{2}_{Z}(d))\to
0.$

The fact that the above long exact sequence is exact at the last term is due
to the fact that $H^{1}(N(L_{1}\cup L_{2}),\mathcal{F})=0$ as its support is
$0$ dimensional. Furthermore, $H^{0}(N(L_{1}\cup L_{2}),\mathcal{F})\cong
H^{0}(N(p),I^{2}_{Z\cap p}(d))$. Hence $h^{0}(N(p),I^{2}_{Z\cap p}(d))\leq
n+1$ (it is either $0$ or $n+1$ depending on whether $p\in Z$).

We note that

$H^{1}(N(L_{1}\cup L_{2}),\pi_{*}\pi^{*}I^{2}_{Z}(d))\cong
H^{1}(N(L_{1}),I^{2}_{Z\cap L_{1}}(d))\oplus H^{1}(N(L_{2}),I^{2}_{Z\cap
L_{2}}(d)).$

Let

$M=Im(H^{0}(N(L_{1}\cup L_{2}),\mathcal{F})\to H^{1}(N(L_{1}\cup
L_{2}),I^{2}(d))).$

Then by the long exact sequence we have that

$h^{1}(N(L_{1}\cup L_{2}),I^{2}_{Z}(d))=h^{1}(N(L_{1}),I^{2}_{Z\cap
L_{1}}(d))+h^{1}(N(L_{2}),I^{2}_{Z\cap L_{2}}(d))+\dim_{\mathbb{F}}M.$

We note that $\dim_{\mathbb{F}}M\leq h^{0}(\mathcal{F})\leq n+1$

We further note that

$0\leq h^{1}(N(L_{i}),I^{2}_{Z\cap
L_{i}}(d))-h^{1}(N(L_{i}),I^{2}_{Z_{i}}(d))\leq n+1.$

Hence

$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L_{1}),I^{2}_{Z_{1}}(d))-h^{1}(N(L_{2}),I^{2}_{Z_{2}}(d))|\leq(n+1)$

in the case when $p\not\in Z$ (the situation of case (3)) and

$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L_{1}),I^{2}_{Z_{1}}(d))-h^{1}(N(L_{2}),I^{2}_{Z_{2}}(d))|\leq
3(n+1)$

in the case when $p\in Z$ (the situation of case (4)).

To finish the proof of (3) and (4) we must show that $\dim M$ only depends on
$|Z_{1}|$ and $|Z_{2}|$. Since $\dim M$ is also equal to

$h^{0}(N(L_{1}\cup L_{2}),\mathcal{F})-\dim\textrm{ker}H^{0}(N(L_{1}\cup
L_{2}),\mathcal{F})\to H^{1}(N(L_{1}\cup L_{2}),I^{2}(d))$ $=h^{0}(N(L_{1}\cup
L_{2}),\mathcal{F})-\dim Im(H^{0}(N(L_{1}\cup
L_{2}),\pi_{*}\pi^{*}(I^{2}(Z(d))))\to H^{0}(N(L_{1}\cup
L_{2}),\mathcal{F})),$

it suffices to show that

$\dim(Im(H^{0}(N(L_{1}\cup L_{2}),\pi_{*}\pi^{*}(I^{2}(Z(d))))\to
H^{0}(N(L_{1}\cup L_{2}),\mathcal{F}))$

depends only on $|Z_{1}|$ and $|Z_{2}|$. In case $p\in Z$, $H^{0}(N(L_{1}\cup
L_{2}),\mathcal{F})=0$, so this is immediate. This establishes (4). We will
now complete the proof of (3).

The map

$H^{0}(N(L_{1}\cup L_{2}),\pi_{*}\pi^{*}(I^{2}(Z(d))))\to H^{0}(N(L_{1}\cup
L_{2}),\mathcal{F})$

is isomorphic to the map

$H^{0}(N(L_{1}),I^{2}_{Z_{1}}(d))\oplus H^{0}(N(L_{2}),I^{2}_{Z_{2}}(d))\to
H^{0}(N(p),\mathcal{O}(d))$

given by restriction. It suffices to show that the image of
$H^{0}(N(L_{1}),I^{2}_{Z_{1}}(d))\to H^{0}(N(p),\mathcal{O}(d))$ is dependent
on $|Z_{1}|$ only. This happens in all but one case. We record what happens
below without proof since it is elementary and just a computation.

If $2|Z_{1}|+1\leq d$, the map $H^{0}(N(L_{1}),I^{2}_{Z_{1}}(d))\to
H^{0}(N(p),\mathcal{O}(d))$ is surjective. We note that if $2|Z_{1}|\geq d+1$
and $|Z_{1}|<d$ the image consists of all vectors vanishing at $p$ and whose
first derivative along the tangent direction to $L_{1}$ also vanishes at $p$.
If $|Z_{1}|\geq d$, $H^{0}(N(L),I^{2}_{Z_{1}}(d))=0$, so the image is also
zero. The remaining case is when $2|Z_{1}|=d,$ In this situation the image is
constrained by the fact that any $f\in H^{0}(N(L_{1}),I^{2}_{Z_{1}}(d))$is
such that $(f(p),\partial_{v}f(p))$ is forced to lie in some fixed one
dimensional linear subspace.

Now the only way that the image of

$H^{0}(N(L_{1}),I^{2}_{Z_{1}}(d))\oplus H^{0}(N(L_{2}),I^{2}_{Z_{2}}(d))\to
H^{0}(N(p),\mathcal{O}(d))$

can have different dimensions for the same value of $|Z_{1}|,|Z_{2}|$ is if
either $2|Z_{1}|=d$ or $2|Z_{2}|=d$.

Assume that $2|Z_{1}|=d$ and $2|Z_{2}|+1\leq d$. Then in this case the map is
surjective, so the dimension of the image is fixed. Similarly, if $2|Z_{1}|=d$
and $|Z_{2}|\geq d$, the image is of dimension $n$ and dependent only on the
sizes of $Z_{1},Z_{2}$. If $2|Z_{1}|=d$ and $2|Z_{2}|\geq d+1$ one gets that
the image is still surjective. In the case when $2|Z_{1}|=2|Z_{2}|=d$ the map
is still surjective and in particular only depends on the sizes of $|Z_{1}|$
and $|Z_{2}|$.

The proof of (5)- (10) are almost identical to that of (3) and (4), so we omit
them. Case (11) is a standard application of Riemann-Roch for an elliptic
curve.

∎

######  Proposition 6.11.

Let $M$ be one of the following:

  1. 1.

The space of twisted cubic curves in $\textrm{Chow}(3)$.

  2. 2.

The space of skew triples of lines in $\textrm{Chow}(3)$.

  3. 3.

The space of pairs of a line and a disjoint conic in $\textrm{Chow}(3)$.

  4. 4.

The space of pairs of a line intersecting a conic at a point in
$\textrm{Chow}(3)$.

  5. 5.

The space of pairs of a line intersecting a conic at two points in
$\textrm{Chow}(3)$.

  6. 6.

The space of smooth planar conics in $\textrm{Chow}(2)$.

  7. 7.

The space of skew lines in $\textrm{Chow}(2)$.

  8. 8.

The space of coplanar lines in $\textrm{Chow}(2)$.

  9. 9.

$\textrm{Chow}(1)$.

Then there exists $\epsilon>0$

$\sum_{|\lambda|\leq N}(-1)^{||\lambda||}W^{M}_{\geq\lambda}=0,$

up to dimension
$-(\frac{3n}{2}+\epsilon)d+h^{0}(\mathbb{P}^{n},\mathcal{O}(d))$. Furthermore
for $k\leq N$,

$\sum_{|\lambda|=k}(-1)^{||\lambda||}w^{M,l}_{\lambda}=0$

up to dimension $-(\frac{3n}{2}+\epsilon)d-l+|\lambda|n$. .

Furthermore for $k\leq N$,

$\sum_{|\lambda|=k}(-1)^{||\lambda||}w^{M,l,n}_{\lambda}=0,$

up to dimension $-(\frac{3n}{2}+\epsilon)d-l+|\lambda|n$.

###### Proof.

The proof of the Proposition 6.11 in these 9 separate cases are almost
identical. However there are some minor differences. We will prove the
Proposition 6.11 in detail in the cases numbered 1,2 and 4. We will indicate
how the proof is to be adjusted in all remaining cases.

Let us begin with the proof in the case when $M$ is the moduli space of
twisted cubic curves, i.e. case (1).

Let $T$ be a fixed twisted cubic curve. Then,

$\bar{w}_{\lambda}^{M,l}=Mw_{\lambda}^{T,l}$

since the family of twisted cubics over $M$ is Zariski locally-trivial. Note
that

$\bar{w}_{\lambda}^{T,l}=\pi_{\lambda}^{-1}(\bar{w}_{|\lambda|}^{M,l}).$

By Proposition 6.10 $\bar{w}_{|\lambda|}^{\\{T\\},l}$ consists of pairs
$(Z,T)$ and $|Z\cap T|=k_{1}$ (where $k_{1}$ is some number depending on $l$
necessarily bigger than $3$ for $d$ big enough). Hence

$\bar{w}_{\lambda}^{T,l}=\sum_{\alpha\subseteq\lambda,|\alpha|=k_{1}}w_{\alpha}(T)w_{\lambda-\alpha}(\mathbb{P}^{n}\setminus
T)$

By Proposition 6.9 $w_{\alpha}(T)w_{\lambda-\alpha}(\mathbb{P}^{n}\setminus
T)$ is a polynomial in $\mathbb{L}$. Thus it suffices to establish that the
image of the above class in $K_{0}(MHS)$ is $0$. By Proposition 4.8 the image
of

$\sum_{|\lambda|=k_{1}+k_{2}}(-1)^{||\lambda||}\sum_{\alpha\subseteq\lambda,|\alpha|=k_{1},|\lambda|=k_{1}+k_{2}}w_{\alpha}(T)w_{\lambda-\alpha}(\mathbb{P}^{n}\setminus
T)$

in $K_{0}(MHS)$ is the class of $H_{*}^{c}(w_{k_{1}}(T)\times
w_{k_{2}}(\mathbb{P}^{n}\setminus T),\pm\mathbb{Q})$. by Lemma 2 in [17] is
$0$, which establishes that

$\sum_{|\lambda|=k_{1}+k_{2}}\bar{w}_{\lambda}^{T,l}(\mathbb{P}^{n})=0.$

By Proposition 6.7

$\bar{w}_{\lambda}^{M,l}(\mathbb{P}^{n})-w_{\lambda}^{M,l}(\mathbb{P}^{n})\subseteq\cup_{d=1}^{\infty}w_{\lambda}^{M^{\prime}(d),l}(\mathbb{P}^{n}),$

where $M^{\prime}(d)$ is the moduli space of degree $d$ curves strictly
containing a twisted cubic. By Proposition 6.6 this is of dimension

$\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d-(n+1)|\lambda|-l.$

Hence

$\sum_{|\lambda|=k_{1}+k_{2}}w_{\lambda}^{T,l}(\mathbb{P}^{n})=0,$

up to dimension

$\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d-(n+1)||\lambda||-l.$

Note that in case (1), for a fixed value of $l$, $w_{\lambda}^{M,l,n}$ is
either equal to $w_{\lambda}^{M,l}$ or is $0$. Hence

$\sum_{|\lambda|=k_{1}+k_{2}}w_{\lambda}^{T,l,n}(\mathbb{P}^{n})=0,$

up to dimension

$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}-\epsilon)d-(n+1)|\lambda|-l.$

Since $W_{\geq\lambda}^{M}=\sum_{l=0}^{\infty}W_{\geq\lambda}^{M,l}$ and
$W_{\geq\lambda}^{M,l}=w_{\geq\lambda}^{M,l}\mathbb{L}^{h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-|\lambda|(n+1)+l}$
case (1) follows.

We will now deal with case when $M$ is the space of skew triples of lines.
Claim: Let $L_{1},L_{2},L_{3}$ be a fixed skew triple of lines. Let
$Z\subseteq L_{1}$ be a subset of size $n$. Let
$f(n)=h^{1}(N(L),I^{2}_{Z}(d))$ (One can check that this quantity only depends
on $|Z|$ ). Let $S_{l}=\\{(a,b,c)|f(a)+f(b)+f(c)=l\\}$. One can check that
$S_{l}$ is a finite set.

Firstly, since any family of skew triples of lines is Zariski locally trivial,

$\sum_{|\lambda|=k}(-1)^{||\lambda||}\bar{w}_{\lambda}^{M,l}(\mathbb{P}^{n})=M\sum_{|\lambda|=k}(-1)^{||\lambda||}\bar{w}_{\lambda}^{L_{1}\cup
L_{2}\cup L_{3},l}(\mathbb{P}^{n}).$

By Proposition 6.10, this equals

$\sum_{|\lambda|=k}(-1)^{||\lambda||}\sum_{(k_{1},k_{2},k_{3})\in
S_{l}}\sum_{\begin{subarray}{c}\alpha_{3}\subseteq\alpha_{2}\subseteq\alpha_{1}\subseteq\lambda\\\
(|\alpha_{1}|,|\alpha_{2}|,|\alpha_{3}|)=(k_{1},k_{2},k_{3})\end{subarray}}\bar{w}_{\alpha_{3}}(L_{3})\bar{w}_{\alpha_{2}-\alpha_{3}}(L_{2})\bar{w}_{\alpha_{1}-\alpha_{2}}(L_{1})\bar{w}_{\lambda-\alpha_{1}}(\mathbb{P}^{n}-L_{1}\cup
L_{2}\cup L_{3}).$

By Proposition 6.9 the right hand side of the above equation is a polynomial
in $\mathbb{L}$. By Proposition 4.8, the image of the right hand side in
$K_{0}(MHS)$ is a sum of terms of the form $H_{*}^{c}(w_{k_{1}}(L_{1})\times
w_{k_{2}}(L_{2})\times w_{k_{3}}(L_{3}))$which by Proposition in [17] is zero.
This implies
$\sum_{|\lambda|=k}(-1)^{||\lambda||}\bar{w}_{\lambda}^{M,l}(\mathbb{P}^{n})=0$
. As in case(1)

$\sum_{|\lambda|=k}(-1)^{||\lambda||}\bar{w}_{\lambda}^{M,l}(\mathbb{P}^{n})-w_{\lambda}^{M,l}(\mathbb{P}^{n})$

is of dimension less than equal to that of
$\cup_{d=1}^{\infty}w_{\lambda}^{M^{\prime}(d),l}(\mathbb{P}^{n}),$ where
$M^{\prime}(d)$ is the moduli space of degree $d$ curves strictly containing a
skew triple of lines. By Proposition 6.6 this is $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}-\epsilon)d-(n+1)||\lambda||-l.$
Case (4) follows.

The proofs of the other cases are nearly identical. In each case we prove that
$\bar{w}_{\lambda}^{M,l}=M\bar{w}_{\lambda}^{\\{C\\},l}$ where $C\in M$, since
$M$ is Zariski locally trivial in all of the cases. We then use Proposition
6.10 to explicitly describe $\bar{w}_{|\lambda|}^{\\{C\\},l}$. In each of our
cases $H^{*}(\bar{w}_{|\lambda|}^{\\{C\\},l},\pm\mathbb{Q})=0$. As a result by
Proposition 4.8
$\sum_{|\lambda|=n}(-1)^{||\lambda||}\bar{w}_{\lambda}^{\\{C\\},l}$ is mapped
to $0$ in $K_{0}(MHS).$

We then note that
$\bar{w}_{\lambda}^{\\{C\\},l}=\pi_{\lambda}^{-1}\bar{w}_{|\lambda|}^{\\{C\\},l}$
is seen to be a polynomial in $\mathbb{L}$. Hence
$\sum_{|\lambda|=n}(-1)^{||\lambda||}\bar{w}_{\lambda}^{\\{C\\},l}=0\in\mathcal{M}$.

We then note that

$\sum_{|\lambda|=n}(-1)^{||\lambda||}(\bar{w}_{\lambda}^{M,l}-w_{\lambda}^{M,l})=\sum_{l}\sum_{M^{\prime},|\lambda|=n}(-1)^{||\lambda||}w_{\lambda}^{M^{\prime},l}.$

where $M^{\prime}$ ranges over the moduli spaces of curves strictly containing
a curve in $M$. However either $M^{\prime}$ consists of curves of degree $\geq
4$ and hence $w_{\lambda}^{M^{\prime},l}$ is of dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}-\epsilon)d-(n+1)|\lambda|-l$
by Proposition 6.6, or $M^{\prime}$ is a case that has already been considered
on this list and hence has dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}-\epsilon)d-(n+1)|\lambda|-l$.
∎

######  Proposition 6.12.

Let $M\subseteq Chow(3)$ be contained in the subset parametrizing three lines.
Then there exists $\epsilon>0$, such that for
$N=\lceil(\frac{3n}{2}+\epsilon)d\rceil$, and $l>0$,
$\sum_{|\lambda|<N}(-1)^{||\lambda||}w_{\lambda}^{M,l}$ is of dimension
$\leq-l+|\lambda|n+-(\frac{3n}{2}+\epsilon)d$

###### Proof.

We note that we may express $M$ as a disjoint union of constructible sets
$M=\amalg_{i=1}^{k}M_{i}$ such that for each $i$, and for each triple of lines
$\\{L_{1},L_{2},L_{3}\\}$ in $M_{i}$ the triple has the same configuration,
i.e. the number of points of intersection of the lines remain constant in
$L_{i}$. It suffices to prove the Proposition for each $M_{i}$ separately.
Thus in what follows we assume that each triple of lines in $M$ has the same
configuration and $M$ is irreducible. We note that under this assumption the
family of triples of lines over $M$ is Zariski locally trivial.

By Proposition 6.7

$\bar{w}_{\lambda}^{M,l}-w_{\lambda}^{M,l}\subseteq\cup_{d^{\prime}\geq
4,l^{\prime}>l}w_{\lambda}^{Chow(d^{\prime}),l^{\prime}}$

and by Proposition 6.6 the right hand side of the above expression is of
dimension $\leq-l+|\lambda|n+-(\frac{3n}{2}+\epsilon)d$. Therefore it suffices
to prove that

$\sum_{|\lambda|<N}(-1)^{||\lambda||}\bar{w}_{\lambda}^{M,l}$

is of dimension $\leq-l+|\lambda|n+-(\frac{3n}{2}+\epsilon)d$.

Let $A=\mathbb{Z}^{7}/S_{3}$ where $S_{3}$ denotes the symmetric group of
three letters and the action of $S_{3}$ on $\mathbb{Z}^{7}$ is induced from
the natural action of $S_{3}$ on the power set of
$\\{1,2,3\\}\setminus\varnothing$. A configuration of points
$Z\subseteq\mathbb{P}^{n}$ and a set of 3 lines $\\{L_{1},L_{2},L_{3}\\}$
naturally gives us $|Z\cap\\{L_{1},L_{2},L_{3}\\}|:=(|Z\cap L_{1}|,|Z\cap
L_{2}|,|Z\cap L_{3}|,|Z\cap L_{1}\cap L_{2}|,\dots|Z\cap L_{1}\cap L_{2}\cap
L_{3}|)$ which is a well defined element of $A$.

Given $a\in A$, let

$\bar{w}_{\lambda}^{M}(a)=\\{(Z,\\{L_{1},L_{2},L_{3}\\})\in
w_{\lambda}^{M}||Z\cap\\{L_{1},L_{2},L_{3}\\}|=a\\}.$

We may now further stratify each $\bar{w}_{\lambda}^{M,l}$ as the disjoint
union of subspaces $\bar{w}_{\lambda}^{M,l}\cap\bar{w}_{\lambda}^{M}(a)$ where
$a\in A$.

However by Proposition 6.10, $\bar{w}_{\lambda}^{M,l}(a)$ is either all of
$\bar{w}_{\lambda}^{M}(a)$ or it is empty. We note that since the family of
triples of lines over $M$ is Zariski locally trivial,
$\bar{w}_{\lambda}^{M}(a)$ is also a locally trivial family over $M$.

Thus since Zariski locally trivial families split into prosucts in
$\mathcal{M}$, we have the following equality:

$[\bar{w}_{\lambda}^{M}(a)]=[M][w_{\lambda}^{c}(a)],$

where $c$ is a point in $M$. Thus it suffices to establish that for a fixed
$c\in M$ and $a\in A$
$\sum_{|\lambda|<N}(-1)^{||\lambda||}w_{\lambda}^{c}(a)=0$.

Since the above expression is polynomial in $\mathbb{L}$, it suffices to prove
that its image in $K_{0}(MHS)$ is $0$. By Proposition 4.8
$\sum_{|\lambda|=r}(-1)^{||\lambda||}w_{\lambda}^{c}(a)$ is mapped to
$H_{*}^{c}(w_{[r]}^{c}(a))$. The topological space $w_{[r]}^{c}(a)$ has a
factor of the form $w_{[r]}^{c}(\mathbb{A}^{1})$ . However by Proposition 4.8
$H_{*}^{c}(w_{[r]}^{c}(\mathbb{A}^{1}),\pm\mathbb{Q})=0$ and thus
$H_{*}^{c}(w_{[r]}^{c}(a))=0$. This concludes the proof. ∎

######  Definition 6.13.

Let $\epsilon>0$. Let $M$ denote the moduli space of smooth plane cubics in
$\mathbb{P}^{n}$. Let $N=(\frac{3n}{2}+\epsilon)d$. Let

$Y(d):=\sum_{l>0}\sum_{|\lambda|<N}((-1)^{||\lambda||})w_{\lambda}^{M,l}\mathbb{L}^{n|\lambda|}(\mathbb{L}^{l}-1).$

We note that a priori this definition of $Y(d)$ depends on our choice of
$\epsilon>0$. However if $\epsilon$ is sufficiently small, different values of
$\epsilon$ will only change the value of $Y(d)$ in very high codimension, and
this is an ambiguity that is acceptable for our purposes.

######  Proposition 6.14.

There exists $\epsilon>0$ such that: Up to dimension
$h^{0}(\mathbb{P}^{n}\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d$,

$Y(d)=\sum_{l>0}\sum_{\lambda}((-1)^{||\lambda||})\bar{w}_{\lambda}^{M,l}\mathbb{L}^{n|\lambda|}(\mathbb{L}^{l}-1).$

Let us now assume $d$ is even. Let $k_{1}(l)=\frac{3d+l}{2}$.

Then up to dimension
$h^{0}(\mathbb{P}^{n}\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d$,

$\sum_{l>0}\sum_{\lambda}((-1)^{||\lambda||})\bar{w}_{\lambda}^{M,l}\mathbb{L}^{n|\lambda|}(\mathbb{L}^{l}-1)$

$=\sum_{l>0,l\textrm{
odd}}\sum_{|\lambda|<N}\sum_{\alpha\subseteq\lambda,|\alpha|=k_{1}(l)}((-1)^{||\lambda||})\\{(Z_{1},Z_{2},C)|C\in
M,Z_{1}\in w_{\alpha}(C),Z_{2}\in
w_{\lambda\setminus\alpha}(\mathbb{P}^{n}\setminus
C)\\}\mathbb{L}^{-(n+1)|\lambda|}(\mathbb{L}^{l}-1).$

###### Proof.

Let $M$ denote the space of plane cubics in $\mathbb{P}^{n}$. We note that

$Y(d)-\sum_{l>0}\sum_{\lambda}((-1)^{||\lambda||})\bar{w}_{\lambda}^{M,l}\mathbb{L}^{n|\lambda|}(\mathbb{L}^{l}-1),$

equals

$\sum_{l>0}\sum_{\lambda}((-1)^{||\lambda||})w_{\lambda}^{M,l}-\bar{w}_{\lambda}^{M,l}\mathbb{L}^{n|\lambda|}(\mathbb{L}^{l}-1).$

By Proposition 6.6 and Proposition 6.7 each difference
$w_{\lambda}^{M,l}-\bar{w}_{\lambda}^{M,l}\mathbb{L}^{n|\lambda|}(\mathbb{L}^{l}-1)$
is of dimension $\leq
h^{0}(\mathbb{P}^{n}\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d$. Thus the entire
sum is of dimension $\leq
h^{0}(\mathbb{P}^{n}\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d.$

If $d$ is even and $l<\frac{3d}{2}$ we have

$barw_{\lambda}^{M,l}=\sum_{\alpha\subseteq\lambda,|\alpha|=k_{1}(l)}((-1)^{||\lambda||})\\{(Z_{1},Z_{2},C)|C\in
M,Z_{1}\in w_{\alpha}(C),Z_{2}\in
w_{\lambda\setminus\alpha}(\mathbb{P}^{n}\setminus C)\\},$

by Proposition $\ref{h1list}(11).$ To see this note that under the assumption
$l<\frac{3d}{2}$, it must be the case that $h^{1}(N(C),I^{2}_{Z\cap
C}(d))=h^{1}(C,\mathcal{O}(d)-2|Z\cap C|)$. Since $C$ is of genus $1$,
$h^{1}(C,\mathcal{O}(d)-2|Z\cap C|)=h^{0}(C,2|Z\cap C|-\mathcal{O}(d))=2|Z\cap
C|-3d$ (by our parity assumption $3d\neq 2|Z\cap C|$ and the case when
$2|Z\cap C|-\mathcal{O}(d)$ is the trivial line bundle does not arise).

We note that we may disregard the terms when $l\geq\frac{3d}{2}$ because they
contribute in too high a codimension.

Thus we have our desired equality. ∎

######  Proposition 6.15.

Let $M$ be the space of plane cubics in $\mathbb{P}^{n}$. Let $d\equiv
1\mod{4}$. Let $k_{0}=\frac{3d+1}{2}$. Then for $\epsilon$ sufficiently small
(the definition of $Y(d)$ implicitly depends on an $\epsilon$), $Y(d)$ has the
same highest weight term as

$H^{*}(\mathcal{M}_{1,1},H_{k_{0}})H^{*}(PGL_{n+1}(\mathbb{C}))\mathbb{L}^{-k_{0}(n+1)+1}$

in $K_{0}(MHS)$. The weight of this highest weight term is
$2(-k_{0}(n+1)+1)+k_{0}+1+(n^{2}-1)$.

###### Proof.

By Proposition 6.14, up to dimension $d(\frac{3n}{2}+\epsilon)$ $Y(d)$

$=\sum_{l>0,l\textrm{
odd}}\sum_{|\lambda|<N}\sum_{\alpha\subseteq\lambda,|\alpha|=k_{1}(l)}((-1)^{||\lambda||})\\{(Z_{1},Z_{2},C)|C\in
M,Z_{1}\in w_{\alpha}(C),Z_{2}\in
w_{\lambda\setminus\alpha}(\mathbb{P}^{n}\setminus
C)\\}\mathbb{L}^{-(n+1)|\lambda|}(\mathbb{L}^{l}-1).$

To finish the proof of the Proposition it suffices to compute the image of the
above expression in $K_{0}(MHS)$ . We note that the image equals
$\sum_{l,k}H^{*}(w_{[k]}^{M,l},\pm\mathbb{Q})\mathbb{L}^{-n|\lambda|+l}$.

For $l=1$, $\sum_{k}H^{*}(w_{[k]}^{M,1},\pm\mathbb{Q})\mathbb{L}^{-nk+l}$ has
highest weight term equal to that of $H^{*}(w_{k_{0}}^{M,1},\pm\mathbb{Q})$,
where $k_{0}=\frac{3d+1}{2}$, there are other terms corresponding to higher
values of $k$, but we will establish (after first dealing with the leading
term)that these have lower weight. This highest weight term corresponds to the
situation where we have $k_{0}$ points on a plane cubic $C$, in this situation
being singular on such a collection of points imposes $(k_{0})(n+1)-1$ linear
conditions.

Now $w_{k_{0}}^{M,1}=\\{(Z,C)\in w_{k_{0}}\times M|Z\subseteq C\\}$(by
Proposition 6.10) and hence
$w_{k_{0}}^{M,1}/PGL_{n+1}(\mathbb{C})=\mathcal{M}_{1,k}$. So we have,

$H^{*}(w_{k_{0}}^{M,1},\pm\mathbb{Q})=H^{*}(PGL_{n+1},\mathbb{Q})\sum_{l,n}H^{*}(\mathcal{M}_{1,k_{0}},\pm\mathbb{Q})$.
By the results of Section 5 of [7] (also see Corollary 2.8 in [8] ) the
highest weight term of this equals that of

$H^{*}(PGL_{n+1}(\mathbb{C}),\mathbb{Q})H^{*}(\mathcal{M}_{1,1},H_{k_{0}}),$

we note that this then implies that
$H^{*}(w_{[k_{0}]}^{M,1},\pm\mathbb{Q})\mathbb{L}^{-nk_{0}+1}$ has the
required highest weight, see [7] for instance.

We will now establish that for $k>k_{0}$, the term
$H^{*}(w_{[k]}^{M,1},\pm\mathbb{Q})\mathbb{L}^{-(n+1)k+l}$ has lower weight
than for $k=k_{0}$. To see this let $\bar{w}_{[k]}^{M,1}=\\{(Z,C)\in
w_{[k]}|h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap C}(d))=1\\}$. We note that
$\bar{w}_{[k]}^{M,1}\setminus w_{[k]}^{M,1}$ is of small dimension by
Proposition 6.6. Thus it suffices to establish that
$H^{*}(\bar{w}_{[k]}^{M,1},\pm\mathbb{Q})\mathbb{L}^{-nk+l}$ has lower weight.
But we note that by Proposition 6.10 $\bar{w}_{[k]}^{M,1}=\\{(Z,C)\in
w_{[k]}\times C||Z\cap C|=k_{0}\\}.$ Now we have a fibration $w_{[k]}^{M,1}\to
M$ with fiber isomorphic to $w_{[k_{0}]}(C)\times
w_{k-k_{0}}(\mathbb{P}^{n}\setminus C)$. We have a further fibration (in the
orbifold sense) $M\to M/PGL_{n+1}(\mathbb{C}).$ The fiber of the composite map
$w_{[k]}^{M,1}\to M/PGL_{n+1}(\mathbb{C})$ is seen to be a copy of
$PGL_{n+1}(\mathbb{C})\times w_{[k_{0}]}(C)\times
w_{k-k_{0}}(\mathbb{P}^{n}\setminus C).$

We may thus use the Leray spectral sequence to bound the weights of
$H_{*}^{c}(w_{[k]}^{M,1}),\pm\mathbb{Q})$ in terms of the dimension of
$M/PGL_{n+1}(\mathbb{C})$ (which is complex 1 dimensional and non compact) and
the weights of $H_{*}^{c}PGL_{n+1}(\mathbb{C})\times w_{[k_{0}]}(C)\times
w_{k-k_{0}}(\mathbb{P}^{n}\setminus C))$. The highest weight term of the
latter is $((n+1)^{2}-1)+k_{0}+1+(2n-1)(k-k_{0})+1$ (this uses a computation
of the Borel-Moore homology of configuration spaces- see Corollary 7.2.6 of
[3] for instance).

Thus the highest weight term of $H_{*}^{c}(w_{[k]}^{M,1}),\pm\mathbb{Q})$ is
at most $1+((n+1)^{2}-1)+k_{0}+1+(2n-1)(k-k_{0})$. Thus the highest weight of

$H_{*}^{c}(w_{[k]}^{M,1}),\pm\mathbb{Q})\mathbb{L}^{-k(n+1)+1}$

is at most $1+((n+1)^{2}-1)+k_{0}+1+(2n-1)(k-k_{0})+2-2k(n+1)$. If we subtract
this quantity from $2(-k_{0}(n+1)+1)+k_{0}+1+((n+1)^{2}-1)$ we obtain:

$3(k-k_{0})-1$

which is always positive for $k>k_{0}$. Thus the possible weights are lower
than the highest term.

We note that for $l>1$ a similar computation shows that all the corresponding
terms $H^{*}(w_{[k]}^{M,l})\mathbb{L}^{-nk+l}$ are of smaller weight and we
may ignore them for the purpose of finding the highest weight term.

This establishes the result.

∎

######  Proposition 6.16.

Let $d\gg n$. Let $L$ be a fixed line in $\mathbb{P}^{n}$. Then there exists,
$\epsilon>0$ such that for $N=(\frac{3n}{2}+\epsilon)d$, we have the following
inequalities/equalities.

  1. 1.

$\\{(f,Z)\in W_{\lambda,\geq N}^{l,n}|\dim\textrm{Sing}(f)=0\\}$

is of dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d$

  2. 2.

$W_{\lambda,\geq N}^{0,n}$is of dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d$

  3. 3.

$\\{(f,Z)\in W_{\lambda,\geq N}^{l,n}|\textrm{Sing}(f)\textrm{contains a curve
of degree}\geq 3\\}$

is of dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d$.

  4. 4.

$W_{\lambda,\geq N}^{l,n}=W_{\lambda,\geq
N}^{l,n}\cap\\{(f,Z)|C\subseteq\textrm{Sing}(f),C\textrm{ is a curve of degree
}\leq 2\\}$ $=\\{(f,L,Z)|(f,Z)\in W_{\lambda,\geq
N}^{l,n},L\subseteq\textrm{Sing}(f),L\textrm{ is a line}\\}$
$+\\{(f,C,Z)|(f,Z)\in W_{\lambda,\geq
N}^{l,n},C\subseteq\textrm{Sing}(f),C\textrm{is an irreducible conic}\\}$

$-\\{(f,L_{1},L_{2},Z)|(f,Z)\in W_{\lambda,\geq
N}^{l,n},L_{i}\subseteq\textrm{Sing}(f)L_{i}\textrm{are distinct lines}\\},$

up to dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d.$

  5. 5.

$\sum_{|\lambda|<N}(-1)^{||\lambda||}\\{(f,L_{1},L_{2},Z)|(f,Z)\in
W_{\lambda,\geq N}^{l,n},L_{i}\subseteq\textrm{Sing}(f)\\}=0$ up to dimension
$\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d$.

  6. 6.

$\sum_{|\lambda|<N}(-1)^{||\lambda||}\\{(f,C,Z)|(f,Z)\in W_{\lambda,\geq
N}^{l,n},C\subseteq\textrm{Sing}(f)\\}=0$ up to dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d$.

  7. 7.

$\sum_{|\lambda|<N}(-1)^{||\lambda||}\\{(f,L,Z)|(f,Z)\in W_{\lambda,\geq
N}^{l,n},L\subseteq\textrm{Sing}(f)L\textrm{ is a line}\\}=0$ up to dimension
$\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d$.

###### Proof.

We begin by proving (1). It suffices to prove (1) in the case when
$|\lambda|=N$. By Proposition 3.1 if $l>0$, then associated to any $Z\in
W_{\geq\lambda}^{n,l}$ is a unique reduced curve $C$ such that
$l=h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap
C}(d))$. Furthermore, if the components of $C$ are $C_{1},\dots C_{k}$ then
$\frac{\deg(C_{k})d-g(C_{k})}{2}\leq|Z\cap C_{k}|$ as otherwise $2|Z|\cap
C_{k}$ imposes linearly independent conditions which is not possible (see
Proposition 3.1). Also, there is some constant $K$ depending on $\deg(C_{k})$
such that if $|Z\cap C_{k}|>\deg(C_{k})d+K,$ a section singular at $Z$ is
singular on all of $C_{k}$. if that happens $Z\not\in W_{\geq\lambda}^{n,l}$.
Thus we must have $|Z\cap C_{k}|<\deg(C_{k})d+K$.

Furthermore there is some number $K^{\prime}$ such that ,
$|l-(n-1)\sum_{k}(|Z\cap C_{k}|)|\leq K^{\prime}$(Refer to Proposition 4.10).
We can therefore bound both the dimension of the space of such $Z$ and $l$,
which gives us (after a computation):

,

$\dim\\{(f,Z)|\dim\textrm{Sing}(f)=0\\}\cap W_{\lambda,\geq N+1}^{l,n}\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d.$

Part (2) is immediate, since

$\dim W_{\lambda,\geq N+1}^{0,n}\leq\dim W_{\geq[N]}^{0,n}\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N.$

Let us now proceed to prove part (3). We first begin with a claim:

Claim: For any fixed curve $C$ of degree $\geq 3$ and a fixed curve
$C^{\prime}$ of degree $r^{\prime}$ possibly empty, there exists
$\epsilon^{\prime}>0$, such that

$\dim\\{(f,Z)\in
W_{\geq\lambda}^{l,n}|C\subseteq\textrm{Sing}(f),h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap(C\cup
C^{\prime})}(d))\\}\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon^{\prime})d$

, here we require $\epsilon^{\prime}$ to be independent of
$C^{\prime},r^{\prime}$.

To prove the claim, we let

$A=\\{(f,Z)|\textrm{Sing}(f)\textrm{contains a curve of degree}\geq 3\\}\cap
W_{\lambda,\geq N}^{l,n}$

and $\tilde{A}=\\{(f,Z,C)|(f,Z)\in A,C\textrm{ is a curve of degree}\geq
3,C\subseteq\textrm{Sing}(f)\\}.$ We note that $\dim\tilde{A}\geq\dim A$. Then

$\dim\tilde{A}\leq\sup_{r\geq 3,C\textrm{is a curve of degree }r}(\dim
Chow(r)+\dim\\{(f,Z)|C\subseteq\textrm{Sing}(f)\\}\cap W_{\lambda,\geq
N}^{l,n}).$

Let us first argue that the claim implies (4). We will establish the claim
afterwards. Let $B_{r^{\prime}}(C)$ be the subset of
$W_{\geq\lambda}^{l,n}\times Chow(r^{\prime})$ consisting of all
$(f,Z,C^{\prime})$ such that $C\subseteq\textrm{Sing}(f)$ and $C^{\prime}$ is
the curve of minimal degree such that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap(C\cup
C^{\prime})}(d))\\}.$

Now we use Proposition 3.1 to argue that

$\dim\\{(f,Z)|C\subseteq\textrm{Sing}(f)\\}\cap W_{\lambda,\geq
N+1}^{l,n})\leq\sup_{r^{\prime}}(\dim Chow(r^{\prime})+\dim
B_{r^{\prime}}(C).$

Note that in the above two expressions, the supremums are actually maximums
for a given $|\lambda|$, there are only finitely many $r,r^{\prime}$ for which
the sets $B_{r^{\prime}}(C)$ can be nonempty. Further, we note that for $d\gg
0$, $\epsilon/10d\gg\dim Chow(r)$ for $r$ in the relevant range. Thus if we
establish that $\dim B_{r^{\prime}}(C)\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d$ we would be
done.

This concludes the proof of (4) assuming the claim, so we will now proceed to
establish the claim.

For this we note the following- for $d\gg\deg(C)$ the space $\\{f\in
H^{0}(\mathbb{P}^{n},\mathcal{O}(d))|C\subseteq\textrm{Sing}(f)\\}$ is of
codimension $h^{0}(N(C),\mathcal{O}(d))$ in
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$. We know that
$h^{0}(N(C),\mathcal{O}(d))\approx n\deg Cd$ by Proposition 4.10. For
$(f,Z)\in W_{\geq\lambda}^{C\cup C^{\prime},l,n}$, let $Z_{1}=Z\cap
C$,$Z_{2}=Z\cap C^{\prime}\setminus Z_{1}$ $Z_{3}=Z\setminus(Z_{1}\cup
Z_{2})$. We now have the following inequalities:

  1. 1.

$\deg(C)d\geq|Z_{1}|$.

  2. 2.

We note that
$|2|Z_{2}|-\deg(C^{\prime})d-h^{0}(\mathbb{P}^{n},I^{2}_{Z_{2}}(d))|$ is
bounded by a constant independent of $d$.

  3. 3.

$\\{f|Z\cup C\subseteq\textrm{Sing}(f)\\}$ is of dimension
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-h^{0}(N(C),\mathcal{O}(d))-h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime}}(d))-(n+1)|Z_{3}|-(n+1)|Z_{2}|$
in .

  4. 4.

But the dimension of the space of $Z$ such that $(f,Z)\in
W_{\geq\lambda}^{C\cup C^{\prime},l,n}$ is less than
$|Z_{1}|+|Z_{2}|+n|Z_{3}|$.

  5. 5.

Hence the dimension of

$\\{(f,Z)\in W_{\geq\lambda}^{C\cup
C^{\prime},l,n}|C\subseteq\textrm{Sing}(f)\\}$
$\leq|Z_{1}|+|Z_{2}|+n|Z_{3}|+h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-h^{0}(N(C),\mathcal{O}(d))+h^{1}(\mathbb{P}^{n},I^{2}_{Z_{2}}(d))-(n+1)|Z_{3}|-(n+1)|Z_{2}|$
$\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+\deg(C)d-n|Z_{2}|-h^{0}(N(C),\mathcal{O}(d))+h^{1}(\mathbb{P}^{n},I^{2}_{Z_{2}}(d))$
$\approx
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(n-1)\deg(C)d-n|Z_{2}|+h^{1}(\mathbb{P}^{n},I^{2}_{Z_{2}}(d))$
$\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(n-1)\deg(C)d$

$\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d.$

For part (4), we note that by (1),

$W_{\lambda,\geq N+1}^{l,n}=X$

up to dimension
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d,$ where

$X:=\\{(f,Z)\in W_{\lambda,\geq N+1}^{l,n}|\textrm{Sing}(f)\textrm{contains a
curve of degree}\leq 2\textrm{and does not contain a curve of degree}\geq
3\\}.$

Standard inclusion-exclusion techniques then imply that the motive of

$X=\\{(f,L,Z)|(f,Z)\in W_{\lambda,\geq
N+1}^{l,n},L\subseteq\textrm{Sing}(f)\\}+\\{(f,C,Z)|(f,Z)\in W_{\lambda,\geq
N+1}^{l,n},C\subseteq\textrm{Sing}(f)\\}$ $-\\{(f,L_{1},L_{2},Z)|(f,Z)\in
W_{\lambda,\geq N+1}^{l,n},L_{i}\subseteq\textrm{Sing}(f)\\}.$

up to
dimension$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d$.

We will omit the proof of parts (5) and (6) because their proofs are similar
to that of part (7) (which we will not omit) and in fact a bit simpler. For
part (7), we note that the family of lines over $Chow(1)$ is Zariski locally
trivial. Hence $\\{(f,L,Z)|(f,Z)\in W_{\lambda,\geq
N+1}^{l,n},L\subseteq\textrm{Sing}(f)\\}$ defines a locally trivial family
over $Chow(1)$ and thus,

$\\{(f,L,Z)|(f,Z)\in W_{\lambda,\geq
N+1}^{l,n},L\subseteq\textrm{Sing}(f)\\}=Chow(1)\\{(f,Z)|(f,Z)\in
W_{\lambda,\geq N+1}^{l,n},L\subseteq\textrm{Sing}(f)\\}$

in $\mathcal{M}$ (Zariski locally trivial families split into products in
$\mathcal{M}$).

We now make the following claim.

Claim:

$\sum_{|\lambda|=k}(-1)^{||\lambda||}\\{(f,Z)|(f,Z)\in W_{\lambda,\geq
N+1}^{l,n},L\subseteq\textrm{Sing}(f)\\}$
$=\sum_{|\lambda|=k}(-1)^{||\lambda||}\sum_{\alpha\subseteq\lambda}w_{\alpha}(L)w_{\lambda\setminus\alpha}(\mathbb{P}^{n}\setminus
L)\mathbb{L}^{-(n+1)(|\lambda|-|\alpha|-h^{0}(N(L),\mathcal{O}(d))}$

up to dimension
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d$, i.e. the sum
is well approximated by the corresponding sum when we ignore the fact that
points in $\mathbb{P}^{n}\setminus L$ need not impose linearly independent
conditions on $H^{0}(\mathbb{P}^{n},I^{2}_{L}(d))$.

The proof of the claim is essentially a dimension computation similar to those
we’ve already seen, if a collection of points $Z$ fails to impose linearly
independent conditions on $H^{0}(\mathbb{P}^{n},I^{2}_{L}(d))$, then a large
number of points of $Z$ lie on a curve, either the curve is of degree $\geq 2$
and the dimension of the resulting error is small (see Proposition 6.6) or the
curve is a line and the alternating sum vanishes for the same reason as in
Proposition 6.11. We will first establish (7) assuming the claim.

To do so we must simply establish that

$\sum_{|\lambda|<N}(-1)^{||\lambda||}\sum_{\alpha\subseteq\lambda}w_{\alpha}(L)w_{\lambda\setminus\alpha}(\mathbb{P}^{n}\setminus
L)\mathbb{L}^{-(n+1)(|\lambda|-|\alpha|-h^{0}(N(L),\mathcal{O}(d))}=0$

up to dimension
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d.$ It is clear
that the sum is a polynomial in $\mathbb{L}$ and thus it suffices to establish
that its image in $K_{0}(MHS)$ is of sufficiently low weight. Furthermore by
Proposition 4.8 the image of the above expression in $K_{0}(MHS)$ is

$\mathbb{L}^{-h^{0}(N(L),\mathcal{O}(d))}\sum_{k<N}\sum_{k^{\prime}\leq
k_{1}}H_{*}^{c}(w_{k_{1}}(\mathbb{P}^{1})\times
w_{k-k_{1}}(\mathbb{P}^{n}\setminus\mathbb{P}^{1}),\pm\mathbb{Q})\mathbb{L}^{-(k-k_{1})(n+1)}.$

To proceed with the proof of (7) we will need the following identity:

$\sum_{k=0}^{\infty}H_{*}^{c}(w_{k}(\mathbb{P}^{1}),\pm\mathbb{Q})=0.$

To establish this we note that
$\sum_{k=0}^{\infty}H_{*}^{c}(w_{k}(\mathbb{P}^{1}),\pm\mathbb{Q})$ is the
image of
$\sum_{\lambda}(-1)^{||\lambda||}w_{\lambda}(\mathbb{P}^{1})=Z_{\mathbb{P}^{1}}^{-1}(0)=0$.
Thus we have established the previous identity.

We also note that in the sum
$\sum_{k=0}^{\infty}H_{*}^{c}(w_{k}(\mathbb{P}^{1}),\pm\mathbb{Q})$ the only
relevant terms are for $k=0,1,2$ as all other terms vanish by Lemma 1.2 of
[17].

We now note that

$\mathbb{L}^{-h^{0}(N(L),\mathcal{O}(d))}\sum_{k<N}\sum_{k^{\prime}\leq
k}H_{*}^{c}(w_{k^{\prime}}(\mathbb{P}^{1})\times
w_{k-k^{\prime}}(\mathbb{P}^{n}\setminus\mathbb{P}^{1}),\pm\mathbb{Q})\mathbb{L}^{-(k-k^{\prime})(n+1)}$

$=\mathbb{L}^{-h^{0}(N(L),\mathcal{O}(d))}\sum_{k^{\prime}=0}^{2}\sum_{k=k^{\prime}}^{N}H_{*}^{c}(w_{k^{\prime}}(\mathbb{P}^{1})\times
w_{k-k^{\prime}}(\mathbb{P}^{n}\setminus\mathbb{P}^{1}),\pm\mathbb{Q})\mathbb{L}^{-(k-k^{\prime})(n+1)}$
$=\mathbb{L}^{-h^{0}(N(L),\mathcal{O}(d))}\sum_{k^{\prime}=0}^{2}H_{*}^{c}(w_{k^{\prime}}(\mathbb{P}^{1}),\pm\mathbb{Q})\sum_{k^{\prime\prime}=0}^{N-k^{\prime}-1}H_{*}^{c}(w_{k^{\prime\prime}}(\mathbb{P}^{n}\setminus\mathbb{P}^{1}),\pm\mathbb{Q})\mathbb{L}^{-(k^{\prime\prime})(n+1)}$

$=\mathbb{L}^{-h^{0}(N(L),\mathcal{O}(d))}\sum_{k^{\prime}=0}^{2}H_{*}^{c}(w_{k^{\prime}}(\mathbb{P}^{1}),\pm\mathbb{Q})\sum_{k^{\prime\prime}=0}^{N-1}H_{*}^{c}(w_{k^{\prime\prime}}(\mathbb{P}^{n}\setminus\mathbb{P}^{1}),\pm\mathbb{Q})\mathbb{L}^{-(k^{\prime\prime})(n+1)}$
$-\mathbb{L}^{-h^{0}(N(L),\mathcal{O}(d))}\sum_{k^{\prime}=0}^{2}H_{*}^{c}(w_{k^{\prime}}(\mathbb{P}^{1}),\pm\mathbb{Q})\sum_{k^{\prime\prime}=N-k^{\prime}-1}^{N-1}H_{*}^{c}(w_{k^{\prime\prime}}(\mathbb{P}^{n}\setminus\mathbb{P}^{1}),\pm\mathbb{Q})\mathbb{L}^{-(k^{\prime\prime})(n+1)}.$

By our previous computation the first term is 0. The second term is easily
seen to be of dimension $\leq-(\frac{3n}{2}+\epsilon)d$. This establishes the
proof of (7) assuming the claim. We will now prove the claim.

Let

$X_{\lambda,j,l}=\\{(f,Z)|(f,Z)\in W_{\lambda,\geq
N}^{l,n},L\subseteq\textrm{Sing}(f),|Z\cap L|=j\\}$

Let
$Y_{\lambda,j,l}=\sum_{|\alpha|=j,\alpha\subseteq\lambda}w_{\alpha}(L)w_{\lambda\setminus\alpha}(\mathbb{P}^{n}\setminus
L)\mathbb{L}^{-(n+1)(|\lambda|-|\alpha|)-h^{0}(N(L),\mathcal{O}(d))}$

We will establish that
$\sum_{|\lambda|<N}(-1)^{||\lambda||}X_{\lambda,j,l}-Y_{\lambda,j,l}$ is of
dimension $\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N$ which will immediately
give us the claim.

We may further stratify $X_{\lambda,j,l}$ into the pieces

$X_{\lambda,j,l,k}=\\{(f,Z)\in
X_{\lambda,j,l}|h^{1}(\mathbb{P}^{n},I^{2}_{Z\cup
L}(d))=h^{0}(N(L),\mathcal{O}(d))+(n+1)(|\lambda|-j)+k\\},$

i.e. $X_{\lambda,j,l,k}$ consists of those $Z$ such that vanishing to order 2
on $Z\cup L$ imposes $k$ fewer linear conditions than expected. Let
$Y_{\lambda,j,l,k}=X_{\lambda,j,l,k}\mathcal{L}^{-}k$. We note that
$\sum_{k}Y_{\lambda,j,l,k}=Y_{\lambda,j,l}$.

We now note that it suffices to establish that
$\sum_{|\lambda|<N}(-1)^{||\lambda||}X_{\lambda,j,l,k}-Y_{\lambda,j,l,k}$ is
of dimension $\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N$. For $k=0$, the
above is exactly $0$.

Let us assume $k>0$. We then note that by an argument similar to that of
Proposition 3.1 there must exist a curve $C^{\prime}$ such that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z\cup L}(d))=h^{1}(N(C^{\prime}\cup
L),I^{2}_{Z\cap(C)\cup L}(d)).$ Further more the above curve $C^{\prime}$ is
uniquely determined by $Z$ assuming it is of minimal degree.

Let

$X_{\lambda,j,l,k}^{r}\subseteq X_{\lambda,j,l,k}\times Chow(r)$

consist of all $(f,Z,C)$ such that

$h^{1}(\mathbb{P}^{n},I^{2}_{Z\cup L}(d))=h^{1}(N(C\cup L),I^{2}_{Z\cap(C)\cup
L}(d)),$

and $C$ is of minimal degree. Thus we have that
$X_{\lambda,j,l,k}=\sum_{r}X_{\lambda,j,l,k}^{r}$

Let $Y_{\lambda,j,l,k}^{r}=X_{\lambda,j,l,k}^{r}\mathbb{L}^{-k}$. Note that
$\sum_{k,r}Y_{\lambda,j,l,k}^{r}=Y_{\lambda,j,l}.$

We now consider the difference $X_{\lambda,j,l,k}^{r}-Y_{\lambda,j,l,k}^{r}$.
It suffices to establish for all fixed values of $j,l,k,r$ that
$\sum_{|\lambda|<N}(-1)^{||\lambda||}X_{\lambda,j,l,k}^{r}-Y_{\lambda,j,l,k}^{r}$
is of dimension $\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N$.

It is an easy computation to note that for $r\geq 2$ the above difference is
of dimension $\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N$. Also for $r=0$,
the above term always vanishes. Thus the only case that we must consider is
when $r=1$. Let $M_{1}$ denote the space of all lines in $\mathbb{P}^{n}$
intersecting $L$ at a single point. Let $M_{2}$ denote the space of all lines
in $\mathbb{P}^{n}$ not intersecting $L$. Let

$X_{\lambda,j,l,k}^{M_{i}}=\\{(f,Z,L^{\prime})\in
X_{\lambda,j,l,k}^{1},L^{\prime}\in M_{i}\\}.$

Let

$Y_{\lambda,j,l,k}^{M_{i}}=X_{\lambda,j,l,k}^{M_{i}}\mathbb{L}^{-k}.$

Let

$\bar{X}_{\lambda,j,l,k}^{M_{i}}=\\{(f,Z,L^{\prime})\in
X_{\lambda,j}|h^{1}(L^{\prime}\cup L,I^{2}_{Z\cap(L^{\prime})\cup
L}(d))=k\\}.$

Let

$\bar{Y}_{\lambda,j,k}^{M_{i}}=\bar{X}_{\lambda,j,k}^{M_{i}}\mathbb{L}^{-k}.$

We claim that:

$\bar{X}_{\lambda,j,l,k}^{M_{i}}-X_{\lambda,j,l,k}^{M_{i}}$ and
$\bar{Y}_{\lambda,j,l,k}^{M_{i}}-Y_{\lambda,j,l,k}^{M_{i}}$ is of dimension
$\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N.$ This follows from the fact that
these differences are contained in the union of $X_{\lambda,j,l,k}^{r}$ for
$r\geq 2$ and we have already remarked that these have dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N.$.

Thus it suffices to establish that

$\sum(-1)^{||\lambda||}\bar{X}_{\lambda,j,l,k}^{M_{i}}=\sum(-1)^{||\lambda||}barY_{\lambda,j,l,k}^{M_{i}}=0.$

To do so we will explicitly describe the sets
$\bar{X}_{\lambda,j,l,k}^{M_{1}}$ and $\bar{X}_{\lambda,j,l,k}^{M_{2}}$.

We note that we have a Zariski locally trivial bundle
$\bar{X}_{\lambda,j,l,k}^{M_{2}}\to Chow(1)$ defined by
$(f,Z,L^{\prime})\mapsto L^{\prime}$. Let us denote the fibre of this map by
$F_{\lambda,j,l,k}^{M_{2}}$, it is clear that
$X_{\lambda,j,l,k}^{M_{2}}=Chow(1)F_{\lambda,j,l,k}^{M_{2}}$ The fibre over
$L^{\prime}$ consist of those $Z$ such that $|Z\cap L|=j$ and
$h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap L^{\prime}\cup
L}(d))=k+h^{1}(\mathbb{P}^{n},I^{2}_{L}(d))$ or equivalently
$h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap}{d})=k$. But as is established in
Proposition 6.10, this just implies that $|Z\cap L|=f(k)$ for some number
$f(k)$ depending on $k$. We may then apply Proposition 6.9 to conclude that
$F_{\lambda,j,l,k}^{M_{2}}$ is a polynomial in $\mathbb{L}$ and it suffices to
establish that the image of
$\sum_{|\lambda|<N}(-1)^{||\lambda||}\bar{F}_{\lambda,j,l,k}^{M_{2}}$ is zero
in $K_{0}(MHS)$.

But by Proposition 4.8 the image of
$\sum(-1)^{||\lambda||}\bar{F}_{\lambda,j,l,k}^{M_{i}}$ in $K_{0}(MHS)$ is

$\sum_{i=0}^{N}H_{*}^{c}(w_{j}(L)\times w_{f(k)}(L^{\prime})\times
w_{i-j-f(k)}(\mathbb{P}^{n}\setminus L\setminus L^{\prime}),\pm,\mathbb{Q}),$

where in all relevant cases $f(k)\geq\frac{d}{2}$ which since $d$ is assumed
to be large is greater than $2$. Thus by Lemma 1.2 in [17] the above
cohomology group vanishes and we have the desired equality.

Similarly

$\sum_{|\lambda|<N}(-1)^{||\lambda||}\bar{X}_{\lambda,j,l,k}^{M_{1}}=0.$

and