A note on $r$ hypersurfaces intersecting in $\mathbb{P}^r$

TL;DR

This paper investigates the structure of $r$-tuples of homogeneous forms in projective space whose common zeros form positive-dimensional sets, revealing that maximal components are either forms vanishing on a line or failing to intersect properly.

Contribution

It characterizes the components of maximal dimension of the locus of forms with positive-dimensional common zeros, identifying specific geometric configurations.

Findings

Maximal components either vanish on a line or have improper intersections.

Provides a classification of the geometric structure of these loci.

Enhances understanding of hypersurface intersections in projective space.

Abstract

We consider the locus of $r$-tuples of homogeneous forms of some fixed degree whose common vanishing locus in $\mathbb{P}^r$ is positive dimensional. We show that any component of maximal dimension of that locus either consists of homogeneous forms all vanishing on some line or homogeneous forms where a proper subset fail to intersect properly.

Equations

$$Z:=\{(F_1,\dots ,F_r)\mid \{F_1=\cdots =F_r=0\}\text{ is positive dimensional}\}$$

$$\{(F_1,\dots ,F_r)\mid \dim (\{F_1=\cdots =F_{r-1}=0\})\ge 2\}$$

$$\{(F_1,\dots ,F_k)\mid \dim (\{F_1=\cdots =F_k=0\})\ge r-k+1\}\subset \prod _{i=1}^k{\mathbb{A}}^{\left(\genfrac{}{}{0ex}{}{r+d_i}{d_i}\right)}$$

$$\{(F_1,\dots ,F_k)\mid \dim (\{F_1=\cdots =F_{k-1}=0\})\ge r-k+2\}\times {\mathbb{A}}^{\left(\genfrac{}{}{0ex}{}{r+d_k}{d_k}\right)}\subset \prod _{i=1}^k{\mathbb{A}}^{\left(\genfrac{}{}{0ex}{}{r+d_i}{d_i}\right)}$$

$$\{(F_1,\dots ,F_k)\mid \dim (\{F_1=\cdots =F_k=0\})\ge r-k+1\}\subset ({\mathbb{A}}^{\left(\genfrac{}{}{0ex}{}{r+d}{d}\right)}{)}^k$$

$${\Phi }_{d_1,\dots ,d_k}^{{\mathbb{P}}^r,a}(X)\subset \prod _{i=1}^k{W}_{r,d_i}$$

$${\Phi }_{d_1,\dots ,d_k}^{{\mathbb{P}}^r,a}(X,\mathrm{Span}(r,b))\subset {\Phi }_{d_1,\dots ,d_k}^{{\mathbb{P}}^r,a}({\mathbb{P}}^r)$$

$${\Phi }_{d_1,\dots ,d_{r+a-1}}^{{\mathbb{P}}^r,a}({\mathbb{P}}^r)\backslash \left({\Phi }_{d_1,\dots ,d_{r+a-2}}^{{\mathbb{P}}^r,a}({\mathbb{P}}^r)\times W_{r,d_{r+a-1}}\right)$$

$${\Phi }_{d_1,\dots ,d_k}^{{\mathbb{P}}^r,a}({\mathbb{P}}^r,\mathrm{Span}(r,b))\backslash \left({\Phi }_{d_1,\dots ,d_{k-1}}^{{\mathbb{P}}^r,a}({\mathbb{P}}^r)\times W_{r,d_k}\right)$$

$$-\dim (\mathbb{G}(b,r))+\min \left\{\sum _{j=1}^{r-b+a}{h}_{b,b-i_j+j}(d_{i_j})\mid 1\le i_1<\cdots <i_{r-b+a}=k\right\}.$$

$$\left\{\Lambda ,(F_1,\dots ,F_k)\mid (F_1{|}_{\Lambda },\dots ,F_k{|}_{\Lambda })\in {\Phi }_{d_1,\dots ,d_k}^{{\mathbb{P}}^r,a+r-b}(\Lambda ,\mathrm{Span}(r,b))\backslash \left({\Phi }_{d_1,\dots ,d_{k-1}}^{{\mathbb{P}}^r,a+r-b}(\Lambda )\times W_{r,d_k}\right)\right\}$$

$${\Phi }_{d_1,\dots ,d_k}^{{\mathbb{P}}^b,a+r-b}({\mathbb{P}}^b,\mathrm{Span}(b,b))\backslash \left({\Phi }_{d_1,\dots ,d_{k-1}}^{{\mathbb{P}}^b,a+r-b}({\mathbb{P}}^b)\times W_{b,d_k}\right)$$

$${\Phi }_{d_1,\dots ,d_k}^{{\mathbb{P}}^r,a}({\mathbb{P}}^r,\mathrm{Span}(r,r))\backslash \left({\Phi }_{d_1,\dots ,d_{k-1}}^{{\mathbb{P}}^r,a}({\mathbb{P}}^r)\times W_{r,d_k}\right)$$

$$\min \left\{\sum _{j=1}^a{h}_{r,r-i_j+j}(d_{i_j})\mid 1\le i_1<\cdots <i_a=k\right\},$$

$$\min \left\{\sum _{j=1}^a{h}_{r,r-i_j+j}(d_{i_j})\mid 1\le i_1<\cdots <i_a\le k\right\}.$$

$$-\dim (\mathbb{G}(1,r))+\sum _{i=1}^{r+a-1}(d_i+1)$$

$${\Phi }_{d_1,\dots ,d_{r+a-1}}^{{\mathbb{P}}^r,a}({\mathbb{P}}^r,\mathrm{Span}(r,b))\backslash \left({\Phi }_{d_1,\dots ,d_{r+a-2}}^{{\mathbb{P}}^r,a}({\mathbb{P}}^r)\times W_{r,d_{r+a-1}}\right)$$

$$-\dim (\mathbb{G}(b,r))+\min \left\{\sum _{j=1}^{r-b+a}{h}_{b,b-i_j+j}(d_{i_j})\mid 1\le i_1<\cdots <i_{r-b+a}=r+a-1\right\}$$

$$-\dim (\mathbb{G}(b,r))+\min \left\{\sum _{j=1}^{r-b+a}{h}_{b,1}(d_{i_j})\mid 1\le i_1<\cdots <i_{r-b+a}=r+a-1\right\},$$

$$-\dim (\mathbb{G}(b,r))+\sum _{j=1}^{r-b+a-1}(b{d}_j+1)+(b{d}_{r+a-1}+1)$$

$$-\dim (\mathbb{G}(b,r))+(-b+1)+\sum _{j=1}^{r+a-1}(d_j+1)+\sum _{j=1}^{r-b+a-1}((b-1)d_j)$$

$$-(b+1)(r-b)+(-b+1)+(b-1)(r-b+a-1)+\sum _{j=1}^{r+a-1}(d_j+1)$$

$$-2(r-1)+a(b-1)+\sum _{j=1}^{r+a-1}(d_j+1)$$

Full text

1 Introduction

A general choice of $r$ hypersurfaces in $\mathbb{P}^{r}$ will intersect in finitely many points. Given a choice of degrees $d_{1}\leq\cdots\leq d_{r}$, we can consider the closed locus

[TABLE]

inside the space $\mathbb{A}^{\binom{r+d_{1}}{d_{1}}}\times\cdots\times\mathbb{A}^{\binom{r+d_{r}}{d_{r}}}$ of all $r$ tuples of homogeneous forms.

Previously, the author showed that if $d_{i}\leq d_{1}+\binom{d_{1}}{2}(i-1)$ and $d_{1}\geq 2$, then the unique component of maximal dimension consists of homogeneous forms all vanishing on some line [Tse18, Theorem 1.3]. The method was then applied to get partial results on the Kontsevich space of rational curves on hypersurfaces of degree $n-1$ in $\mathbb{P}^{n}$ [Tsea] and on the space of hypersurfaces with positive dimensional singular locus [Tse18, Theorem 1.6] extending work of Slavov [Sla15].

The purpose of this note is to give the following result about $Z$ that holds for all choices of degrees.

Theorem 1**.**

For all choices of degrees $d_{1}\leq\cdots\leq d_{r}$, the unique component of $Z$ of maximal dimension not contained in

[TABLE]

consists of homogeneous forms all vanishing on some line.

While the statement of Theorem 1 is qualitative, the proof uses the same methods as [Tse18] and involves some numerical bounds working out right. Finally, like in [Tse18], the proof also applies for the locus of $r+a-1$ homogeneous forms whose common vanishing locus is positive dimensional for $a\geq 1$, and we state this slightly more general form in Theorem 9.

2 Further questions

Given Theorem 1, one might ask for the largest components of the locus (1). More precisely, we can ask

Question 2**.**

Fix $k\leq r$ and $d_{1}\leq\cdots\leq d_{k}$. Does a component of maximal dimension of

[TABLE]

not contained in

[TABLE]

consist of homogeneous forms all vanishing on some dimension $r-k+1$ linear space?

Theorem 1 says the answer to 2 is always yes when $k=r$. When $k<r$ this is no longer true. For example, fix $k=2$ and $(d_{1},d_{2})=(2,2)$. One can quickly verify that (2) has two components for every $r\geq 2$: the locus where $F_{1}$ and $F_{2}$ both vanish on some hyperplane and the locus where $F_{1}$ and $F_{2}$ both vanish on some quadric hypersurface.

Also, by setting up an incidence correspondence, we check the two components have codimensions $r^{2}$ and $\frac{r^{2}+3r}{2}$ respectively in $\mathbb{A}^{\binom{r+2}{2}}\times\mathbb{A}^{\binom{r+2}{2}}$.

[TABLE]

In particular, we see in this case for $r=2$ the the answer to 2 is yes as predicted by Theorem 1, for $r=3$ the answer is still yes but the largest component is no longer unique, and for $r\geq 4$ the answer is no. Given this example, one can also ask the following preliminary question where we set all the $d_{i}$’s to be equal.

Question 3**.**

Fix $k\leq r$ and a degree $d$. Is it true that a component of maximal dimension of

[TABLE]

must consist either of homogeneous forms vanishing on some dimension $r-k+1$ linear space or homogeneous forms that are linearly dependent?

If we fix $k\leq r$ and let $d>>0$, the author can show that there is a unique largest component and the first possibility occurs as a special case of a more general problem [Tseb]. If $k=r$, then again the locus of homogeneous forms vanishing on some line is the unique component of maximal dimension either by [Tse18, Theorem 1.3] or by applying Theorem 1 and permuting the hypersurfaces.

2.1 Acknowledgements

The author would like to thank his advisor Joe Harris for helpful conversations.

3 Definitions

We will follow the notation in [Tse18], since we will rely fundamentally on its main argument. As a trade off, the notation will be more cumbersome. Finally, the reader is referred to [Tse18, Section 2] for a worked example of the key argument, without the notational baggage. We will work over an algebraically closed field of arbitrary characteristic.

Definition 4** ([Tse18, Definition 3.7]).**

Let $W_{r,d}\simeq\mathbb{A}^{\binom{r+d}{d}}$ be the affine space whose underlying vector space is $H^{0}(\mathbb{P}^{r},\mathscr{O}_{\mathbb{P}^{r}}(d))$.

Definition 5** ([Tse18, Definition 3.9]).**

Given a tuple $(d_{1},\ldots,d_{k})$ of positive integers, $a$ a positive integer, and a subscheme $X\subset\mathbb{P}^{r}$, define

[TABLE]

to be the locus of tuples $(F_{1},\ldots,F_{k})$ of homogeneous forms of degrees $(d_{1},\ldots,d_{k})$ such that the vanishing locus $\{F_{1}=\cdots=F_{k}=0\}\cap X$ has dimension at least $\dim(X)-k+a$.

3.1 Definitions used in proof

For the proof of Theorem 9, we will also need the following definitions

Definition 6** ([Tse18, Definitions 3.14 and 4.1]).**

Given $b\leq r$, let

[TABLE]

be the locus of forms $(F_{1},\ldots,F_{k})$ such that $\{F_{1}=\cdots=F_{k}=0\}\cap X$ contains an integral subscheme of dimension $\dim(X)-k+a$ with span exactly a dimension $b$ plane.

The locus $\Phi^{\mathbb{P}^{r},a}_{d_{1},\ldots,d_{k}}(X,\operatorname{Span}(r,b))$ is a constructible subset given the discussion around [Tse18, Definitions 3.14 and 4.1].

Definition 7** ([Tse18, Definition 4.1]).**

Define $h_{r,a}(d):=(r-a)\binom{d+a-1}{d-1}+\binom{d+a}{d}$.

The following elementary lemma can be found in [Par15, Theorem 1.3], for example, and it is proven by taking hyperplane slices and applying [Har82, Lemma 3.1].

Lemma 8**.**

If $Z\subset\mathbb{P}^{r}$ is a nondegenerate integral scheme of dimension $a$, then the Hilbert function $h_{Z}(d)$ of $Z$ is bounded below by $h_{r,a}(d)$.

4 Proof of Main Theorem

Theorem 1 follows from Theorem 9 when $a=1$. We will prove Theorem 9.

Theorem 9**.**

If $a\geq 1$ and $1\leq d_{1}\leq\cdots\leq d_{r+a-1}$ are integers, the unique component of maximal dimension of

[TABLE]

consists of $r+a-1$ tuples $(F_{1},\ldots,F_{r+a-1})$ of homogeneous forms of degree $(d_{1},\ldots,d_{r+a-1})$ where $\{F_{1}=\cdots=F_{r+a-1}=0\}$ contains some line.

The key input from [Tse18] is Lemma 10.

Lemma 10**.**

We have the codimension of

[TABLE]

in $\prod_{i=1}^{k}{W_{r,d_{i}}}$ is at least

[TABLE]

Proof of Lemma 10.

This follows from the proof of [Tse18, Lemma 4.2]. Since there are no new ideas not contained in [Tse18], we will give an informal proof, recapping the key idea in [Tse18] and mentioning the small modification necessary to show Lemma 10.

First, we can reduce to the case $b=r$ by a standard incidence correspondence. The locus

[TABLE]

inside of $\mathbb{G}(b,r)\times\prod_{i=1}^{k}{W_{r,d_{i}}}$ surjects onto (3) when forgetting the $\mathbb{G}(b,r)$ factor. When we forget the $\prod_{i=1}^{k}{W_{r,d_{i}}}$ factor, the fibers of the projection of (4) to $\mathbb{G}(b,r)$ are all isomorphic and all have codimension in $\prod_{i=1}^{k}{W_{r,d_{i}}}$ equal to the codimension of

[TABLE]

in $\prod_{i=1}^{k}{W_{b,d_{i}}}$. Therefore, the codimension of (3) is the codimension of (5) minus the dimension of $\mathbb{G}(b,r)$.

To compute the codimension of (5) in $\prod_{i=1}^{k}{W_{b,d_{i}}}$, it suffices to prove the statement of Lemma 10 when $b=r$, so we want to bound the codimension of

[TABLE]

in $\prod_{i=1}^{k}{W_{r,d_{i}}}$ from below. To do this, suppose $(F_{1},\ldots,F_{k})$ is in (6). Then, there exist $a$ values (possibly more) $1\leq i_{1}<\cdots<i_{a}=k$ where for $i=i_{j}$, $F_{i_{j}}$ must vanish on one of the nondegenerate components $Y$ of $\{F_{1}=\cdots=F_{i_{j}-1}=0\}$. Since that component $Y$ is dimension at least $r-i_{j}+j$, it is at least $h_{r,r-i_{j}+j}(d_{i_{j}})$ conditions for $F_{i_{j}}$ to vanish on $Y$. Taking this over all possibilities for $1\leq i_{1}<\cdots<i_{a}=k$ yields the bound

[TABLE]

which is what we wanted. ∎

Remark*.*

The part of the proof of Lemma 10 that can be made more precise is the very end, where we argue there are $a$ instances where the next hypersurface must contain a nondegenerate component of the intersection of the previous hypersurfaces and thus obtain a bound on the codimension of (6). The arguments in [Tse18] make this part rigorous, but we choose not to repeat the arguments here for the sake of clarity and length.

If the reader wants a more detailed proof of the bound (7), see [Tse18, Lemma 4.2] for the lower bound for the codimension of $\Phi^{\mathbb{P}^{r},a}_{d_{1},\ldots,d_{k}}(\mathbb{P}^{r},\operatorname{Span}(r,r))$ in $\prod_{i=1}^{k}{W_{r,d_{i}}}$ given as

[TABLE]

The only difference is we minimized over all $1\leq i_{1}<\cdots<i_{a}\leq k$ instead of over $1\leq i_{1}<\cdots<i_{a}=k$. See also [Tse18, Section 2] for an example of how the argument works without the heavy notation.

Proof of Theorem 9.

By setting up the usual incidence correspondence,

{\{(F_{1},\ldots,F_{r+a-1}),\ell\mid F_{i}\text{ restricts to 0 on the line }\ell\text{ for all }i\}}$${\mathbb{G}(1,r)}$${\prod_{i=1}^{r+a-1}W_{r,d_{i}}}

one sees that the locus of forms $(F_{1},\ldots,F_{r+a-1})$ where $\{F_{1}=\cdots=F_{r+a-1}=0\}$ contains some line (e.g. $\Phi^{\mathbb{P}^{r},a}_{d_{1},\ldots,d_{r+a-1}}(\mathbb{P}^{r},\operatorname{Span}(r,1))$) is codimension

[TABLE]

in $\prod_{i=1}^{r+a-1}W_{r,d_{i}}$. Now, we want to apply Lemma 10 and show the codimension of

[TABLE]

is always greater than the codimension of $\Phi^{\mathbb{P}^{r},a}_{d_{1},\ldots,d_{r+a-1}}(\mathbb{P}^{r},\operatorname{Span}(r,1))$ if $b>1$.

From Lemma 10

[TABLE]

is a lower bound for the codimension of (9). We can bound (10) from below by

[TABLE]

which is

[TABLE]

which is always greater than (8) since $b>1$. ∎