Demailly's conjecture on Waldschmidt constants for sufficiently many very general points in $\mathbb{P}^n$

TL;DR

This paper advances the understanding of Demailly's conjecture on Waldschmidt constants for very general points in projective space, providing new conditions under which the conjecture holds, especially for large sets of points.

Contribution

The authors improve existing results by establishing broader conditions for Demailly's conjecture to hold for very general points in projective space.

Findings

Demailly's conjecture verified for all m when s is a perfect n-th power.

Conjecture holds if s exceeds certain bounds related to n, assuming Nagata-Iarrobino conjecture.

Enhanced conditions for the conjecture's validity for large s and specific geometric configurations.

Abstract

Let $Z$ be a finite set of $s$ points in the projective space $\mathbb{P}^n$ over an algebraically closed field $F$. For each positive integer $m$, let $\alpha(mZ)$ denote the smallest degree of nonzero homogeneous polynomials in $F[x_0,\ldots,x_n]$ that vanish to order at least $m$ at every point of $Z$. The Waldschmidt constant $\widehat{\alpha}(Z)$ of $Z$ is defined by the limit \[ \widehat{\alpha}(Z)=\lim_{m \to \infty}\frac{\alpha(mZ)}{m}. \] Demailly conjectured that \[ \widehat{\alpha}(Z)\geq\frac{\alpha(mZ)+n-1}{m+n-1}. \] Recently, Malara, Szemberg, and Szpond established Demailly's conjecture when $Z$ is very general and \[ \lfloor\sqrt[n]{s}\rfloor-2\geq m-1. \] Here we improve their result and show that Demailly's conjecture holds if $Z$ is very general and \[ \lfloor\sqrt[n]{s}\rfloor-2\ge \frac{2\varepsilon}{n-1}(m-1), \] where $0\le \varepsilon<1$ is the fractional part of $\sqrt[n]{s}$. In particular, for $s$ very general points where $\sqrt[n]{s}\in\mathbb{N}$ (namely $\varepsilon=0$), Demailly's conjecture holds for all $m\in\mathbb{N}$. We also show that Demailly's conjecture holds if $Z$ is very general and \[ s\ge\max\{n+7,2^n\}, \] assuming the Nagata-Iarrobino conjecture $\widehat{\alpha}(Z)\ge\sqrt[n]{s}$.

Equations

$$\hat{\alpha }(Z)=\underset{m\to \infty }{\lim }\frac{\alpha (mZ)}{m}.$$

$$\hat{\alpha }(Z)\ge \frac{\alpha (mZ)+n-1}{m+n-1}.$$

$$\lfloor \sqrt[n]{s}\rfloor -2\ge m-1.$$

$$\lfloor \sqrt[n]{s}\rfloor -2\ge \frac{2\epsilon }{n-1}(m-1),$$

$$s\ge \max \{n+7,2^n\},$$

$$\alpha (J)=\min \{\mathrm{deg}f\mid f\text{ is a nonzero homogeneous polynomial in }J\}.$$

$$I^{(m)}=⟨f\in F[x_0,\dots ,x_n]\mid f\text{ is homogeneous and vanishes to order ≥m at each }{p}_i⟩$$

$$\hat{\alpha }(I)=\underset{m\ge 1}{\inf }\frac{\alpha (I^{(m)})}{m}.$$

$$\hat{\alpha }(I)=\underset{m\to \infty }{\lim }\frac{\alpha (I^{(m)})}{m}.$$

$$\hat{\alpha }(I)\ge \frac{\alpha (I^{(m)})+n-1}{m+n-1}$$

$$\lfloor \sqrt[n]{s}\rfloor -2\ge m-1.$$

$$\frac{\alpha (I^{(m)})+n-1}{m+n-1}\le \sqrt[n]{s}.$$

$$\frac{\alpha (I^{(m)})+n-1}{m+n-1}\le \lfloor \sqrt[n]{s}\rfloor .$$

$$\lfloor \sqrt[n]{s}\rfloor -2\ge \frac{2\epsilon }{n-1}(m-1),$$

$$\delta$$

$$=\sqrt[n]{s}\cdot m+(\sqrt[n]{s}-1)(n-1)-1$$

$$\ge \sqrt[n]{s}\cdot m+(\sqrt[n]{s}-1)(n^{\prime }-1)-1$$

$$=\sqrt[n]{s}(m+n^{\prime }-1)-n^{\prime }+1-1$$

$$=\sqrt[n]{s}(m+n^{\prime }-1)-n^{\prime }.$$

$$\delta +n^{\prime }$$

$$(\delta +n)(\delta +n-1)\dots (\delta +1)>s(m+n-1)(m+n-2)\dots m,$$

$$\left(\genfrac{}{}{0ex}{}{\delta +n}{n}\right)>\left(\genfrac{}{}{0ex}{}{m+n-1}{n}\right)\cdot s.$$

$$\alpha (I^{(m)})\le \delta =\lfloor \sqrt[n]{s}(m+n-1)\rfloor -n+1.$$

$$\frac{\alpha (I^{(m)})+n-1}{m+n-1}\le \frac{\lfloor \sqrt[n]{s}(m+n-1)\rfloor }{m+n-1}\le \sqrt[n]{s}.$$

$$k-2\ge \frac{2\epsilon }{n-1}(m-1),$$

$$\left(\genfrac{}{}{0ex}{}{k(m+n-1)+1}{n}\right)>\left(\genfrac{}{}{0ex}{}{m+n-1}{n}\right)\cdot s.$$

$$\left(\genfrac{}{}{0ex}{}{k(m+n-1)+1}{n}\right)>\left(\genfrac{}{}{0ex}{}{m+n-1}{n}\right)\cdot (k+\epsilon {)}^n,$$

$$\bigl{(}k(m+n-1)+1\bigr{)}\cdots\bigl{(}k(m+n-1)+1-(n-1)\bigr{)}>(m+n-1)\cdots m\cdot(k+\varepsilon)^{n}.$$

$$\bigl{(}k(m+n-1)+1-i\bigr{)}\bigl{(}k(m+n-1)+1-(n-1)+i\bigr{)}\\ >(k+\varepsilon)^{2}(m+n-1-i)(m+i)$$

$$\text{LHSeqrefE:i-th inequ}-\text{RHSeqrefE:i-th inequ}=A{m}^2+Bm+C_i,$$

$$A=k^2-(k+\epsilon {)}^2=-2k\epsilon -{\epsilon }^2\le 0,$$

$$B=(n-1)(k^2-(1+2\epsilon )k-{\epsilon }^2)+2k,$$

Full text

[TABLE]

Demailly conjectured that

[TABLE]

Malara, Szemberg, and Szpond [10] established Demailly’s conjecture when $Z$ is very general and

[TABLE]

Here we improve their result and show that Demailly’s conjecture holds if $Z$ is very general and

[TABLE]

where $0\leq\varepsilon<1$ is the fractional part of $\sqrt[n]{s}$. In particular, for $s$ very general points where $\sqrt[n]{s}\in\mathbb{N}$ (namely $\varepsilon=0$), Demailly’s conjecture holds for all $m\in\mathbb{N}$. We also show that Demailly’s conjecture holds if $Z$ is very general and

[TABLE]

assuming the Nagata-Iarrobino conjecture $\widehat{\alpha}(Z)\geq\sqrt[n]{s}$.

1. Introduction

Let $Z=\{p_{1},\ldots,p_{s}\}$ be a set of $s$ points in the projective space $\mathbb{P}^{n}$ over an algebraically closed field $F$, and let $I\subseteq F[x_{0},\ldots,x_{n}]$ be the homogeneous ideal of $Z$. For any homogeneous ideal $J\subseteq F[x_{0},\ldots,x_{n}]$, we write

[TABLE]

Given a positive integer $m$, an interesting question is how large the degree of a hypersurface has to be in order for it to pass through each point of $Z$ with multiplicity at least $m$. In more algebraic terms, this is asking about $\alpha(I^{(m)})$, where

[TABLE]

is the $m$-th symbolic power of $I$.

An asymptotic invariant closely related to this question is the Waldschmidt constant $\widehat{\alpha}(I)$ of $I$, defined by

[TABLE]

It is known [1, Lemma 2.3.1] that, in fact,

[TABLE]

Waldschmidt constants have been studied in various branches of mathematics, such as complex analysis [13, 11], commutative algebra [8], and algebraic geometry [5]. By definition, $\widehat{\alpha}(I)\leq\alpha(I^{(m)})/m$ for all $m\in\mathbb{N}$. An interesting conjecture of Demailly predicts an inequality in the reverse direction.

Demailly’s Conjecture** ([2]).**

Let $I$ be the homogeneous ideal of any finite set of $s$ points in $\mathbb{P}^{n}$. Then

[TABLE]

for all $m\in\mathbb{N}$.

Demailly’s conjecture is known for $n=2$ over an algebraically closed field of characteristic [math] [5]. Demailly’s conjecture for $m=1$ (also called the Chudnovsky’s conjecture) is known for very general points over an algebraically closed field of characteristic [math] [7], and for $s\geq 2^{n}$ very general points over an arbitrary algebraically closed field [4].

Recently, Malara, Szemberg, and Szpond [10] established Demailly’s conjecture for $s\geq(m+1)^{n}$ very general points in $\mathbb{P}^{n}$.111There is a preprint that improves this to $s\geq m^{n}$: see [3, Theorem 4.8]. For later comparison with our Corollary 2, note that the condition $s\geq(m+1)^{n}$ is equivalent to $\lfloor\sqrt[n]{s}\rfloor\geq m+1$, or

[TABLE]

We now state our main theorem in this paper.

Theorem 1**.**

Let $I$ be the homogeneous ideal of any finite set of $s$ points in $\mathbb{P}^{n}$.

1. (a)

For all $m\in\mathbb{N}$,

[TABLE] 2. (b)

If $\lfloor\sqrt[n]{s}\rfloor-2\geq\frac{2\varepsilon}{n-1}(m-1)$, where $0\leq\varepsilon<1$ is the fractional part of $\sqrt[n]{s}$, then

[TABLE]

Since $\widehat{\alpha}(I)\geq\lfloor\sqrt[n]{s}\rfloor$ if $I$ is the homogeneous ideal of $s$ very general points in $\mathbb{P}^{n}$ [6, 4], Theorem 1 (b) implies the following Corollary 2, which is an improvement of the aforementioned result in [10] by Malara, Szemberg, and Szpond.

Corollary 2**.**

Demailly’s conjecture holds for $s$ very general points in $\mathbb{P}^{n}$ as long as

[TABLE]

where $0\leq\varepsilon<1$ is the fractional part of $\sqrt[n]{s}$. In particular, for $s$ very general points where $\sqrt[n]{s}\in\mathbb{N}$ (namely $\varepsilon=0$), Demailly’s conjecture holds for all $m\in\mathbb{N}$.

It was conjectured by Iarrobino [9] (and by Nagata [12] for $n=2$) that $\widehat{\alpha}(I)\geq\sqrt[n]{s}$ if $I$ is the homogeneous ideal of $s\geq\max\{n+7,2^{n}\}$ very general points in $\mathbb{P}^{n}$. Hence Theorem 1 (a) implies the following

Corollary 3**.**

Demailly’s conjecture holds for $s\geq\max\{n+7,2^{n}\}$ very general points in $\mathbb{P}^{n}$, provided that the Nagata-Iarrobino conjecture holds.

2. Proof of Theorem 1

(a) Let $\delta=\lfloor\sqrt[n]{s}(m+n-1)-n+1\rfloor.$ Since $0\leq(\sqrt[n]{s}(m+n-1)-n+1)-\delta<1$,

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Therefore, we have

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and then

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which implies

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It thus follows from dimension count that there exists a homogeneous polynomial of degree $\delta$ in $F[x_{0},\ldots,x_{n}]$ vanishing at any given $s$ points in $\mathbb{P}^{n}$ to order at least $m$. Hence

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Thus

[TABLE]

(b) Let $k=\lfloor\sqrt[n]{s}\rfloor\in\mathbb{N}$. We want to show that if

[TABLE]

then $\alpha(I^{(m)})\leq k(m+n-1)-n+1$, that is, $I^{(m)}$ contains a homogeneous polynomial of degree $k(m+n-1)-n+1$. By dimension count, it suffices to show that

[TABLE]

Since $\sqrt[n]{s}=k+\varepsilon$, the inequality (2) can be written as

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which is equivalent to

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So to establish (2), it is sufficient to show that

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for each $i=0,1,\ldots,\lfloor\frac{n-1}{2}\rfloor$.

Write the left-hand side minus the right-hand side of (3) as a polynomial in $m$:

[TABLE]

where

[TABLE]

[TABLE]

and

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Viewed as a quadratic function of $i$, $C_{i}$ attains its minimum at $i=\frac{n-1}{2}$, and the minimum value is

[TABLE]

Here $C>0$ because $k\geq 2$ by assumption (1). Since $C_{i}\geq C$ for all $i$, to establish (3), it suffices to show that $Am^{2}+Bm+C>0$.

If $\varepsilon=0$, then $A=0$, $B>0$, and $C>0$, so $Am^{2}+Bm+C>0$ for all $m\in\mathbb{N}$. If $\varepsilon>0$, then the assumption (1) can be rewritten as

[TABLE]

To show that $Am^{2}+Bm+C$ is positive under this assumption, it suffices to show that $Am^{2}+Bm+C>0$ at $m=\frac{n-1}{2\varepsilon}(k-2)+1$, because $A<0$ and $C>0$.

Set $m=\frac{n-1}{2\varepsilon}(k-2)+1$, and denote $N=n-1$. Then

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where

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and

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Note that $f(k,\varepsilon)$ and $g(k,\varepsilon)$ both tend to [math] as $\varepsilon\to 1$, and their partial derivatives with respect to $\varepsilon$ are negative on $0<\varepsilon<1$, $k\geq 2$, which implies that they are both positive on $0<\varepsilon<1$ and $k\geq 2$:

[TABLE]

[TABLE]

Therefore $Am^{2}+Bm+C>0$ at $m=\frac{n-1}{2\varepsilon}(k-2)+1$, and the proof is complete.