An Eisenbud-Goto-type Upper Bound for the Castelnuovo-Mumford Regularity of Fake Weighted Projective Spaces

TL;DR

This paper establishes an Eisenbud-Goto-type upper bound for the Castelnuovo-Mumford regularity of certain projective toric varieties, specifically focusing on very ample lattice simplices and fake weighted projective spaces.

Contribution

It introduces a new upper bound for the regularity of specific classes of projective toric varieties, extending Eisenbud-Goto bounds to fake weighted projective spaces.

Findings

Upper bound for the $k$-normality of very ample lattice simplices

Eisenbud-Goto-type bound for certain projective toric varieties

Application to fake weighted projective spaces

Abstract

We will give an upper bound for the $k$-normality of very ample lattice simplices, and then give an Eisenbud-Goto-type bound for some special classes of projective toric varieties.

Equations

$$\mathrm{reg}(X)\le \mathrm{deg}(X)-\mathrm{codim}(X)+1?$$

$$k_X\le \dim X+\lfloor \frac{\dim X}{2}\rfloor -1.$$

$$\mathrm{reg}(X)\le \mathrm{deg}(X)-\mathrm{codim}(X)+\lfloor \frac{\dim X}{2}\rfloor .$$

$$H^i(X,\mathcal{F}(k-i))=0$$

$$\underset{k\text{ times}}{\underbrace{P\cap M+\cdots +P\cap M}}\to kP\cap M$$

$$\mathrm{reg}(X)=\max \{k_P,\mathrm{deg}(P)\}+1.$$

$$\max \{\mathrm{deg}(P),k_P\}\le \mathrm{Vol}(P)-|P\cap M|+d+1?$$

$$\mathrm{deg}(P)\le \mathrm{Vol}(P)-|P\cap M|+d+1.$$

$$k_P\le \mathrm{Vol}(P)-|P\cap M|+d+1.$$

$${\mathrm{Ehr}}_P(t)=\sum _{k=0}^{\infty }{\mathrm{ehr}}_P(k)t^k=\frac{h_P^{\ast }(t)}{(1-t{)}^{d+1}}$$

$$h_0^{\ast }=1,h_1^{\ast }=|P\cap M|-d-1,h_d^{\ast }=|P^0\cap M|,\text{ and }\sum _{i=0}^d{h}_i^{\ast }=\mathrm{Vol}(P).$$

$$\mathrm{deg}(P)=\{\begin{array}{ll}d& \text{if }|P^0\cap M|\ne 0.\\ \min \left\{i\in {\mathbb{Z}}_{\ge 0}|(kP{)}^0\cap M=\varnothing \text{ for all }1\le k\le d-i\right\}& \text{otherwise}.\end{array}$$

$$x+(k-1)v_i=\sum _{j=1}^{2k-1}{u}_j$$

$$\sum _{i=0}^d{a}_i{v}_i+x=\sum _{i=1}^{2k-1}{u}_i$$

$$\sum _{i=0}^d{a}_i{v}_i+x=\sum _{i=1}^{2s-1}{u}_i$$

$$(s-1)v_0+x=\sum _{i=1}^{2s-1}{w}_i$$

$$\sum _{i=0}^d{a}_i{v}_i+x=\sum _{i=0}^d{a}_i{v}_i+\sum _{i=1}^{s-1}{t}_i+w_{2s-1},$$

$$\underset{a_0-1+{\sum }_{i=1}^d{a}_i=s-2}{\underbrace{(a_0-1)v_0+\sum _{i=1}^d{a}_i{v}_i}}+y=\sum _{i=1}^{2(s-1)-1}{w}_i^{\prime }$$

$$\sum _{i=0}^d{a}_i{v}_i+x$$

$$=w_1+w_2+y+(a_0-1)v_0+\sum _{i=1}^d{a}_i{v}_i$$

$$=w_1+w_2+\sum _{i=0}^{2(s-1)-1}{w}_i^{\prime }.$$

$$(k+1)P\cap M=P\cap M+kP\cap M.$$

$$(k+1)P\cap M=\mathcal{V}+kP\cap M.$$

$$k_P\le {\nu }_P+d_P-1.$$

$$p=x+\sum _{i=1}^{{\nu }_P-d_P}{u}_i+\sum _{i=0}^d{a}_i{v}_i$$

$$x+\sum _{i=0}^d{a}_i{v}_i=\sum _{i=1}^{d_P+k-{\nu }_P}{u}_i^{\prime }$$

$$p=\sum _{i=1}^{{\nu }_P-d_P}{u}_i+\sum _{i=1}^{d_P+k-{\nu }_P}{u}_i^{\prime }.$$

$$\mathcal{V}+kP\cap M\twoheadrightarrow (k+1)P\cap M.$$

$$x=v_0+(a_0-1)v_0+\sum _{i=1}^d{a}_i{v}_i=v_0+\left((a_0-1)v_0+\sum _{i=1}^d{a}_i{v}_i\right)\in \mathcal{V}+kP\cap M.$$

$$k_P\le \mathrm{Vol}(P)-|P\cap M|+d+\lfloor \frac{d}{2}\rfloor .$$

$$k_P$$

Full text

1. Introduction

The study of the Castelnuovo-Mumford regularity for projective varieties has been greatly movitated by the Eisenbud-Goto conjecture ([EG84]) which asks for any irreducible and reduced variety $X$, is it always the case that

[TABLE]

The Eisenbud-Goto conjecture is known to be true for some particular cases. For example, it holds for smooth surfaces in characteristic zero ([Laz87]), connected reduced curves ([Gia05]), etc. Inspired by the conjecture, there are also many attempts to give an upper bound for the Castelnuovo-Mumford regularity for various types of algebraic and geometric structures ([Stu95], [Kwa98], [Miy00], [DS02], etc).

For toric geometry, suppose that $(X,L)$ is a polarized projective toric varieties such that $L$ is very ample. Then there is a corresponding very ample lattice polytope $P:=P_{L}$ associated to $L$ such that $\Gamma(X,L)=\bigoplus_{m\in P\cap M}\mathbb{C}\cdot\chi^{m}$ ([CLS11, Section 5.4]). Therefore, by studying the $k$-normality of $P$ (cf. Definition 2.2), we can obtain the $k$-normality and also the regularity of the original variety $(X,L)$. For the purpose of this article, we will focus on the case that $X$ is a fake weighted projective $d$-space and $P_{L}$ a $d$-simplex.

For any fake weighted projective $d$-space $X$ embedded in $\mathbb{P}^{r}$ via a very ample line bundle, Ogata ([Oga05]) gives an upper bound for its $k$-normality:

[TABLE]

In this article, we will improve Ogata’s bound by giving a new upper bound for the $k$-normality of very ample lattice simplices and show that

[TABLE]

Recently, McCullough and Peeva showed some counterexamples to the Eisenbud-Goto conjecture and that the difference $\operatorname{reg}(X)-\deg(X)+\operatorname{codim}(X)$ can be arbitrary large ([MP17, Counterexample 1.8]). However, for any fake weighted projective space $X$ embedded in $\mathbb{P}^{r}$ via a very ample line bundle, it follows from (2) that $\operatorname{reg}(X)-\deg(X)+\operatorname{codim}(X)$ is bounded above by $\dim(X)/2$. Furthermore, we will show that the Eisenbud-Goto conjecture holds for any projective toric variety corresponding to a very ample Fano simplex.

Acknowledgments

We would like to thank Milena Hering for reading the drafts of this article and for some valuable suggestions.

2. Background Material

2.1. Toric Varieties and Lattice Simplices

We begin this section by recalling the definition of the Castelnuovo-Mumford regularity:

Definition 2.1**.**

Let $X\subseteq\mathbb{P}^{r}$ be an irreducible projective variety and $\mathscr{F}$ a coherent sheaf over $X$. We say that $\mathscr{F}$ is $k$-regular if

[TABLE]

for all $i>0$. The regularity of $\mathscr{F}$, denoted by $\operatorname{reg}(\mathscr{F})$, is the minimum number $k$ such that $\mathscr{F}$ is $k$-regular. We say that $X$ is $k$-regular if the ideal sheaf $\mathcal{I}_{X}$ of $X$ is $k$-regular and use $\operatorname{reg}(X)$ to denote the regularity of $X$ (or of $\mathcal{I}_{X}$).

As the main object of the article is to find an upper bound for $k$-normality of very ample lattice simplices, it is important for us to revisit the definition of $k$-normality of lattice polytopes.

Definition 2.2**.**

A lattice polytope $P$ is $k$-normal if the map

[TABLE]

is surjective. The $k$-normality of $P$, denoted by $k_{P}$, is the smallest positive integer $k_{P}$ such that $P$ is $k$-normal for all $k\geq k_{P}$.

Suppose now that $X$ is a fake weighted projective $d$-space embedded in $\mathbb{P}^{r}$ via a very ample line bundle. Then the polytope $P$ corresponding to the embedding is a very ample lattice $d$-simplex. Furthermore, $\operatorname{codim}(X)=|P\cap M|-(d+1)$, where $M$ is the ambient lattice, and $\deg(X)=\operatorname{Vol}(P)$, the normalized volume of $P$.

We have a combinatorial interpretation of $\operatorname{reg}(X)$ in terms of $k_{P}$ and $\deg(P)$ ([Tra18, Proposition 4.1.5]) as follows:

[TABLE]

From this, we obtain a combinatorial form of the Eisenbud-Goto conjecture: for very ample lattice polytope $P\subset M_{\mathbb{R}}$, is it always true that

[TABLE]

The first inequality was confirmed to be true recently ([HKN17, Proposition 2.2]); namely,

[TABLE]

Therefore, in order to verify the Eisenbud-Goto conjecture for the polarized toric variety $(X,L)$, it suffices to check if

[TABLE]

2.2. Ehrhart Theory

We now recall some basic facts about Ehrhart theory of polytopes and the definition of their degree.

Let $P$ be a lattice polytope of dimension $d$. We define $\operatorname{ehr}_{P}(k)=|kP\cap M|$, the number of lattice points in $kP$. Then from Ehrhart’s theory ([Ehr62, Sta80]),

[TABLE]

for some polynomial $h_{P}^{*}\in\mathbb{Z}_{\geq 0}[t]$ of degree less than or equal to $d$. Let $h_{P}^{*}(t)=\sum_{i=0}^{d}h_{i}^{*}t^{i}$. We have

[TABLE]

Definition 2.3** ([BN07, Remark 2.6]).**

Let $P$ be a lattice polytope of dimension $d$. We define the degree of $P$, denoted by $\deg(P)$, to be the degree of $h_{P}^{*}(t)$. Equivalently,

[TABLE]

3. $k$-normality of Very Ample Simplices

The following lemma by Ogata is crucial to the main result of this article:

Lemma 3.1** ([Oga05, Lemma 2.1]).**

Let $P=\operatorname{conv}(v_{0},\ldots,v_{d})$ be a very ample lattice $n$-simplex. Suppose that $k\geq 1$ is an integer and $x\in kP\cap M$. For any $i=0,\ldots,d$, we have

[TABLE]

for some $u_{j}\in P\cap M$.

Using the ideas in [Oga05, Lemma 2.5], we generalize the above lemma as follows.

Lemma 3.2**.**

Suppose that $P=\operatorname{conv}(v_{0},\ldots,v_{d})$ is a very ample $d$-simplex. Let $k\in\mathbb{N}_{\geq 1}$. Then for any $x\in kP\cap M$, $a_{0},\ldots,a_{d}\in\mathbb{Z}_{\geq 0}$ such that $\sum_{i=0}^{d}a_{i}=k-1$, we have

[TABLE]

for some $u_{i}\in P\cap M$.

Proof.

We will use induction in this proof. The case $k=1$ is trivial. Suppose that the lemma holds for $k=s-1$. We will now show that it holds for $k=s$; i.e., for any $x\in sP\cap M$, $a_{1},\ldots,a_{d}\in\mathbb{Z}_{\geq 0}$ such that $\sum_{i=0}^{d}a_{i}=s-1$, we have

[TABLE]

for some $u_{i}\in P\cap M$. Without loss of generality, we can take $a_{0}$ to be positive and move $v_{0}$ to the origin. By Lemma 3.1,

[TABLE]

for some $w_{i}\in P\cap M$. Since $v_{0}=0$, we can write $x=\sum_{i=1}^{2s-1}w_{i}$. If $w_{i}+w_{j}\in P\cap M$ for any $i\neq j$, then we can let $t_{i}=w_{2i-1}+w_{2i}$ for $i=1,\ldots,s-1$ and have $x=t_{1}+\cdots+t_{s-1}+w_{2s-1}$, which lies in $\sum_{i=1}^{s}P\cap M$. Therefore,

[TABLE]

which satisfies (6). Conversely, without loss of generality, suppose that $w_{1}+w_{2}\notin P\cap M$. Then since $x=w_{1}+w_{2}+(w_{3}+\cdots+w_{2s-1})\in sP\cap M$, we have $y:=w_{3}+\cdots+w_{2s-1}\in(s-1)P\cap M$ and $v_{0}+x=w_{1}+w_{2}+y$. Using the induction hypothesis,

[TABLE]

for some $w^{\prime}_{i}\in P\cap M$. It follows that

[TABLE]

The conclusion follows. ∎

Now define the invariants $d_{P}$ and $\nu_{P}$ as in [Tra18, Definition 2.2.8]:

Definition 3.3**.**

Let $P$ be a lattice polytope with the set of vertices $\mathcal{V}=\{v_{0},\ldots,v_{n-1}\}$. We define $d_{P}$ to be the smallest positive integer such that for every $k\geq d_{P}$,

[TABLE]

We also define $\nu_{P}$ to be the smallest positive integer such that for any $k\geq\nu_{P}$,

[TABLE]

Notice that for $P$ an $n$-simplex, $d_{P}\leq\nu_{P}\leq n-1$.

Proposition 3.4**.**

Let $P=\operatorname{conv}(v_{0},\ldots,v_{d})$ be a very ample $d$-simplex. Then

[TABLE]

Proof.

For any $k\geq d_{P}+\nu_{P}-1$ and $p\in kP\cap M$, by the definition of $d_{P}$ and $\nu_{P}$, we have

[TABLE]

for some $x\in d_{P}P\cap M$, $u_{i}\in P\cap M$, $\sum_{i=0}^{d}a_{i}=k-\nu_{P}$. By assumption, $k-\nu_{P}\geq d_{P}-1$, so it follows from Lemma 3.2 that

[TABLE]

for some $u^{\prime}_{i}\in P\cap M$. Substitute (8) into (7), we have

[TABLE]

The conclusion follows. ∎

Remark 3.5*.*

This bound is stronger than [Oga05, Proposition 2.4] since $\nu_{P}\leq d$ ([Tra18, Proposition 2.2]) and $d_{P}\leq d/2$ ([Oga05, Proposition 2.2]).

4. An Eisenbud-Goto-Type Upper Bound for Very Ample Simplices

Suppose that $P$ is a very ample simplex. If $P$ is unimodularly equivalent to the standard simplex $\Delta_{d}=\operatorname{conv}(0,e_{1},\ldots,e_{d})$ then (5) holds. Now consider the case $P$ is not unimodularly equivalent to $\Delta_{d}$.

The following lemma is a rewording of [Her06, Proposition IV.10] to fit our purpose. We provide a proof for the sake of completeness.

Lemma 4.1**.**

Let $\mathcal{V}=\{v_{0},\ldots,v_{d}\}$ and suppose that $P=\operatorname{conv}(\mathcal{V})$ is a lattice simplex not unimodularly equivalent to $\Delta_{d}$. Then $\deg(P)\geq\nu_{P}$.

Proof.

Since $\nu_{P}\leq d$, it suffices to show that for any $d\geq k\geq\deg(P)$,

[TABLE]

Indeed, any $x\in(k+1)P\cap M$ can be written as $x=\sum_{i=0}^{d}a_{i}v_{i}$ such that $a_{i}\geq 0$ and $\sum_{i=0}^{d}a_{i}=k+1$. If $a_{i}<1$ for all $i$, then $d>k$ and the point $\sum_{i=0}^{d}(1-a_{i})v_{i}$ is an interior lattice point of $(d-k)P$, a contradiction since $d-k\leq d-\deg(P)$. Hence, $a_{i}\geq 1$ for some $i$, say $a_{0}\geq 1$. Then

[TABLE]

Hence, $k\geq\nu_{P}$. The conclusion follows. ∎

Proposition 4.2**.**

Let $P=\operatorname{conv}(v_{0},\ldots,v_{d})$ be a very ample simplex. Then

[TABLE]

Proof.

Form Proposition 3.4, (4), and Lemma 4.1,

[TABLE]

By [Oga05, Proposition 2.2], $d_{P}\leq\frac{d}{2}$. Therefore, since $k_{P}$, $\operatorname{Vol}(P)$, and $|P\cap M|$ are all integers,

[TABLE]

∎

Remark 4.3*.*

We show some cases that the result of Proposition 4.2 is stronger than [Oga05, Proposition 2.4]:

1. (1)

$\operatorname{Vol}(P)\leq|P\cap M|+2$. In this case,

[TABLE]

Example 4.4*.*

Let $\Delta_{d}$ be the standard $d$-simplex. Then

[TABLE]

This is clearly a better bound compared to $d+\left\lfloor\frac{d}{2}\right\rfloor-1$. 2. (2)

$P^{0}\cap M=\emptyset$ or equivalently $\deg(P)\leq d-1$. Indeed, in this case,

[TABLE]

We will show in next section that this is the only case that we still need to consider in order to verify the Eisenbud-Goto conjecture for very ample simplices.

Example 4.5*.*

Consider $P=2\Delta_{d}$ for $d\geq 4$, where $\Delta_{d}$ is the standard $d$-simplex. Then $\deg(P)=2$ and by Proposition 3.4,

[TABLE]

Theorem 4.6**.**

Suppose that $X$ is a fake weighted projective space embedded in $\mathbb{P}^{r}$ via a very ample line bundle. Then

[TABLE]

Proof.

Let $P$ be the corresponding polytope of the embedding. From (3), (4), and Proposition 4.2, it follows that

[TABLE]

∎

5. The Eisenbud-Goto Conjecture for Non-hollow Very Ample Simplices

In this section, we will improve the bound of $k$-normality for non-hollow very ample simplices.

Definition 5.1**.**

A lattice polytope is hollow if it has no interior lattice points.

We now show that the inequality (5) holds for non-hollow very ample simplices.

Proposition 5.2**.**

Let $P\subseteq M_{\mathbb{R}}$ be a non-hollow very ample lattice $d$-simplex. Then

[TABLE]

Proof.

We will consider two cases, namely $|P\cap M|=d+2$ and $|P\cap M|\geq d+3$. For the first case, we have the following lemma:

Lemma 5.3**.**

Suppose that $P=\operatorname{conv}(v_{0},\ldots,v_{d})$ is a very ample lattice $d$-simplex with $u$ is the only lattice point beside the vertices. Then $P$ is normal.

Proof.

Assume that $d_{P}\geq 2$. Then there exists a point $p\in d_{P}P\cap M$ such that $p$ cannot be written as $p=x+w$ for some $x\in(d_{P}-1)P\cap M$ and $w\in P\cap M$. Since $P$ is a simplex, $u$ and $p$ can be uniquely written as

[TABLE]

and

[TABLE]

respectively. It follows from the condition of $p$ that $\lambda_{i}<1$ for all $0\leq i\leq d$ and there exists $0\leq i\leq d$ such that $\lambda_{i}<\lambda_{i}^{*}$, say $i=0$. By Lemma 3.1,

[TABLE]

for some $a_{i},b\in\mathbb{Z}_{\geq 0}$ such that $b+\sum_{i=0}^{d}a_{i}=2d_{P}-1$. Replacing $p$ by $\sum_{i=0}^{d}\lambda_{i}v_{i}$ and $u$ by $\sum_{i=0}^{d}\lambda_{i}^{*}v_{i}$ yields

[TABLE]

Since $\lambda_{0}<\lambda_{0}^{*}$ and $\lambda_{i}<1$ for all $0\leq i\leq d$, it follows that $a_{0}=a_{2}=\cdots=a_{d}=0$ and $b=0$. Then $p=d_{P}v_{1}$, a contradiction to the choice of $p$. Therefore, $P$ is normal. ∎

As a consequence, $1=k_{P}\leq\operatorname{Vol}(P)-|P\cap M|+d+1=\operatorname{Vol}(P)-1$. Now we consider the case $|P\cap M|\geq d+3$. By the hypothesis, $|P\cap M|-(d+1)\geq 2$. Consider the Ehrhart vector $h^{*}=(h_{0}^{*},\cdots,h_{d}^{*})$ of $P$. We have

[TABLE]

By [Hib94, Theorem 1.1], $2\leq h_{1}^{*}\leq h_{i}^{*}$ for all $1\leq i<d$. Therefore,

[TABLE]

By [Oga05, Proposition 2.4],

[TABLE]

for all $d\geq 3$. The conclusion follows. ∎

Let us now recall the definition of Fano polytopes:

Definition 5.4**.**

A Fano polytope is a convex lattice polytope $P\subseteq M_{\mathbb{R}}$ such that $P^{0}\cap M=\{0\}$ and each vertex $v$ of $P$ is a primitive point of $M$.

From Proposition 5.2, we obtain the following corollary:

Corollary 5.5**.**

The Eisenbud-Goto conjecture holds for any projective toric variety corresponding to a very ample Fano simplex.

6. Final Remarks

We start with a remark that Proposition 3.4 fails in general.

Example 6.1* ([GB09]).*

Consider the polytope $P$ which is the convex hull of the vertices given by the columns of the following matrix

[TABLE]

with $s\geq 4$. Then $d_{P}=\nu_{P}=2$, and by [BDGM15, Theorem 3.3], $k_{P}=s-1$. It is clear that $k_{P}>d_{P}+\nu_{P}-1$ for all $s\geq 6$.

Furthermore, it can be shown that $P$ cannot be covered by very ample simplicies ([Tra18, Proposition 4.3.3]) ; hence, it is very unlikely that we can apply Proposition 3.4 to find a bound of the $k$-normality of generic very ample polytopes.

6.1. What About Hollow Very Ample Simplices

Finally, we would love to see a classification of hollow very ample lattice simplices. For dimension $2$, Rabinotwiz [Rab89, Theorem 1] showed that any such simplex is unimodularly equivalent to either $T_{p,1}:=\operatorname{conv}(0,(p,0),(0,1))$ for some $p\in\mathbb{N}$ or $T_{2,2}=\operatorname{conv}(0,(2,0),(0,2))$. Now we will show a way to obtain some hollow very ample simplices in any dimension with arbitrary volume.

We recall the definition of lattice pyramids as in [Nil08]:

Definition 6.2**.**

Let $B\subseteq\mathbb{R}^{k}$ be a lattice polytope with respect to $\mathbb{Z}^{k}$. Then $\operatorname{conv}(0,B\times\{1\})\subseteq\mathbb{R}^{k+1}$ is a lattice polytope with respect to $\mathbb{Z}^{k+1}$, called the ($1$-fold) standard pyramid over $B$. Recursively, we define for $l\in\mathbb{N}_{\geq 1}$ in this way the $l$-fold standard pyramid over $B$. As a convention, the [math]-fold standard pyramid over $B$ is $B$ itself.

Proposition 6.3**.**

Let $P$ be a lattice polytope. Then the 1-fold pyramid over $P$ is very ample if and only if $P$ is normal.

Proof.

Let $Q=\operatorname{conv}(0,P\times\{1\})$ be the 1-fold pyramid over $P$. Then it is easy to see that if $P$ is normal then so is $Q$. Now suppose that $Q$ is very ample. We have $k_{Q}\geq k_{P}$ ([Tra18, Lemma 4.2.2]) and each lattice point of $k_{Q}Q\cap M$ sits in $(tP\cap M)\times\{t\}$ for some $0\leq t\leq k_{Q}$. In particular, suppose that $(x,t)\in(tP\cap M)\times\{t\}\subseteq k_{Q}Q\cap M$. Then

[TABLE]

for some $u_{i}\in P\cap M$. It follows that $x=\sum_{i=1}^{t}u_{i}$. Hence, $P$ is $t$-normal for all $k_{Q}\geq t\geq 1$. Since $k_{Q}\geq k_{P}$, it follows that $P$ is normal. The conclusion follows. ∎

From Proposition 6.3, if we take any $(d-2)$-fold pyramid over either $T_{p,1}$ with $p\in\mathbb{Z}_{\geq 1}$ or $T_{2,2}$, which are all normal, then we obtain a hollow normal (hence very ample) $d$-simplex with normalized volume $p$. The Eisenbud-Goto conjecture holds for these.

Example 6.4*.*

We give here an example to demonstrate the case that if $Q$ is very ample but not normal then the $1$-fold pyramid over $Q$ is not very ample. Let $Q$ be the convex polytope given by taking $s=4$ in Example 6.1. Then $Q$ is very ample; however, the $1$-fold pyramid of $Q$, which is given by the convex hull of

[TABLE]

is not very ample.