January 01, 1970
We prove that every \(n\)-dimensional lattice simplex \(P\) whose lattice length \(L(P)\ge n-1\) is integrally closed. As an application, we obtain a simple criterion for the projective normality of ample line bundles on \(\mathbb{Q}\)-factorial toric Fano varieties with Picard number one. We further obtain a refinement of this result in terms of the invariant \(\Gamma_{P}\) introduced in [1].
Let \(M\) be a free \(\mathbb{Z}\)-module of rank \(n\ge 1\) and let \(M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}\) be the associated \(n\)-dimensional vector space. A polytope \(P\subseteq M_{\mathbb{R}}\) is defined as the convex hull of a finite set \(\{u_0,\dots,u_m\}\subseteq M_{\mathbb{R}}\), denoted by \(P=\textrm{Conv}(u_0,\cdots,u_m)\). After removing redundant elements, we may assume that \(u_0,\dots,u_m\) are the vertices of \(P\). We call \(P\) a lattice polytope if \(u_i\in M\) for all \(i\). Throughout this paper, all polytopes are assumed to be full-dimensional.
Let \(P,\;P_1,\;P_2\) be polytopes in \(M_{\mathbb{R}}\) and \(r\in \mathbb{R}\). The Minkowski sum \(P_1+P_2:=\{u_1+u_2\: | \: u_1\in P_1, u_2\in P_2\}\) and the dilation by scalar \(rP:=\{ru \: | \: u\in P\}\) . When \(r\) is a natural number, \(rP=\sum^r_{i=1} P\).
A fundamental problem concerning lattice polytopes is:
Problem 1. For which lattice polytope \(P\) does the equality \[\label{eq:32integrally32closed32P} P\cap M+(rP)\cap M=(r+1)P\cap M\qquad{(1)}\] hold for all positive integers \(r\)?
A lattice polytope that satisfies the equality ?? for all \(r\in \mathbb{Z}_{>0}\) is called integrally closed. From the perspective of algebraic geometry, the integral closedness of \(P\) is equivalent to the projective normality of the polarized toric variety \((X,L)\) associated with \(P\).
Let us briefly recall some basics in toric geometry (cf. [2], [3]). Let \(N=\text{Hom}(M,\mathbb{Z})\). Given a lattice polytope \(P\subseteq M_{\mathbb{R}}\), one associates to \(P\) a normal fan \(\Delta_P\), which defines a projective toric variety \(X\) together with an ample Cartier divisor \(D=D_P\). The associated ample line bundle is \(L=\mathcal{O}_X(D)\). Denote by \(T\) the algebraic torus in \(X\). For \(u\in M\), let \(\chi^u\) denote the corresponding regular function on \(T\), also viewed as a rational function on \(X\). Then one can describe the global sections of \(L\) through the lattice points in \(P\): \[\label{eq:32global32section32of32L} H^0(X,L)\cong \bigoplus_{u\in P\cap M} \mathbb{C}\cdot \chi^u.\tag{1}\] Moreover, the dilated polytope \(rP\) corresponds to the r-fold tensor product \(L^{\otimes r}\), and the multiplication map \[\label{eq:32multiplication32map} H^0(X,L)\otimes H^0(X,L^{\otimes r})\to H^0(X,L^{\otimes (r+1)})\tag{2}\] sends \(\chi^{u_1}\otimes \chi^{u_2}\), where \(u_1\in P\cap M\) and \(u_2\in rP\cap M\), to \(\chi^{u_1+u_2}\) under the identification 1 . Therefore, the equality ?? amounts to the surjectivity of the multiplication map 2 . Recall that a basepoint free line bundle \(L\) is said to be projectively normal if the multiplication map 2 is surjective for all \(r\in \mathbb{Z}_{>0}\). Hence, \(L\) is projectively normal if and only if \(P\) is integrally closed.
Projective normality for a general smooth projective variety is already highly nontrivial even for surfaces. For instance, Mukai’s conjecture (cf. [4]) remains widely open at the level of projective normality. Inspired by the Mukai conjecture, the first and third authors proposed the following conjecture in the toric setting, see [5].
Conjecture 2. Let \(P\subseteq M_{\mathbb{R}}\) be an \(n\)-dimensional lattice polytope. If \(L(P)\ge n-1\), then the equality ?? holds for all \(r\in \mathbb{Z}_{>0}\), i.e., \(P\) is integrally closed.
Here \(L(P)\) denotes the lattice length of \(P\) (see Definition 1). In the toric setting, we can interpret the lattice length in terms of intersection numbers. More precisely, each edge \(e\) of \(P\) corresponds to a \(T\)-invariant curve \(C\) on \(X\), and the intersection number \(L\cdot C\) is equal to the lattice length of \(e\), where \(L\) is the line bundle associated with \(P\) as above. By Definition 1, the lattice length of \(P\) is the minimum of the lattice lengths of all edges of \(P\). Therefore, Conjecture 2 can be rephrased as follows: if \(L\cdot C\ge n-1\) for every \(T\)-invariant curve \(C\), then \(L\) is projectively normal.
Several partial results toward this conjecture are known. In [6], Ogata and Nakagawa proved that every dilated polytope \(P=sQ\) with \(s\ge n-1\) is integrally closed. By Gubeladze [7], \(P\) is projectively normal provided that \(L(P)\ge 4n(n+1)\), and the condition can be relaxed to \(L(P)\ge n(n+1)\) for lattice simplices. However, it is desirable to have a condition that is linear in dimension. In [5], by special covering, we proved the conjecture in dimension \(3\) for every lattice simplex and in dimension \(4\) for most lattice simplices.
In this paper, we prove Conjecture 2 for lattice simplices.
Theorem 3. Let \(P\subseteq M_{\mathbb{R}}\simeq\mathbb{R}^n\) be a lattice simplex. Suppose \(L(P)\ge n-1\). Then \(P\) is integrally closed.
As explained in [8], every \(\mathbb{Q}\)-factorial toric Fano variety with Picard number one can be obtained from a lattice simplex, so we have
Corollary 1. Let \(X\) be an \(n\)-dimensional \(\mathbb{Q}\)-factorial toric Fano variety with Picard number one and \(L\) be an ample line bundle on \(X\). Suppose \(L\cdot C\ge n-1\) for any \(T\)-invariant curve \(C\) on \(X\). Then \(L\) is projectively normal. 0◻
In [1], González and the third author introduced an invariant \(\Gamma_P\) associated with \(P\) to study the k-jet ampleness of line bundles on toric varieties. The invariant \(\Gamma_P\) is a nonnegative rational number that measures the singularity of \(P\): the more singular \(P\) is, the larger \(\Gamma_P\) becomes. In general, \(\Gamma_P\le n-2\) whenever \(n\ge 2\).
The same line of reasoning as in the proof of Theorem 3 yields a more refined statement when \(P\) is “sufficiently singular".
Theorem 4. Let \(P\) be an \(n\)-dimensional lattice simplex in \(M_{\mathbb{R}}\). Assume that either
\(n=2\), or
\(n\ge 3\) and \(\Gamma_P\ge (\frac{2}{3})n-1\).
If \(L(P)\ge \Gamma_P+1\), then \(P\) is integrally closed.
Acknowledgments. During the preparation of the article, L.S. was partially supported by NSFC grants No. 12471043, No. 12371063 and Guangdong Basic and Applied Basic Research Foundation No. 2025A1515012258, and Z.Z. was partially supported by NSFC grant No. 12101423.
In this section, we introduce some definitions and fix notation.
Let \(P=\textrm{Conv}(u_0,\dots,u_m)\) be a lattice polytope in \(M_{\mathbb{R}}\), with vertices \(u_0,u_1,\ldots,u_m\).
Definition 1. For any pair of distinct vertices \(u_i, u_j\) of \(P\), let \(e_{ij}=\{\lambda u_i+(1-\lambda)u_j\;|\;0\le \lambda\le 1\}\) denote the line segment joining \(u_i\) and \(u_j\). The lattice length* of \(e_{ij}\) is defined by \[l_{ij}:=\#(e_{ij}\cap M)-1.\] We refer to \(l_{ij}\) as the lattice length of \(e_{ij}\), regardless of whether \(e_{ij}\) is an edge of \(P\). The lattice length of \(P\) is defined by \[L(P):=\min \{l_{ij}\mid e_{ij}\text{ is an edge of } P\}.\]*
For \(i\ne j\), set \[u_{ij}:=u_j-u_i \text{ and }\hat{u}_{ij}:=\frac{u_{ij}}{l_{ij}}.\] Thus \(\hat{u}_{ij}\) is the primitive lattice vector in the direction from \(u_i\) to \(u_j\).
For each vertex \(u_i\) of \(P\), let \(u_{j_1},\cdots,u_{j_k}\) be the vertices adjacent to \(u_i\). We define \(C(P,u_i)\) to be the cone generated by \(u_{ij_1},\cdots, u_{ij_k}\). Equivalently, \(C(P, u_i)\) is the tangent cone of \(P\) at \(u_i\), translated to the origin. Its primitive ray generators are \(\hat{u}_{ij_1},\cdots,\hat{u}_{ij_k}\).
Next we define the nonnegative invariant \(\Gamma_P\) associated to the lattice polytope \(P\).
Definition 2. Let \(Q\) be a strictly convex rational polyhedral cone in \(M_{\mathbb{R}}\), and let \(w_1,\cdots,w_s\) be the primitive ray generators. Let \(\beta_1,\cdots,\beta_t\) be the Hilbert basis of the affine semigroup \(Q\cap M\). Every \(u\in Q\cap M\) admits an expression in terms of either the ray generators \(w_i\) or the Hilbert basis elements \(\beta_j\). Define \[\textrm{ht}(u)=\max\{\sum_{i=1}^s a_i\;|\;u=\sum_{i=1}^s a_i w_i \text{ and } a_i\in \mathbb{R}_{\ge 0} \text{ for all } i\}\] and \[\textrm{ord}(u)=\max\{\sum_{j=1}^t \lambda_j\mid u=\sum_{j=1}^t \lambda_j \beta_j\text{ and } \lambda_j\in \mathbb{Z}_{\ge 0} \text{ for all } j\}.\] Let \(S_Q=\{\sum_{i=1}^s a_i w_i\;|\;0\le a_i<1 \text{ for all } i\}\). We define \[\Gamma_Q:=\max\{\textrm{ht}(u)-\textrm{ord}(u)\mid u\in S_Q\cap M\}.\]
Definition 3. Given a lattice polytope \(P\) with vertices \(u_0, \ldots, u_m\), define \[\Gamma_P:=\max\{\Gamma_{C(P,u_i)}\mid 0\le i\le m\}.\]
We will use the following bound.
Lemma 1 ([1]). For any lattice polytope \(P\) of dimension \(n\ge 2\), we have \(0\le \Gamma_P\le n-2\). Furthermore, if \(P\) is smooth, then \(\Gamma_P=0\).
In this section, we present a useful combinatorial partition lemma that provides an effective method of decomposing points. In the following, for a nonnegative integer \(l\), let \([l]=\{1, 2, \cdots, l\}\).
Lemma 2. Let \(1<r\) be a real number and let \(x_0,x_1,\dots,x_l\) be nonnegative real numbers. Let \(a\) and \(a'\) be real numbers such that \(a\le r-1, a'\le 1\), and \[a+a'+\sum\limits^l_{i=0}x_i=r.\] Assume that \(x_i\le x_0\) for each \(1\le i\le l\). Then there exists a subset \(I\) of \([l]\) satisfying the property \[a+\sum_{i\in I}x_i\le r-1 \text{ and } a'+\sum_{i\in [l]\backslash I}x_i\le 1.\]
Remark 5. In practice below, \(r\) is an integer, and \(a, a'\ge 0\).
Proof. First we observe that either \(a+x_1\le r-1\) or \(a'+x_1\le 1\). Indeed, if neither inequality holds, we have \[a+a'+2x_1>(r-1)+1=r,\] which implies \[x_1>r-(a+a'+x_1)=\sum^l_{i=2} x_i+x_0\ge x_0,\] which is a contradiction to the assumption. Thus if \(a+x_1\le r-1\), we can take \(I_1=\{1\}\); if \(a+x_1>r-1\), then \(a'+x_1\le 1\), and we can take \(I_1=\emptyset\).
Suppose for some \(1\le k\le l-1\), we have found a subset \(I_k\subseteq [k]\) satisfying \[a+\sum_{i\in I_k} x_i\le r-1, \text{ and } a'+\sum_{i\in [k]\backslash I_k} x_i\le 1.\] We claim that either \[a+\left(\sum_{i\in I_k} x_i\right)+x_{k+1}\le r-1 \text{ or } a'+\left(\sum_{i\in [k]\backslash I_k}x_i\right)+x_{k+1}\le 1,\] because otherwise, it follows that \[a+a'+\left(\sum^k_{i=1} x_i\right)+2x_{k+1}>(r-1)+1=r,\] implying \[x_{k+1}>r-(a+a'+\sum^{k+1}_{i=1}x_i)=\left(\sum^l_{i=k+2} x_i\right)+x_0\ge x_0,\] which is a contradiction to the assumption.
Therefore if \(a+\left(\sum_{i\in I_k} x_i\right)+x_{k+1}\le r-1\), we can take \(I_{k+1}=I_k\cup \{k+1\}\); and if \(a+\left(\sum_{i\in I_k} x_i\right)+x_{k+1}> r-1\), then it occurs that \(a'+\left(\sum_{i\in [k]\backslash I_k} x_i\right)+x_{k+1}\le 1\) and we can take \(I_{k+1}=I_k\).
So after \(l\) steps, we reach \(I=I_l\subseteq [l]\) satisfying the desired property. ◻
In this section, we shall prove Theorems 3 and 4. Theorem 3 follows from Propositions 6 and 9. The case \(r\ge 3\) is relatively straightforward, while the case \(r=2\) involves Hilbert bases and will be treated separately.
The core idea is that every lattice point in \(rP\) admits a unique representation as a convex combination of vertices when \(P\) is a simplex. We will apply Lemma 2 to partition the combination coefficients into two subsets, which yields a splitting of the sum over the lattice points.
Proposition 6. Let \(P\) be a lattice simplex in \(M_{\mathbb{R}}\) with \(L(P)\ge n-1\). We have \(rP=(r-1)P+P\cap M\) for all \(r\ge 3\).
Remark 7. Note that \(rP=(r-1)P+P\cap M\) implies that \(rP\cap M=(r-1)P\cap M+P\cap M\). It is also important to note that Proposition 6 is false for \(r=2\). Consider the following example. Let \(P\) be the 3-simplex with \(u_0=(0, 0, 0), u_1=(2, 0, 0), u_2=(0, 2, 0)\) and \(u_3=(0, 0, 2)\). Then \[\begin{align} P\cap M&=&\{(0, 0, 0), (2, 0, 0), (0, 2, 0), (0, 0, 2), (1, 1, 0), \\ &&(1, 0, 1), (0, 1, 1), (1, 0, 0), (0, 1, 0), (0, 0, 1)\}. \end{align}\] Let \(x=(\frac{14}{15}, \frac{14}{15}, \frac{14}{15})\in 2P\). But for any \(y\in P\cap M\), \(x-y\notin P\).
Proposition 6 is a direct consequence of the following result by taking \(m=n\).
Proposition 8. Let \(P\) be a lattice polytope of dimension \(n\), with \(m+1\) vertices. Suppose \(l_{ij}\ge n-1\) for any \(i\neq j\). Then we have \[rP=(r-1)P+P\cap M\] for \(r\ge \lceil\frac{m}{n-1}\rceil+1\).
Proof. Given \(u\in rP\), put \(u=\sum^m_{i=0}x_iu_i\) with \(x_i\ge 0\) and \(\sum^m_{i=0} x_i=r\). We can assume that \(x_0\) is the largest among \(\{x_i\}\).
For each \(1\le i\le m\), there exists a unique non-negative integer \(t_i\) such that \[\frac{t_i}{l_{0i}}\le x_i, \text{ while }\frac{t_i+1}{l_{0i}}> x_i.\] Put \(\tilde{x}_i=x_i-\frac{t_i}{l_{0i}}\). By assumption, \[\label{eq:32key32formula321} \sum_{i=1}^m \tilde{x}_i<\sum_{i=1}^m \frac{1}{l_{0i}}\le \frac{m}{n-1}\le r-1 .\tag{3}\] Applying Lemma 2 (by taking \(a=\sum_{i=1}^m \tilde{x}_i\) and \(a'=0\)), we obtain a set \(I\subseteq [m]\) satisfying the property that \[\sum^m_{i=1}\tilde{x_i}+\sum_{i\in I}\frac{t_i}{l_{0i}}\le r-1, \text{ and } \sum_{i\in [m]\backslash I}\frac{t_i}{l_{0i}}\le 1.\] Consequently, we can write \(u=v_1+v_2\), where \[\begin{align} v_1 &=& (r-1)u_0+\sum^m_{i=1}\tilde{x}_i(l_{0i}\hat{u}_{0i})+\sum_{i\in I}\frac{t_i}{l_{0i}}(l_{0i}\hat{u}_{0i})\in (r-1)P,\\ v_2 &=& u_0+\sum_{i\in [m]\backslash I}\frac{t_i}{l_{0i}}(l_{0i}\hat{u}_{0i})\in P\cap M. \end{align}\] This finishes the proof. ◻
Proposition 9. Let \(P\) be a lattice simplex with \(L(P)\ge n-1\). We have \[2P\cap M=P\cap M+P\cap M.\]
Lemma 3. Let \(P\) be a lattice polytope. Suppose \(L(P)\ge \Gamma_P+1\). Then at any vertex \(u_i\), the elements of the minimal Hilbert basis for \(C(P, u_i)\) lie in \((P-{u_i})\cap M\).
Proof. We can assume \(i=0\). Let \(y\) be an element of the minimal Hilbert basis of \(C(P,u_0)\). Let \(u_1,\cdots, u_m\) be the vertices of \(P\) adjacent to \(u_0\). We write \(y=\sum^m_{j=1} a_j\hat{u}_{0j}\) for some \(a_j\in \mathbb{Q}\cap [0, 1)\).
By the definitions of \(\text{ht}(y),\;\Gamma_{C(P,u_0)}\), and \(\Gamma_P\), and using \(\text{ord}(y)=1\), we have \[\sum^m_{j=1} a_j\le\text{ht}(y)\le \Gamma_{C(P,u_0)}+1\le \Gamma_P+1.\] As a result, \[\sum^m_{j=1}\frac{a_j}{l_{0j}}\le \left(\frac{1}{\Gamma_P+1}\right)\sum^m_{j=1} a_j\le 1.\] Finally, noting \(y=\sum^m_{j=1}\frac{a_j}{l_{0j}}(l_{0j}\hat{u}_{0j})\), we have \(y+u_0\in P\cap M\). ◻
Proof of Proposition 9. To begin with, we introduce the following notation. For \(v \in C(P,u_0)\), write \(v=\sum\limits_{i=1}^n x_i u_{0i}\), and define \[|v|=\sum\limits_{i=1}^n x_i.\]
Step 0 (setup): Given \(u=\sum^n_{i=0}x_iu_i\in 2P\cap M\), we can assume that \(x_0=\max\limits_{0\le i\le n}\{x_i\}\), in particular \(x_0\ge \frac{2}{n+1}\). We write \[u=2u_0+\sum^n_{i=1} x_i(l_{0i}\hat{u}_{0i}).\]
Step 1 (reduction): For each \(1\le i\le n\), there exists a unique non-negative integer \(t_i\) such that \[\frac{t_i}{l_{0i}}\le x_i<\frac{t_i+1}{l_{0i}}.\] Put \(\tilde{x}_i=x_i-\frac{t_i}{l_{0i}}\). Then \[u=2u_0+\sum^n_{i=1} \tilde{x}_i(l_{0i}\hat{u}_{0i})+\sum^n_{i=1} \frac{t_i}{l_{0i}}(l_{0i}\hat{u}_{0i}).\]
Note for all \(i>0\), \(\frac{t_i}{l_{0i}}\le x_i\le x_0\), so by Lemma 2, if we are able to write \(\tilde{x}:=\sum\limits^n_{i=1} \tilde{x}_i(l_{0i}\hat{u}_{0i})=\tilde{v}+\tilde{v}'\) such that \(|\tilde{v}|, |\tilde{v}'|\le 1\) and \(\tilde{v}, \tilde{v}'\in (P-u_0)\cap M\), then there exists a subset \(I\subseteq [n]\) such that \[|\tilde{v}|+\sum_{i\in I}\frac{t_i}{l_{0i}}\le 1, \text{ and } |\tilde{v}'|+\sum_{i\in [n]\backslash I}\frac{t_i}{l_{0i}}\le 1.\] As a result, we can write \(u=v_1+v_2\), where \[\begin{align} v_1 &=& u_0+\tilde{v}+\sum_{i\in I}\frac{t_i}{l_{0i}}(l_{0i}\hat{u}_{0i})\in P\cap M,\\ v_2 &=& u_0+\tilde{v}'+\sum_{i\in [n]\backslash I}\frac{t_i}{l_{0i}}(l_{0i}\hat{u}_{0i})\in P\cap M, \end{align}\] as asserted.
Step 2: In view of the above step, we can replace \(x_i\) by \(\tilde{x}_i\). So it remains to consider the case where \(x_i<\frac{1}{l_{0i}}\) for all \(i>0\).
Take \(y_1, \cdots, y_m\) (not necessarily distinct) in the minimal Hilbert basis for \(C(P, u_0)\cap M\) such that \[u-2u_0=\sum^m_{i=1} y_i.\] We have \[\label{upper32bound32for32modulus} \sum^m_{i=1} |y_i|=\sum^n_{i=1}x_i<\sum^n_{i=1}\frac{1}{l_{0i}}\le \frac{n}{n-1}.\tag{4}\] Therefore \[\label{eq:32key32formulu322} 2-\sum^m_{i=1}|y_i|>2-\frac{n}{n-1}=\frac{n-2}{n-1}.\tag{5}\]
We shall proceed by cases.
Case (i): Assume that \(|y_j|\le 2-\sum\limits^m_{i=1}|y_i|\) for all \(1\le j\le m\).
Then by Lemma 2, there exists a subset \(J\subseteq [m]\) such that \(\sum\limits_{j\in J} |y_j|\le 1\) and \(\sum\limits_{j\in [m]\backslash J} |y_j|\le 1\). Therefore, \(u=v_1+v_2\), where \[\begin{align}
v_1&=& u_0+\sum_{j\in J} y_j\in P\cap M, \\
v_2&=& u_0+\sum_{j\in [m]\backslash J} y_j\in P\cap M.
\end{align}\] We are done.
Case (ii): For some \(s\), it holds that \(|y_s|>2-\sum\limits_{i=1}^m |y_i|\). Without loss of generality, we may assume that \(s = m\).
Then we claim \[\label{equality}
\sum^{m-1}_{i=1}|y_i|\le 1.\tag{6}\]
For otherwise, \[\sum^m_{i=1}|y_i|>1+|y_m|>1+(2-\sum_{i=1}^m |y_i|)>1+\frac{n-2}{n-1},\] where the last inequality follows from 5 . This immediately contradicts (4 ) provided \(n\ge 3\). When \(n=2\), we have \[|y_m|> 2-\sum^m_{i=1}|y_i|=x_0\ge \frac{2}{3},\] which implies that \[\sum^{m-1}_{i=1}|y_i|\le 2-\frac{2}{3}-|y_m|< 2-2\times \frac{2}{3}<1.\] Thus in any event, we have established inequality (6 ).
Finally, combining (6 ) together with \(|y_m|\le 1\), which is guaranteed by Lemmas [le: bound of Gamma_X] and 3, we can write \(u=v_1+v_2\), where \[\begin{align} v_1&=& u_0+y_m\in P\cap M,\\ v_2 &=& u_0+\sum^{m-1}_{i=1} y_i\in P\cap M. \end{align}\] This completes the proof. ◻
By adapting the argument in the proof of Theorem 3, we obtain the following more general sufficient condition for the integral closedness of a lattice simplex \(P\) in terms of the invariant \(\Gamma_P\) associated with \(P\).
Now we prove Theorem 4.
Proof. When \(n=2\), Lemma 1 implies that \(\Gamma_P = 0\). Hence the statement follows from Theorem 3.
When \(n\ge3\), the proof is analogous to those of Propositions 8 and 9. In the setting of Proposition 8, we take \(m=n\) and rewrite 3 as \[\sum_{i=1}^n \tilde{x}_i <\sum_{i=1}^n \frac{1}{l_{0i}}\le \frac{n}{\Gamma_P+1}\le \frac{3}{2},\] where the last two inequalities follow from our assumption. Consequently, if \(r-1\ge \frac{3}{2}\), and hence \(r\ge 3\), then \[\label{eq:32r62613} rP=(r-1)P+P\cap M.\tag{7}\]
It remains to show that \(P\cap M+P\cap M=2P \cap M\). Following the proof of Proposition 9, it suffices to reconsider Step 2. Under the present assumption, we have \[\label{eq:32upper32bound} \sum^m_{i=1} |y_i|=\sum^n_{i=1}x_i<\sum^n_{i=1}\frac{1}{l_{0i}}\le \frac{n}{\Gamma_P+1},\tag{8}\] and hence, \[\label{eq:32lower32bound} 2-\sum_{i=1}^m |y_i|> 2-\frac{n}{\Gamma_P+1}.\tag{9}\] These inequalities are analogous to 4 and 5 . We now proceed by cases. The argument for Case (i), i.e., \(|y_j|\le 2-\sum\limits^m_{i=1}|y_i|\) for all \(1\le j\le m\), is identical to that in Proposition 9. For Case (ii), i.e., \(|y_m|>2-\sum\limits_{i=1}^m|y_i|\), we still claim that \[\sum_{i=1}^{m-1} |y_i|\le 1.\] Otherwise, \[\sum_{i=1}^m |y_i|>1+|y_m|>1+(2-\sum_{i=1}^m |y_i|)\ge 3-\frac{n}{\Gamma_P+1},\] where the last inequality follows from 9 . This contradicts 8 provided that \(\Gamma_P\ge \frac{2}{3} n-1\). Therefore, we deduce that \(u=(u_0+y_m)+(u_0+\sum\limits_{i=1}^{m-1}y_i)\). Hence we have proved \[\label{eq:32r612} P\cap M+P\cap M=2P\cap M.\tag{10}\] This completes the proof. ◻
In[9], Oda asked whether any smooth lattice polytope \(P\) is integrally closed. Since smooth polytopes satisfy \(\Gamma_P=0\), it is natural to consider the following general question:
Problem 10. Let \(P\subseteq M_{\mathbb{R}}\) be a lattice polytope. If \(L(P)\ge \Gamma_P+1\), then is \(P\) integrally closed?
The problem can be viewed as a unifying framework that includes both Oda’s conjecture and Conjecture 2 as special cases: indeed, in light of Lemma 1, an affirmative answer would immediately imply the validity of both conjectures. Theorem 4 provides some evidence in this direction.
Finally, in the toric setting, integral closedness is equivalent to projective normality, which amounts to Property \(N_p\) (cf. [10]) for \(p=0\). Inspired by Mukai’s conjecture, the first and third authors propose the following conjecture.
Conjecture 11. Let \(L\) be an ample line bundle on a projective toric variety \(X\). Suppose \(L\cdot C\ge n-1+p\) for any \(T\)-invariant curve \(C\). Then \(L\) satisfies Property \(N_p\).
The conjecture is true for \(n=2\) by [11], and for arbitrary \(n\) but \(L=A^{\otimes k}\) with \(k\ge n-1+p\) by [12]. In a forthcoming paper [13], the conjecture is proved for a class of smooth projective toric varieties, including the blowup of \(\mathbb{P}^{n}\) at \(k\) points in general position, for \(0\le k\le n+1\); the blowup of \(\mathbb{P}^{n}\) along \(\mathbb{P}^{k}\), for \(0\le k\le n-1\); and projective bundles \(\mathbb{P}(\mathscr{O}_{\mathbb{P}^{n}}^{\oplus r_1}\oplus \mathscr{O}_{\mathbb{P}^{n}}(1)^{\oplus r_2})\), for \(r_1, r_2\ge 1\). Moreover, the class is closed under taking finite products.
School of Mathematics, Sun Yat-sen University
W. 135 Xingang Rd., Guangzhou, Guangdong 510275, P.R. China
E-mail address: songlei3@mail.sysu.edu.cn
School of Mathematics, Sun Yat-sen University
W. 135 Xingang Rd., Guangzhou, Guangdong 510275, P.R. China
E-mail address: wenhq7@mail2.sysu.edu.cn
Academy for Multidisciplinary Studies, Capital Normal University
No. 105 West 3rd Ring Road, Beijing 100048, P.R. China
E-mail address: zhixian@cnu.edu.cn