Sectional genus and the volume of a lattice polytope
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Sectional genus and the volume of a lattice polytope Ryo Kawaguchi1 Received: 30 October 2019 / Accepted: 27 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract For a convex lattice polytope having at least one interior lattice point, a lower bound for its volume is derived from Hibi’s lower bound theorem for the h ∗ -vector. On the other hand, it is known that the sectional genus of a polarized variety has an upper bound, which is an extension of the Castelnuovo bound for the genus of a projective curve. In this paper, we prove the equivalence of these two bounds. Namely, a polarized toric variety has maximal sectional genus if and only if its associated polytope has minimal volume. This is a generalization of the known fact that polytopes corresponding to the anticanonical bundles of Gorenstein toric Fano varieties are reflexive polytopes (whose typical examples are minimal volume polytopes with only one interior lattice point). Keywords Lattice polytope · Discrete volume · Toric variety · Polarized variety · Sectional genus Mathematics Subject Classification Primary 14M25 · 52B11; Secondary 14C20 · 52B20 · 52C07
1 Introduction A lattice polytope in Rn (i.e., a polytope all whose vertices lie in Zn ) is said to be convex if it is the convex hull of a finite number of points in Zn . For an n-dimensional convex lattice polytope P and a positive integer k, we denote by Ik and Bk the number of lattice points in the interior and on the boundary of the dilated polytope kP = {k x | x ∈ P}, and moreover, define I0 = 0 and B0 = 1. Ehrhart [5] proved that Ik + Bk is a polynomial in k of degree n whose constant term is one. Equivalently, its generating
Supported by the Grant-in-Aid for Scientific Research (C) 20K03562 from JSPS.
B 1
Ryo Kawaguchi [email protected] Department of Mathematics, Nara Medical University, Kashihara, Nara 634-8521, Japan
123
Journal of Algebraic Combinatorics
function can be written as a rational function: 1+
k≥1
(Ik + Bk )t k =
h ∗0 + h ∗1 t + · · · + h ∗n t n . (1 − t)n+1
The vector (h ∗0 , . . . , h ∗n ) is called the h ∗ -vector (alternatively, Ehrhart h-vector) of P. The nonnegativity of h i∗ ’s has been proved by Stanley [17]. It is well known that h ∗0 = 1, h ∗1 = I1 + B1 − n − 1, h ∗n = I1 , and n!vol(P) =
n j=0
h ∗j .
(1)
Moreover, if P has at least one interior lattice point, then h ∗k ≥ h ∗1 hold for k = 2, . . . , n − 1, which is known as Hibi’s lower bound theorem (cf. [11]). As a simple consequence of this theorem and (1), we obtain the following bound for the volume, which also we call the Hibi’s lower bound (for the volume) in this paper. Proposition 1.1 Let P be an n-dimensional convex lattice polytope with I1 ≥ 1. Then, n!vol(P) ≥ n I1 + (n − 1)B1 − n 2 + 2, and the equality holds if and only if the h ∗ -vector of P is ⎧ ⎨1 h i∗ = I1 + B1 − n − 1 ⎩ I1
(i = 0), (i = 1, . . . , n − 1), (i = n).
Polytopes with I1 = 1 whose volume achieves this bound are typical examples of so-called reflexive polytopes, which have bee
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