80 research outputs found
Volume and lattice points of reflexive simplices
We prove sharp upper bounds on the volume and the number of lattice points on
edges of higher-dimensional reflexive simplices. These convex-geometric results
are derived from new number-theoretic bounds on the denominators of unit
fractions summing up to one. The main algebro-geometric application is a sharp
upper bound on the anticanonical degree of higher-dimensional Q-factorial
Gorenstein toric Fano varieties with Picard number one, where we completely
characterize the case of equality.Comment: AMS-LaTeX, 19 pages; paper reorganized, introduction added,
bibliography updated; typos correcte
Complete toric varieties with reductive automorphism group
We give equivalent and sufficient criteria for the automorphism group of a
complete toric variety, respectively a Gorenstein toric Fano variety, to be
reductive. In particular we show that the automorphism group of a Gorenstein
toric Fano variety is reductive, if the barycenter of the associated reflexive
polytope is zero. Furthermore a sharp bound on the dimension of the reductive
automorphism group of a complete toric variety is proven by studying the set of
Demazure roots.Comment: AMS-LaTeX, 20 pages with 1 figur
A Simple Combinatorial Criterion for Projective Toric Manifolds with Dual Defect
We show that any smooth lattice polytope P with codegree greater or equal
than (dim(P)+3)/2 (or equivalently, with degree smaller than dim(P)/2), defines
a dual defective projective toric manifold. This implies that P is Q-normal (in
the terminology of a recent paper by Di Rocco, Piene and the first author) and
answers partially an adjunction-theoretic conjecture by Beltrametti and
Sommese. Also, it follows that smooth lattice polytopes with this property are
precisely strict Cayley polytopes, which completes the answer of a question of
Batyrev and the second author in the nonsingular case.Comment: 12 page
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