55 research outputs found
Higher order selfdual toric varieties
The notion of higher order dual varieties of a projective variety, introduced
in \cite{P83}, is a natural generalization of the classical notion of
projective duality. In this paper we present geometric and combinatorial
characterizations of those equivariant projective toric embeddings that satisfy
higher order selfduality. We also give several examples and general
constructions. In particular, we highlight the relation with Cayley-Bacharach
questions and with Cayley configurations.Comment: 21 page
Classifying smooth lattice polytopes via toric fibrations
We define Q-normal lattice polytopes. Natural examples of such polytopes are
Cayley sums of strictly combinatorially equivalent lattice polytopes, which
correspond to particularly nice toric fibrations, namely toric projective
bundles. In a recent paper Batyrev and Nill have suggested that there should be
a bound, N(d), such that every lattice polytope of degree d and dimension at
least N(d) decomposes as a Cayley sum. We give a sharp answer to this question
for smooth Q-normal polytopes. We show that any smooth Q-normal lattice
polytope P of dimension n and degree d is a Cayley sum of strictly
combinatorially equivalent polytopes if n is greater than or equal to 2d+1. The
proof relies on the study of the nef value morphism associated to the
corresponding toric embedding.Comment: Revised version, minor changes. To appear in Advances in Mat
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