2013/03/08 by Lorenz, Benjamin, Nill, Benjamin
#14J45 #14M25 #52B20 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1303.2138
A Gorenstein polytope of index r is a lattice polytope whose r-th dilate is a reflexive polytope. These objects are of interest in combinatorial commutative algebra and enumerative combinatorics, and play a crucial role in Batyrev's and Borisov's computation of Hodge numbers of mirror-symmetric generic Calabi-Yau complete intersections. In this paper, we report on what is known about smooth Gorenstein polytopes, i.e., Gorenstein polytopes whose normal fan is unimodular. We classify d-dimensional smooth Gorenstein polytopes with index larger than (d+3)/3. Moreover, we use a modification of Oebro's algorithm to achieve classification results for smooth Gorenstein polytopes in low dimensions. The first application of these results is a database of all toric Fano d-folds whose anticanonical divisor is divisible by an integer r larger than d-8. As a second application we verify that there are only finitely many families of Calabi-Yau complete intersections of fixed dimension that are associated to a smooth Gorenstein polytope via the Batyrev-Borisov construction.