Gorenstein Simplices and Even Binary Self-Complementary Codes

Abstract

It is known that if a Gorenstein simplex of dimension \(d\) and degree \(s\) is not a lattice pyramid, then \(d \leq 2s-1\). In this paper, we study the extremal case \(d=2s-1\). More precisely, we characterize Gorenstein simplices of dimension \(2s-1\) and degree \(s\) which are not lattice pyramids in terms of even binary self-complementary codes. As an application, combining this characterization with existing classification results on reflexive simplices, we classify Gorenstein simplices of degree \(3\) and \(4\). Equivalently, we classify polarized \(d\)-dimensional Gorenstein fake weighted projective spaces \((X,L)\) satisfying $-K_X=(d-2)L$ or $-K_X=(d-3)L$, where \(-K_X\) is the anticanonical divisor of \(X\) and \(L\) is a Cartier divisor on \(X\).