Singular divisors and syzygies of polarized abelian threefolds

Abstract

We provide numerical conditions for a polarized abelian threefold $(A,L)$ to have simple syzygies, in terms of property $(N_p)$ and the vanishing of Koszul cohomology groups $K_{p,1}$. We rely on a reduction method of Lazarsfeld-Pareschi-Popa, convex geometry of Newton-Okounkov bodies, inversion of adjunction techniques from work on Fujita's conjecture, and the use of differentiation by Ein-Lazarsfeld-Nakamye. As a by-product, we construct effective divisors in any ample class of high self-intersetion, whose singularities are all concentrated on an abelian subvariety. This can be seen as the dual picture considered by Ein-Lazarsfeld for theta divisors.

Rewrote the introduction and simplified parts of the exposition. Added work on globally generatedness for polarized abelian threefolds and included this property in the more general setting of studying syzygies

References in corpus (1)

• A vanishing theorem for weight one syzygies

Cited by in corpus (5)

• The basepoint-freeness threshold and syzygies of abelian varieties
• Higher syzygies on general polarized abelian varieties of type $(1,\dots ,1,d)$
• The Infinitesimal Torelli Theorem for hypersurfaces in abelian varieties
• Seshadri constants of indecomposable polarized abelian varieties
• Multiplicities of irreducible theta divisors