Syzygies of toric varieties.

Abstract

Studying the equations defining the embedding of a projective variety and the higher relations (syzygies) between them is a classical problem in algebraic geometry. We give criteria for ample line bundles on toric varieties to give rise to a projectively normal embedding whose ideal is generated by quadratic equations and whose first <italic>q</italic> syzygies are linear. We illustrate the interactions with the combinatorics of lattice polytopes, and we study the related question of when the homogeneous coordinate ring of the embedding is Koszul. We obtain these results by exploiting the connection between the regularity of an ample line bundle <italic>L</italic> on a projective variety and the syzygies of embeddings induced by powers of <italic>L</italic>. Much of this has also appeared in a preprint with H. Schenck and G. Smith.