On the n-th row of the graded Betti table of an n-dimensional toric variety

TL;DR

This paper derives explicit formulas for the first non-zero entry and the entire n-th row of the graded Betti table of n-dimensional projective toric varieties, including special cases like Veronese embeddings, over any field.

Contribution

It provides new explicit formulas for the Betti table's n-th row of toric varieties, confirming a special case of a conjecture by Ein and Lazarsfeld.

Findings

Explicit formula for the first non-zero entry in the n-th Betti row.

Explicit formula for the entire n-th Betti row when the interior is one-dimensional.

Results hold over arbitrary fields.

Abstract

We prove an explicit formula for the first non-zero entry in the n-th row of the graded Betti table of an n-dimensional projective toric variety associated to a normal polytope with at least one interior lattice point. This applies to Veronese embeddings of projective space where we prove a special case of a conjecture of Ein and Lazarsfeld. We also prove an explicit formula for the entire n-th row when the interior of the polytope is one-dimensional. All results are valid over an arbitrary field k.