Basepoint-freeness thresholds and higher syzygies on abelian threefolds

TL;DR

This paper investigates the relationship between a certain invariant and the degrees of abelian subvarieties on abelian threefolds, providing bounds that advance understanding of basepoint-freeness and syzygies in algebraic geometry.

Contribution

It establishes an upper bound for the invariant on abelian threefolds using degrees of subvarieties, answering open questions in the field.

Findings

Derived an upper bound for the invariant on abelian threefolds.

Connected the invariant to degrees of abelian subvarieties.

Confirmed some open questions about abelian varieties in three dimensions.

Abstract

For a polarized abelian variety, Z. Jiang and G. Pareschi introduce an invariant and show that the polarization is basepoint free or projectively normal if the invariant is small. Their result is generalized to higher syzygies by F. Caucci, that is, the polarization satisfies property $(N_p)$ if the invariant is small. In this paper, we study a relation between the invariant and degrees of abelian subvarieties with respect to the polarization. For abelian threefolds, we give an upper bound of the invariant using degrees of abelian subvarieties. In particular, we affirmatively answer some questions on abelian varieties asked by the author, V. Lozovanu and Caucci in the three dimensional case.