On the Seshadri constants of adjoint line bundles

TL;DR

This paper investigates the values of Seshadri constants for adjoint line bundles, establishing explicit lower bounds based on dimension, characterizing possible values in specific cases, and exploring multi-point scenarios.

Contribution

It provides new lower bounds for Seshadri constants of adjoint line bundles and characterizes their possible values on surfaces, extending understanding of their geometric properties.

Findings

Lower bounds depend only on the dimension of the variety.

On surfaces, the range between 1/2 and 1 has few possible Seshadri constants.

More restrictive bounds are obtained for very ample adjoint line bundles.

Abstract

In the present paper we are concerned with the possible values of Seshadri constants. While in general every positive rational number appears as the local Seshadri constant of some ample line bundle, we point out that for adjoint line bundles there are explicit lower bounds depending only on the dimension of the underlying variety. In the surface case, where the optimal lower bound is 1/2, we characterize all possible values in the range between 1/2 and 1 -- there are surprisingly few. As expected, one obtains even more restrictive results for the Seshadri constants of adjoints of very ample line bundles. Our description of the border case in this situation makes use of adjunction-theoretical results on surfaces. Finally, we study Seshadri constants of adjoint line bundles in the multi-point setting.