Seshadri constants on principally polarized abelian surfaces with real multiplication

TL;DR

This paper explores the complex behavior of Seshadri constants and the Seshadri function on principally polarized abelian surfaces with real multiplication, revealing both fractal-like complexity and global invariance properties.

Contribution

It characterizes the Seshadri function on these surfaces, showing its intricate structure and invariance under automorphisms, advancing understanding beyond Picard number one cases.

Findings

Seshadri function exhibits Cantor-like fractal structure

Function is invariant under an infinite automorphism group

Provides new insights into line bundle positivity on abelian surfaces

Abstract

Seshadri constants on abelian surfaces are fully understood in the case of Picard number one. Little is known so far for simple abelian surfaces of higher Picard number. In this paper we investigate principally polarized abelian surfaces with real multiplication. They are of Picard number two and might be considered the next natural case to be studied. The challenge is to not only determine the Seshadri constants of individual line bundles, but to understand the whole \emph{Seshadri function} on these surfaces. Our results show on the one hand that this function is surprisingly complex: On surfaces with real multiplication in $\mathbb Z[\sqrt e]$ it consists of linear segments that are never adjacent to each other -- it behaves like the Cantor function. On the other hand, we prove that the Seshadri function it is invariant under an infinite group of automorphisms, which shows that it does have interesting regular behavior globally.