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 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 is invariant under an infinite group of automorphisms, which shows that it does have interesting regular behavior globally.
Highlights
The purpose of this paper is to contribute to the study of Seshadri constants on abelian surfaces
Seshadri constants are fully understood in the case of Picard number ρ = 1 [5]
In contrast to the case of ρ = 1, the challenge on these surface is to determine the Seshadri constant of one ample line bundle, but to understand the behavior of the Seshadri function, ε : Amp(X ) → R, L → ε(L), which associates to each ample line bundle its Seshadri constant
Summary
The purpose of this paper is to contribute to the study of Seshadri constants on abelian surfaces. There are only few known cases where one has effective computational access to the Seshadri constants of all line bundles on the surface (the self-product E × E of a general elliptic curve being an exception again). Our methods provide such computational access for the surfaces studied here. Theorem C There is an algorithm that computes the Seshadri constant of every given ample line bundle on principally polarized abelian surfaces with real multiplication.
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