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Seshadri constants on principally polarized abelian surfaces with real multiplication

Thomas Bauer, Maximilian Schmidt

TL;DR

The paper investigates Seshadri constants on principally polarized abelian surfaces with real multiplication (Picard number two). It develops Pell divisors and submaximal curves to understand the Seshadri function, proving it is broken linear with a Cantor-type, non-adjacent segment structure, and simultaneously globally regular under an infinite automorphism group action; it also provides an algorithm to compute the Seshadri constant for any ample line bundle and shows the constant depends only on the endomorphism ring. The results reveal a deep link between number-theoretic Pell data and geometric positivity, yielding a computable framework and explicit examples illustrating one- and two-curve submaximal behavior. The work advances understanding of the Seshadri function beyond Picard number one, offering both local interval analyses and global cone decompositions that stabilize across endomorphism types, with implications for explicit plots and computational techniques. Overall, it establishes a precise, endomorphism-driven picture of Seshadri constants on this natural next class of abelian surfaces and opens avenues for further classification via Pell-type structures.

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.

Seshadri constants on principally polarized abelian surfaces with real multiplication

TL;DR

The paper investigates Seshadri constants on principally polarized abelian surfaces with real multiplication (Picard number two). It develops Pell divisors and submaximal curves to understand the Seshadri function, proving it is broken linear with a Cantor-type, non-adjacent segment structure, and simultaneously globally regular under an infinite automorphism group action; it also provides an algorithm to compute the Seshadri constant for any ample line bundle and shows the constant depends only on the endomorphism ring. The results reveal a deep link between number-theoretic Pell data and geometric positivity, yielding a computable framework and explicit examples illustrating one- and two-curve submaximal behavior. The work advances understanding of the Seshadri function beyond Picard number one, offering both local interval analyses and global cone decompositions that stabilize across endomorphism types, with implications for explicit plots and computational techniques. Overall, it establishes a precise, endomorphism-driven picture of Seshadri constants on this natural next class of abelian surfaces and opens avenues for further classification via Pell-type structures.

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 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.

Paper Structure

This paper contains 6 sections, 28 theorems, 33 equations, 4 figures.

Table of Contents

  1. Pell divisors and submaximal curves on abelian surfaces
  2. Submaximal curves on intervals
  3. Abelian surfaces with real multiplication.
  4. Seshadri function on abelian surfaces with real multiplication
  5. Fundamental cone and sample plots for Seshadri functions
  6. Distinguishing the cases of one and two submaximal curves

Key Result

Theorem A

Let $X$ be a principally abelian surface, whose endomorphism ring is isomorphic to $\mathbb Z[\sqrt e]$ for some non-square integer $e>0$. Then the Seshadri function of $X$ is broken linear.

Figures (4)

  • Figure 1: The Seshadri function of an abelian surface with real multiplication in $\mathbb Z[\sqrt{2}]$.
  • Figure 2: The Seshadri function of an abelian surface with real multiplication in $\mathbb Z[\sqrt{5}]$.
  • Figure 3: The Seshadri function of an abelian surface with real multiplication in $\mathbb Z[{\frac{1}{2}} + {\frac{1}{2}}\sqrt{5}]$.
  • Figure 4: The Seshadri function of an abelian surface with real multiplication in $\mathbb Z[{\frac{1}{2}} + {\frac{1}{2}}\sqrt{33}]$.

Theorems & Definitions (37)

  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary D
  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Lemma 2.1
  • Theorem 2.2
  • ...and 27 more