具有实乘的主极化阿贝尔曲面上的 Seshadri 常数
Seshadri constants on principally polarized abelian surfaces with real multiplication
AI总结：
本文研究 Picard 数为 2 的具有实乘的主极化阿贝尔曲面上的 Seshadri 函数，证明其局部呈现 Cantor 函数式的复杂分段线性结构，同时整体上具有无限自同构群不变性。
摘要：
阿贝尔曲面上的 Seshadri 常数在 Picard 数为 1 的情形已被完全理解。对于具有更高 Picard 数的简单阿贝尔曲面，目前所知甚少。本文研究具有实乘的主极化阿贝尔曲面。它们的 Picard 数为 2，可被视为下一个自然需要研究的情形。挑战在于不仅要确定单个线丛的 Seshadri 常数，还要理解这些曲面上的整个 Seshadri 函数。我们的结果一方面表明该函数出人意料地复杂：在具有 Z[√e] 中实乘的曲面上，它由彼此永不相邻的线性段组成——其行为类似于 Cantor 函数。另一方面，我们证明了 Seshadri 函数在一个无限自同构群下保持不变，这表明它在整体上确实具有有趣的规则行为。
英文摘要：
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.
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