Indecomposability of derived categories in families
Università degli Studi di Bari Aldo Moro（巴里阿尔多·莫罗大学）；Utrecht University（乌得勒支大学）；Osaka University（大阪大学）；Scuola Internazionale Superiore di Studi Avanzati (SISSA)（国际高等研究院）
Francesco Bastianelli, Pieter Belmans, Shinnosuke Okawa, Andrea T. Ricolfi
本文通过引入光滑射影族的半正交分解模空间，提出研究导出范畴不可分解性的新方法，探讨了族中不可分解性的行为，并通过对称幂曲线、一般型正则曲面和希尔伯特方案等例子，分析了典范基 locus 的行为。
摘要：
利用第二、第三和第四作者引入的光滑射影族的半正交分解模空间，我们提出了一种研究导出范畴不可分解性的新方法。在自然猜想的假设下，我们给出了普遍结果，并讨论了对称幂曲线等例子中不可分解性的行为，将其与族中典范基 locus 的行为联系起来。对称幂曲线的不可分解性已通过其他方法解决，其他情况仍待解决，我们预期通过分析基 locus 将对找到无条件证明起到关键作用。
英文摘要：
Using the moduli space of semiorthogonal decompositions in a smooth projective family, introduced by the second, the third and the fourth author, we propose a novel approach to indecomposability questions for derived categories. Modulo a natural conjecture on the structure of the moduli space, we give both general results, and discuss interesting explicit examples of the behaviour of indecomposability in families, by relating it to the behaviour of the canonical base locus in families. These examples are symmetric powers of curves, certain regular surfaces of general type with large canonical base locus, and Hilbert schemes of points on surfaces. Indecomposability for symmetric powers of curves has been settled via other means, the other cases remain open and we expect that our analysis of the base locus will prove instrumental in finding unconditional proofs.
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