基础代数超几何笔记。Hilbert 和 Picard 超概形
Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes
AI总结：
本文系统地将代数几何的经典概念扩展到超概形几何，首次发展了超概形的 Grothendieck 相对对偶性，构造了更一般的 Hilbert 超概形和 Picard 超概形，并建立了超曲线到主极化阿贝尔超概形的周期映射。
摘要：
这些笔记旨在提供代数超几何某些基础方面的一个完整且系统的说明，即，将构成代数几何一般骨干的许多经典概念、技术和结果（其中大部分源于 Grothendieck 的工作）扩展到超概形的几何中。特别地，我们将射射态射和固有态射、上同调的有限性、向量丛和射影丛、上同调基变换、半连续性定理、相对对偶性、Castelnuovo-Mumford 正则性、平坦化、Hilbert 概形和 Quot 概形、忠实平坦下降、商 étale 关系（特别是 Picard 概形）等概念扩展到代数超几何中。某些结果可能在别处找到，特别地，与 Moosavian 和 Zhou 最近的预印本有一些重叠。然而，许多技术和构造是首次在此呈现，特别是首次对超概形的固有态射发展 Grothendieck 相对对偶性，在比已知更一般的情形下构造 Hilbert 超概形（这特别允许处理超格拉斯曼流形的子超概形情形），以及为具有几何积分纤维的 noetherian 超概形的局部超射影态射严格构造 Picard 超概形。此外，这里给出的一些证明也是新的，即使限制在普通概形上也是如此。在最后一节中，我们构造了一个从固有且光滑超曲线的模的一个开子栈到主极化阿贝尔超概形的模栈的周期映射。
英文摘要：
These notes aim at providing a complete and systematic account of some foundational aspects of algebraic supergeometry, namely, the extension to the geometry of superschemes of many classical notions, techniques and results that make up the general backbone of algebraic geometry, most of them originating from Grothendieck's work. In particular, we extend to algebraic supergeometry such notions as projective and proper morphisms, finiteness of the cohomology, vector and projective bundles, cohomology base change, semicontinuity theorems, relative duality, Castelnuovo-Mumford regularity, flattening, Hilbert and Quot schemes, faithfully flat descent, quotient étale relations (notably, Picard schemes), among others. Some results may be found elsewhere, and, in particular, there is some overlap with a recent preprint by Moosavian and Zhou. However, many techniques and constructions are presented here for the first time, notably, a first development of Grothendieck relative duality for proper morphisms of superschemes, the construction of the Hilbert superscheme in a more general situation than the one already known (which in particular allows one to treat the case of sub-superschemes of supergrassmannians), and a rigorous construction of the Picard superscheme for a locally superprojective morphism of noetherian superschemes with geometrically integral fibres. Moreover, some of the proofs given here are new as well, even when restricted to ordinary schemes. In a final section we construct a period map from an open substack of the moduli of proper and smooth supercurves to the moduli stack of principally polarized abelian superchemes.
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