Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes
Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes
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DOI：
10.1016/j.aim.2023.108890
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发表时间：
2020-08
影响因子：
1.7
通讯作者：
U. Bruzzo;D. H. Ruipérez;A. Polishchuk
中科院分区：
文献类型：
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作者：
U. Bruzzo;D. H. Ruipérez;A. Polishchuk
这些笔记旨在为代数超几何的一些基本方面提供一个完整而系统的说明，即，将许多经典概念、技术和结果的超方案扩展到几何中，这些概念、技术和结果构成了代数几何的一般主干，其中大部分源于格罗滕迪克的工作。特别地，我们将这些概念推广到代数超几何中，如射影和固有态射、上同调的有限性、向量和射影束、上同调基变化、半连续性定理、相对对偶、Castelnuovo-Mumford正则、平坦化、Hilbert和Quot格式、忠实平坦下降、商的可变关系（特别是Picard格式）等。有些结果可能在其他地方发现，特别是与[51]有一些重叠。然而，许多技术和构造在这里是第一次提出，值得注意的是，超方案的固有态射的Grothendieck相对对偶性的第一次发展，Hilbert超方案在比已知的更一般的情况下的构造（特别是允许人们处理超格拉斯曼的次超方案的情况），以及具有几何积分纤维的noether超格式的局部超射影态射的Picard超格式的严格构造。此外，这里给出的一些证明也是新的，即使在普通方案中也是如此。在最后一节中，我们构造了一个从固有光滑超曲线模的开子堆到主极化阿贝尔格式模堆的周期映射。
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 [51]. 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 schemes.