Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes
Abstract
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, CastelnuovoMumford 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 subsuperschemes 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2020
 arXiv:
 arXiv:2008.00700
 Bibcode:
 2020arXiv200800700B
 Keywords:

 Mathematics  Algebraic Geometry;
 High Energy Physics  Theory;
 Mathematical Physics;
 14K10;
 32D11;
 14D22;
 14D23;
 14H10;
 83E30
 EPrint:
 86 pages. v2: 88 pages. Added a comparison of the superperiod map with the D'HokerPhong superperiod matrix. Minor corrections