Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes
Ugo Bruzzo, Daniel Hernandez Ruiperez, Alexander Polishchuk
TL;DR
This work develops a comprehensive Grothendieck-style foundation for algebraic supergeometry, extending core concepts such as projectivity, cohomology, duality, and moduli to superschemes. It introduces and analyzes projective superspaces, supergrassmannians, and CM-regularity, proving the existence of Hilbert and Picard superschemes under broad hypotheses and establishing relative duality via derived categories. A key achievement is the rigorous construction of the Hilbert superscheme and the Picard superscheme, including generalizations of the classical Hilbert and Picard theories to the super setting and their applications to period maps for supercurves. The results lay a solid groundwork for supermoduli theory, enabling systematic treatment of families, deformations, and period mappings in algebraic supergeometry. The framework is poised to impact superstring-inspired geometry and moduli problems by providing robust tools for navigating the interplay between even and odd geometric data.
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, 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.