Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes

These notes aim at providing a complete and systematic account of some foundational aspects algebraic supergeometry, namely, the extension to geometry superschemes many classical notions, techniques results that make up general backbone geometry, most them originating from Grothendieck's work. In particular, we extend supergeometry such notions as projective proper morphisms, finiteness cohomology, vector bundles, cohomology base change, semicontinuity theorems, relative duality, Castelnuovo-Mumford regularity, flattening, Hilbert Quot schemes, faithfully flat descent, quotient étale relations (notably, Picard schemes), among others. Some may be found elsewhere, and, in there is overlap with [51]. However, constructions are presented here for first time, notably, development Grothendieck duality morphisms superschemes, construction superscheme more situation than one already known (which particular allows treat case sub-superschemes supergrassmannians), rigorous locally superprojective morphism noetherian geometrically integral fibres. Moreover, proofs given new well, even when restricted ordinary schemes. final section construct period map an open substack moduli smooth supercurves stack principally polarized abelian