These notes aim at providing a complete and systematic ac-count of some foundational aspects of algebraic supergeome-try, 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 orig-inating 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 etale relations (notably, Picard schemes), among oth-ers. 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, no-tably, 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 rig-orous 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.(c) 2023 Elsevier Inc. All rights reserved.
