Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Notes on fundamental algebraic supergeometry. Hilbert and Picard
  superschemes   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 \'etale 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.

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The categories of this paper are: chao-dyn
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The categories of this paper are: adap-org
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: solv-int
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The categories of this paper are: hep-lat
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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: stat