# Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes.

@article{Bruzzo2020NotesOF, title={Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes.}, author={Ugo Bruzzo and Daniel Hern{\'a}ndez Ruip{\'e}rez and Alexander Polishchuk}, journal={arXiv: Algebraic Geometry}, year={2020} }

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… Expand

#### One Citation

The moduli space of stable supercurves and its canonical line bundle

- Mathematics
- 2020

We prove that the moduli of stable supercurves with punctures is a smooth proper DM stack and study an analog of the Mumford's isomorphism for its canonical line bundle.

#### References

SHOWING 1-10 OF 73 REFERENCES

Construction of Hilbert and Quot Schemes

- Mathematics
- 2005

This is an expository account of Grothendieck's construction of Hilbert and Quot Schemes, following his talk 'Techniques de construction et theoremes d'existence en geometrie algebriques IV : les… Expand

Projective superspaces in practice

- Mathematics, Physics
- Journal of Geometry and Physics
- 2018

Abstract This paper is devoted to the study of supergeometry of complex projective superspaces P n | m . First, we provide formulas for the cohomology of invertible sheaves of the form O P n | m ( l… Expand

The Grothendieck duality theorem via Bousfield’s techniques and Brown representability

- Mathematics
- 1996

Grothendieck proved that if f: X ) Y is a proper morphism of nice schemes, then Rf* has a right adjoint, which is given as tensor product with the relative canonical bundle. The original proof was by… Expand

Fundamental Algebraic Geometry

- Mathematics
- 2005

Grothendieck topologies, fibered categories and descent theory: Introduction Preliminary notions Contravariant functors Fibered categories Stacks Construction of Hilbert and Quot schemes:… Expand

Nested Hilbert schemes on surfaces: Virtual fundamental class

- Mathematics
- 2017

We construct virtual fundamental classes on nested Hilbert schemes of points and curves in complex nonsingular projective surfaces. These classes recover the virtual classes of Seiberg-Witten theory… Expand

Super Riemann surfaces: Uniformization and Teichmüller theory

- Mathematics
- 1988

Teichmüller theory for super Riemann surfaces is rigorously developed using the supermanifold theory of Rogers. In the case of trivial topology in the soul directions, relevant for superstring… Expand

Moduli and periods of supersymmetric curves

- Mathematics, Physics
- Advances in Theoretical and Mathematical Physics
- 2019

Supersymmetric curves are the analogue of Riemann surfaces in super geometry. We establish some foundational results about complex Deligne-Mumford superstacks, and we then prove that the moduli… Expand

Supergeometry of Π-projective spaces

- Mathematics, Physics
- 2018

Abstract In this paper we prove that Π -projective spaces P Π n arise naturally in supergeometry upon considering a non-projected thickening of P n related to the cotangent sheaf Ω P n 1 . In… Expand

Super Atiyah classes and obstructions to splitting of supermoduli space

- Mathematics, Physics
- 2013

The first obstruction to splitting a supermanifold S is one of the three components of its super Atiyah class, the two other components being the ordinary Atiyah classes on the reduced space M of the… Expand

Global structures for the moduli of (punctured) super riemann surfaces

- Mathematics
- 1997

Abstract A fine moduli superspace for algebraic super Riemann surfaces with a level- n structure is constructed as a quotient of the split superscheme of local spin-gravitivo fields by an etale… Expand