Seminar on deformations and moduli spaces in algebraic geometry and applications

Institut Fourier. Université Joseph Fourier, Grenoble, France.

Organizers:

• Louis-Clément LEFÈVRE - http://louisclement.lefevre.perso.math.cnrs.fr/
• Pedro MONTERO - http://pmontero.mat.utfsm.cl/

There are 7 regular participants.

Description

The goal of this seminar is to study the book Deformation Theory of R. Hartshorne,between us PhD students and postdocs. We have a one hour talk every week, followed by thirty minutes of discussions. We suppose some familiarity with algebraic geometry (for example, the book Algebraic Geometry of R. Hartshorne) but we take time to make recalls.

• Description of the seminar

Next talks

After 15 talks the seminar is now finished. We thank all the regular participants and we will meet next year for new adventures.

Previous talks

• 21/04/2015: Pedro, Some representable functors (Hartshorne, section 4.25). We introduce some new ideas on representable functors coming from moduli problems and we apply them to the Hilbert functor, the Quot functor and the deformations of maps. Notes.
• 7/04/2015: Louis-Clément, Introduction to moduli questions (Hartshorne, section 4.23). We begin the study of moduli spaces, in relation with deformation theory. Notes.
• 10/03/2015: Pedro, Comparison of embedded and abstract deformations (Hartshorne, section 3.20). We compare embedded and abstract deformations. Notes.
• 3/03/2015: Louis-Clément, Miniversal and universal deformations of schemes (Hartshorne, section 3.18). We apply the previous theory to the functor of deformations of a given scheme. Notes.
• 24/02/2015: Pedro, Schlessinger's criterion and applications (Hartshorne, sections 3.16 and 3.17). We study Schlessinger's criterion and apply it to the local Hilbert functor and to the Picard functor. Notes.
• 10/02/2015: Louis-Clément, Functors of Artin rings (Hartshorne, sections 3.15 and 3.16). We turn to a more abstract and powerfull point of view. A deformation problem is now a functor from Artin rings to sets. Notes.
• 3/02/2015: Pedro, Plane curve singularities (Hartshorne, section 3.14). We give the definitions of versal, miniversal and universal families and study the case of deformations of plane curves singularities. Notes.
• 27/01/2015: Pedro, Higher-order deformations of schemes (Hartshorne, sections 2.10). We study obstructions to the global extensions of deformations of schemes. Notes.
• 20/01/2015: Louis-Clément, Higher-order deformations, general principles (Hartshorne, section 2.6). We summarize all what we have done, we explain general principles on extensions of deformations and study the cases of line bundles and closed subschemes. Notes.
• 16/12/2014: Louis-Clément, Complements and comparisons with the classical Kodaira-Spencer theory. We talk about first order deformations of a scheme and then we turn to families of complex manifolds and Ehresmann's theorem. We study the role of the cohomology group H^1(X, TX) in both cases. Notes.
• 9/12/2014: Pedro, Deformations of rings (Hartshorne, section 1.5). We discuss the relations between T^i functors, smoothness, local complete intersections and deformations of affine schemes. Notes.
• 2/12/2014: Louis-Clément, The infinitesimal lifting property (Hartshorne, section 1.4). We define the deformations of a scheme, study the infinitesimal lifting property, and show how it is related to smoothness. Notes.
• 18/11/2014: Pedro, The T^i functors (Hartshorne, section 1.3). We construct some functors (introduced by Lichtenbaum and Schlessinger) that we are going to use to study the deformations of rings and schemes. Notes.
• 12/11/2014: Louis-Clément, Deformations over the dual numbers (Hartshorne, section 1.2). We still study flatness and we define first ordrer deformations of a subscheme in a fixed scheme, and of a coherent sheaf. Notes.
• 4/11/2014: Pedro, Flat morphisms and Hilbert schemes (Hartshorne, section 1.1 and recalls). We recall properties of flatness, then we define Hilbert polynomials and the Hilbert schemes. Notes.

More references

• Artin, Lectures on deformations of singularities.
• Hartshorne, Algebraic Geometry. For recalls on algebraic geometry.
• Hartshorne, Deformation Theory. The main book for the moment.
• Manetti, Deformation theory via differential graded Lie algebras.
• Manetti, Lectures on deformation of complex manifolds. Good complements for the Kodaira-Spencer theory.
• Schlessinger, Functors of Artin rings. The original article.
• Sernesi, Deformations of algebraic schemes. Very good complement to Hartshorne's book, with more algebra and more proofs.
• Vistoli, The deformation theory of local complete intersections.