Moduli spaces are varieties that parametrize other geometric or algebraic objects (e.g., the complex projective space parametrizes lines through the origin in C^n). Deformation theory deals with the local description of these spaces, specifically studying the structure of the neighborhood of a chosen point. Its global counterpart is moduli theory, which explains how to construct these spaces. This course serves as an introduction to these theories. The target audience consists of master's students and doctoral candidates interested in algebra and algebraic geometry. We will do many examples.
Textbooks and sources:
"Deformation Theory", R. Hartshorne,
"The geometry of schemes", D. Eisenbud, J. Harris,
"Deformations of Algebraic Schemes", E. Sernesi,
Fundamental Algebraic Geometry explained, Fantechi et.al.
Foundations of Algebraic Geometry, Vakil.
Lectures
9.10. General overview: functors, moduli spaces, question of representability, functors on Artin rings. Representable functors: open subfunctors, open covers, Zariski descent, main theorem on representability [VI.2.1, Geometry of schemes]. Representability and smoothness of the Grassmannian.
16.10. Hilbert scheme of points: representability on any affine scheme and on a quasi-projective scheme over a field. Functor Map(X, Y), example of representability: tangent "bundle".
23.10. Tangent space to the Grassmannian, Hilbert scheme and Quot scheme. Tangent bundle to the Grassmannian, tangent bundle to PV, Euler sequence (mostly following Eisenbud-Harris, VI). Flatness following Vakil (and EGA?), including local criterion for flatness and locally free of finite rank = finitely presented, flat.
30.10. Flatness in the projective setting, Hilbert polynomial. The general definition of Hilbert scheme. Examples: Hilbert polynomials, constant, t+1, 3t+1.
6.11. Fogarty's proof of smoothness of the Hilbert scheme of points on a surface, case of projective plane.
13.11. Fibered categories: definitions, morphisms, homotopies. Examples. Principal G-bundles. Fibered products of fibered categories.
20.11. Functors embed into fibered categories. Fibered products of fibered categories. The fiber product of two copies of Spec(k) over BG. Glueing and stack condition. Representable diagonals, Artin stacks and Deligne-Mumford stacks. Further examples of stacks.
27.11. Generalities on Gmult actions on affine schemes and fiberwise actions on affine morphisms. The stack InvShv of invertible sheaves is isomorphic to BGmult. The stack of invertible sheaves with n-sections is isomorphic to [A^n/G_m]. Automorphisms group of k-points of stacks. Properties of stacks: smoothness, dimension, finite type etc.
11.12. Maps from BG to a stack X correspond to k-points of X together with a G-action: proof. The proof is not required on the exam.
18.12. Valuative criteria and infinitesimal lifting criteria (without proofs). Example: Hilb_d(A^{d-1}) to FinAlg_d is smooth surjective. Deformations of smooth affine schemes are trivial. Deformations over k[e]/e^2 of a smooth X are in bijection with H^1(T_X), formulated. Consequences for curves: deformations of P^1, elliptic curves and higher genus curves over k[e]/e^2 have dimension 0, 1, 3g-3, respectively.
8.01. Tangent space to deformations of a smooth scheme. Tangent spaces to several other deformation problems. Extending from tangent space to k[e]/e^n: examples.
15.01. Complete local rings. Lifting maps from Artinian schemes to Spec(R), where R=S/J is a complete local ring. Obstruction theories. Consequences for dimension. The obstruction group dual(J/mJ) is the smallest one.
22.01. Obstruction theory for embedded deformations. Schlessinger's T^2 functor. Different versions of cotangent complexes.
29.01. Deformations of complexes of vector spaces. Differential graded Lie algebras: Maurer-Cartan, deformations functors, tangent spaces and obstruction spaces. Informations about derived deformations in characteristic zero.
Exam
The oral exams will be scheduled individually (you should have received an email about it). The main focus of the exam will be on overall understanding, examples and formulations, much less about the proofs. Before taking your oral exam, you should send to me your written solutions of the "homework" assignment, which is posted here.
Each student is responsible for their own assessment by analyzing their understanding of the material. If the assessment is unfavorable, you should consult with peers, or me.
A formal assessment will be conducted at the end of the semester. I do not anticipate any mid-term exams. There will be, however, points for exercises (30% of the final grade, max number of points for declaring half of total number of exercises, lenient) and an exam (70% of the grade) in the form of "homework" assignment (to be completed in about a week) followed by an oral exam. The grading scale is as follows: ≥97% earns 5!, ≥90 earns 5, ≥85 earns 4.5, ≥80 earns 4, ≥70 earns 3.5, and ≥60 earns 3. These thresholds may change, but only to benefit the students.
Modified: 06 lutego 2025
