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This course will be an introduction to the combinatorial structure and construction of Hilbert schemes, including local and global structure. We will focus on Hilbert schemes parametrizing subschemes of projective space. We will use deformation theory, Groebner deformations, generic initial ideals, cohomology, and other methods, to understand the combinatorial structure of these schemes (e.g. that there is always a smooth point, connectedness, and the radius of the graph of irreducible components). There are many open problems, we will mention many of them throughout the semester! A tentative list of topics (not completely in order):
  • Hilbert functions and polynomials
  • Introduction to the basic idea of Grothendieck
  • Castelnuovo-Mumford regularity of sheaves and modules
  • Existence of the Hilbert scheme; representable functors
  • Groebner degenerations, generic initial ideals, and Borel fixed ideals
  • Deformation theory
  • Piene-Schlessinger theorem about the Hilbert scheme of the twisted cubic curve in projective 3-space
  • Smoothness of the lexicographic point
  • Connectedness theorem of Hartshorne, and Reeve's theorem on radius of the graph of irreducible components
  • Local equations of the Hilbert scheme
  • The hilbert scheme of points in P^n
  • The Hilbert scheme of curves in projective 3-space of a given degree and genus.
  • Vakil's theorem: Murphy's law for Hilbert schemes


Online via zoom, MWF 10:20 am - 11:10 am


Mike Stillman (503 Malott Hall, Office Hours: Probably we will set up a lunch time office hours zoom meeting. You can join in, have your lunch, and just talk about Hilbert schemes, and related topics.


We will assume some basic algebraic geometry, at the level of Hartshorne chapters 1-3, although we will review those aspects that we really need. This includes the notion of flatness, Hilbert functions and series, coherent sheaves on projective varieties and schemes, and their cohomology.


We will not follow any one textbook, but we will often use Hartshorne "Deformation theory", and Sernesi "Deformations of algebraic schemes". We will also consult a number of papers on the topic.


We will use my Macaulay2 software to aid us in exploring Hilbert schemes. (You may also use Macaulay2 on the web).


S/U only

Course notes:

I will ask class participants to take turns taking notes, and latex'ing them. Class videos and class notes will be available once ready.


I will hand out suggested homework problems, (which will NOT be handed in), and we will discuss these in class. Some problems will be simple, making sure you understand the concepts, some will be computational, and some will be more challenging, to allow you to understand the material more deeply. Finally, some might have you explore some of the techniques that we will not have time to cover in class.


Near the end of the semester, students will do (short) projects, and present them to the class. We will discuss possible project topics in October.