Math 245A: Moduli spaces in algebraic geometry

 Winter 2022

Mondays, Wednesdays, Fridays 1:30-2:30 (undoubtedly with exceptions, announced on the email list) online (and hopefully at some point in 320-220)
https://stanford.zoom.us/j/99675848262?pwd=N3pOVDVYZk1GcUxkSWwzYmh2ZFlDdz09

I will give topics courses in both winter and spring quarters this year. The winter quarter course will deliberately be a "second course in algebraic geometry", and I want to move at a pace where I can keep as many people with me as possible, with as complete an understanding as possible. The topic of the course will be the theory of "moduli spaces" in algebraic geometry. The precise topics of what we cover will depend on the people attending, but we will develop the topic from the beginning: defining what we mean by moduli spaces, finding out when they exist (as varieties or schemes). By starting with some important basic cases (the Grassmannian, hypersurfaces, Hilbert schemes of points), we will appreciate the importance of the Cohomology and Base Change Theorem, and we'll construct the Hilbert scheme and the Quot scheme. From there, we can go in several directions.

To follow this class, you'll have to be prepared to ask questions, and reserve some time outside of class for thinking, reading, and proving.

All are welcome. To attend the zoom meetings, you don't have to be enrolled in the course, although if you are allowed to enrol, it is helpful if you do so. If you are not enrolled and want to attend, I'll give you the link. One rule I'd like to try is people leave their cameras on unless they have told me that they have a reason not to (they don't have to give me the reason). The videos for the lectures will not be publicly available, because I want people to be comfortable asking questions and making comments without worrying that they might sound silly.

References. Later on, useful references will be Mumford's "Curves on an algebraic surface", "FGA explained", and (depending on what topics we discuss) Alper's notes on stacks, and Harris and Morrison's book on curves and moduli.

Notes. I may try to post slides and/or notes for the course here, at least as far as I am able. The slides are probably incomprehensible without the lectures, I'm afraid.

  • Mon. Jan. 3. What is a moduli space? Moduli functor (FUNCTOR = contravariant functor from schemes to sets), examples, representability, Yoneda's Lemma. (slides)
  • Wed. Jan. 5. Several possible choices for the Grassmannian FUNCTOR. Starting to show that it is representable, in a hands-on way. (slides)
  • Fri. Jan. 7. Conclusion of the argument that the Grassmannian FUNCTOR is representable. The Grassmannian is smooth projective and irreducible of dimension k(n-k). Homework: think through the black magic in the proof. (slides)
  • Mon. Jan. 10. Open subFUNCTORS. (Aside, which later turns into more than aside: representable maps of FUNCTORS.) An open subFUNCTOR of a representable FUNCTOR is also representable. The intersection of two (open) subFUNCTORs. The definition of when a FUNCTOR is a SHEAF. Every representable FUNCTOR is a SHEAF. An open subFUNCTOR of a SHEAF is also a SHEAF. Making sense of the union of two open subSHEAVES of a SHEAF. The union of two open subSHEAVES of a SHEAF (and why we need SHEAFiness, and not just FUNCTORiness). Often FUNCTORs want to be "geometric in nature. Even more so, SHEAVES want to be geometric. (slides)
  • Wed. Jan. 12. Big theorem of the day: if F is a SHEAF which has a cover by open subFUNCTORS (=open subSHEAVES) that are representable, then F itself is representable. Grassmannian bundles. Various candidates for the Hilbert functor. (slides)
  • Fri. Jan. 14. fppf. The Hilbert FUNCTOR (any candidate) of projective space is the disjoint union of those with "constant Hilbert polynomial". The representability of many of our choices of Hilbert FUNCTORS of hypersurfaces. (slides)
  • Mon. Jan. 17: no class (Martin Luther King day).
  • Wed. Jan. 19. Class didn't work out because of technology issues in the classroom.
  • Fri. Jan. 21. Flatness. The flattening stratification for finitely presented quasicoherent sheaves on arbitrary schemes. (slides)
  • Mon. Jan. 24. Cohomology and base change discussion; pushing forward flat coherent sheaves with no higher cohomology. (slides)
  • Wed. Jan. 26. Existence of the flattening stratification (most of it). (slides)
  • Fri. Jan. 28. We finish the proof of the flattening stratification (except for one subtle detail), and begin to discuss Castelnuovo-Mumford regularity. (slides)
  • Mon. Jan. 31. We conclude our discussion of Castelnuovo-Mumford regularity, with a proof of a bound on the regularity of a coherent sheaf on projective n-space, that is a subsheaf of rho copies of the structure sheaf, with Hilbert polynomial p(t). (slides)
  • Wed. Feb. 2. Cohomology and base change: preliminary discussion, and setting up Eric Larson's proof. (slides)
  • Fri. Feb. 4. Eric Larson's proof of cohomology and base change, and the resulting sense of "why" it is true. (slides)
  • Here are the notes on working with cohomology that I mentioned.
  • Mon. Feb. 7. Ben Church explains the last subtle issue in our proof of the existence of the flattening stratification. Quick outline of next day's proof of the existence of the Quot scheme. (slides)
  • Wed. Feb. 9. Proof of the existence of the Quot scheme, and that it is quasiprojective over the integers. (slides)
  • Fri. Feb. 11. Checking that the Quot scheme (with given numerical invariants, which we know is a locally closed subscheme of some Grassmannian) is projective. We explore a statement allowing us to get Quot schemes in surprising generality. (slides)
  • Mon. Feb. 14. We prove the statement discussed in the previous lecture. We then discussed the consequences of a statement that will allow us to show existence/representability of a moduli space of morphisms, isomorphisms, automorphisms, etc. (slides)
  • Wed. Feb. 16. The locus where a family of maps (of flat proper finitely presented schemes) is flat (or an isomorphism) is open. A class of schemes where we can talk about families of line bundles: proper, flat, O-connected, geometrically connected fibers, geometrically reduced fibers. (slides)
  • Fri. Feb. 18. class postponed to Monday.
  • Mon. Feb. 21: This week we discuss why, given a line bundle on a "reasonable" family of varieties, the locus on the base where the line bundle is "trivial" above it is a closed subscheme. (slides)
  • Wed. Feb. 23. Continuing our discussion of our generalization of the SeeSaw Lemma. (slides)
  • Fri. Feb. 25. Spontaneous side topic (coupled with Yi Hu's algebraic geometry seminar talk): the moduli space of point-line incidence configuration spaces, and Mnev's universality theorem. (slides)
  • Mon. Feb. 28. A bit more on Mnev. Completion of proof of SeeSaw. Also, here is a linkage to Daniel Litt's nonreduced tweetstorm. (slides)
  • Wed. Mar. 2. (from Benson Farb's office) Construction of the moduli space of curves, and of the Picard variety of a curve, through quotients. (slides)
  • Fri. Mar. 4. More on the definition of the Picard variety of a curve. The Zariski tangent space of the Hilbert scheme at a point corresponding to "smooth in smooth" is the vector space of sections of the normal bundle. Statement of lower bound of dimension of Hilbert scheme as h^0(N)-h^1(N). Discussion and some applications. (slides)
  • Mon. Mar. 7. We begin to discuss the deformations of closed subschemes, considering the affine case in some detail. (slides)
  • Wed. Mar. 9. Some thoughts on globalizing and generalizing the deformation theory discussed last day. (slides)
  • Fri. Mar. 11. Summary of what we did in the course, to solidify it in your mind. (slides)

    Still to come: I want to post some excellent ideas of Ben Church. He has sent me pdfs.

    Back to my home page.