# Math 245: Topics in algebraic geometry: Curves on an algebraic surface

Fall 2008
Wednesday 10-11:30 and Friday 10-11 in 383-N

In this course, we'll discuss some more advanced topics in algebraic geometry, working through much of Mumford's classic book Curves on an algebraic surface. I hope to discuss the construction of the Hilbert scheme, possibly the Picard scheme, and of course curves on an algebraic surface. I'll assume as background a solid first course in scheme theory (such as last year's 216). If you're not sure if you are adequately prepared, I'm happy to chat about it, and if you are close, I'm happy to spend extra time making the course comprehensible to you.

Reference: Curves on an Algebraic Surface by David Mumford.

I'm happy to discuss this material (or anything else), and office hours are by appointment.

I will list the topics covered in each lecture on this page.

• Class 1: Setting the stage: schemes, quasicoherent sheaves, morphisms, and properties thereof. Yoneda's lemma. Representability of (contravariant) functors (from schemes to sets). Fibered products, projective space, proj, moduli spaces, the Grassmannian.
• Class 2: Open sub(contravariant)functors(from schemes to sets). Locally closed sub(c)functors(fsts). Open sub(c)functors(fsts) of a representable (c)functor(fsts) are representable. A (c)functor(fsts) covered by representable open sub(c)functors(fsts) is representable. The Grassmannian functor is representable (cheaply). The Grassmannian is proper.
• Class 3: The Grassmannian is projective. Plucker coordinates. Lessons for representability of the Hilbert scheme.
• Class 4: a generalization of the Quot scheme; reduction of the question of existence to a special case: quotients of O^p on P^n.
• Class 5: review of coherent sheaves on projective schemes, and their cohomology (including higher pushforwards).
• Class 6: various cohomology and base change theorems; the cohomology and base change theorem; introduction to Castelnuovo-Mumford regularity, and why we want it.
• Class 7: m-regularity implies higher-regularity, and generated in degree m; the uniform bound on regularity of a quotient of O^p on P^n depending only on the Hilbert polynomial; why a flattening stratification is all that remains between us and the existence of the Quot scheme.
• Class 8: flattening stratification definition and strategy; n=0 case; generic flatness.
• Class 9: proof of flattening stratification.
• Class 10: the Quot scheme is projective (using the valuative criterion of closedness for locally closed subsets of a Noetherian scheme, not the valuative criterion of properness); representability of other moduli problems: Mor and Isom in good circumstances. Toward other moduli spaces: what to do when they are not representable?
• Class 11: Functors as "geometric spaces". Examples of moduli problems (such as line bundles) that aren't functors, but aren't so terrible. How to capture this information?
• Class 12: categories fibered in groupoids.
• Class 13: stacks on a topological space, or on the Zariski site.
• Class 14: (2-)fibered products of stacks; representable morphisms e.g. open substacks; Grothendieck topologies; sheaves etc. on a Grothendieck topology
• Class 15: etale morphisms; the etale and smooth topologies (big and small sites); algebraic spaces.
• Classes 16 and 17: Deligne-Mumford and Artin stacks; why you should think of them as geometric objects, and not horribly formal things (or formally horrible things). Why nice examples are of these forms. Coarse moduli spaces and "stacky points". Examples. Quotients stacks, moduli of curves and stable maps.