Curriculum

1/20,1/25,1/27, 2/1: Proof of the equivalence between (1) nonsingular projective curves over the complex numbers, (2) complex function fields, and (3) compact Riemann surfaces.

Source: Foster, Lectures on Riemann surfaces

2/3: Review of compact vs. proper vs. projective. Proof that proper curves are necessarily projective.

2/8, 2/13: Review of Poincare duality and the comparison isomorphism between de Rham cohomology and sheaf cohomology for manifolds. Proof of Serre duality and Riemann-Roch theorem for curves.

Source (for Serre duality and Riemann-Roch): Belmans, Grothendieck duality

2/15: Every curve of genus 1 is a plane cubic which can be put into Weierstrass form. A curve of genus 1 is isomorphic to the degree 1 part of its Picard group. The coarse moduli space for elliptic curves is the affine line.

Source: Hartshorne IV.4

2/17: Every curve of genus 2 is a hyperelliptic curve, a double cover of the projective line branched at 6 points. The coarse moduli space of genus 2 curves is a 3-dimensional rational variety.

Source: Mathew, Genus two curves

2/24, 3/1, 3/3: Systematic discussion of the moduli space of curves of genus g. Artin stacks and Deligne-Mumford stacks. Explicit presentation of M_0 and M_1 as Artin stacks, and M_2 as a DM stack. Sketch of the uniformization of M_g in terms of the tricanonical embedding and Hilbert schemes.

Source: Stacks project (for generalities on stacks).

3/15, 3/17, 3/22: Introduction to Surfaces. Existence of nontrivial birational maps between nonsingular projective surfaces. The intersection product on Pic X. Computation of Pic X as a lattice for P^2, P^1 x P^1, Bl_P P^2, and related examples. Adjunction formula 2g-2 = C.(C+K) for a curve on a surface. Riemann-Roch formula.

Source: Hartshorne V

3/24, 3/29, 3/31: Hodge index theorem, and criterion for when a divisor is effective. Neron-Severi group, and the theorem of the base.

Source: Hartshorne V

4/5, 4/7: Every nonsingular cubic surface in P^3 is rational, and has 27 lines. Discussion of del Pezzo surfaces (meaning -K is ample): these are either P^1 x P^1 or P^2 blown up at 0,1,...,8 points.

Source: Hartshorne V

4/12, 4/14, 4/19: Elliptic surfaces. Relation between Neron-Severi group, trivial lattice, and Mordell-Weil theorem (Shioda-Tate formula). Types of bad fibers (= Kodaira symbols), and relation to the ADE root lattices.

Source: Schütt, Shioda, Elliptic Surfaces

4/21: A rational elliptic surface is always P^2 blown up at 9 points. Generically its Mordell-Weil group has rank 8. An extremal rational elliptic surface has rank 0, and these have all been classified (examples: Legendre and Hesse pencils)

Source: Schütt, Shioda, Elliptic Surfaces

4/26, 4/28, 5/3: K3 surfaces: definition and basic properties, including Hodge diamond. Examples: quartic surfaces in P^3, Kummer surfaces, and elliptic surfaces with Weierstrass equation satisfying deg a_i(t) \leq 2i. Torelli theorem and moduli space of polarized K3 surfaces of a given degree.

Source: Huybrechts, Lectures on K3 Surfaces