Math 595: Algebraic curves and surfaces

Material from section V.3 (blowing up a surface at a point; embedded resolution of singularities of curves).

We reviewed problem 1.5.2. Material from section V.2 (normalized presentation of a ruled surface as a projective bundle; examples) and review of blow-ups at a point. Look at these exercises. If you have a chance to work on them before Wednesday, we can discuss them then.

Material from section V.2 (Picard group of a ruled surface; normalized presentation of a ruled surface as a projective bundle).

Material from section V.1 (Nakai--Moishezon criterion for ample divisors on a surface); material from section V.2 (review of projective bundles; definition and examples of ruled surfaces).

Problem session. We discussed exercises from 17, 19, and 21 April. Material from section V.1 (Riemann--Roch for surfaces; Hodge Index Theorem). Look at these exercises.

Material from section V.1 (properties of the intersection pairing; computing some examples; the adjunction formula). Look at these exercises. (For those who were away at the conference: I tried to focus on examples rather than new theorems and proofs, but we needed to cover a few new results in order to do any interesting examples. Try to look over at least the statements of these. Here are my handwritten notes from the lecture. It is an exercise to fill in the bits that were cut off by the scanner.)

Material from section IV.6 (examples of curves of degree 7 and degree 9 in P³); material from section V.1 (!) (introducing the intersection pairing for divisors on a surface). Look at these exercises.

(Lecture given by Tom Nevins.) Material from section IV.6 (Castelnuovo's theorem giving a bound on the genus of a degree d curve in P³).

Problem session. We discussed exercises from 10 April and 12 April. Material from section IV.6 (curves in P³ with special/non-special hyperplane section).

Material from section IV.4 (Clifford's theorem; moduli of curves of genus g; examples in genus 3, 4, 5). Look at these exercises.

Parts of section IV.5 (canonical embeddings for non-hyperelliptic curves; hyperelliptic curves and their canonical morphisms; the statement of Clifford's theorem). Look at these exercises; and please fill out this mid-semester feedback form if you haven't already.

Problem session. We discussed exercises from 3 April and 5 April. Material from section IV.4 (elliptic curves over k are classified by their j-invariant).

Material from section IV.3 (strange curves; birational maps from smooth curves to nodal curves in P²). Look at these exercises.

Parts of section IV.3 (every curve can be embedded in P³). Look at these exercises.

Problem session. We discussed exercises from 17 March, 27 March, and 29 March.

Material from section IV.2 (Frobenius morphism; purely inseparable finite morphisms of curves); review of material from section II.7 (linear systems and maps into projective spaces). Look at these exercises.

Parts of section IV.2 (Hurwitz's theorem; étale coverings; review of dual projective space and dual curves). Look at this exercise.

Problem session. We discussed exercises from Monday and Wednesday's lectures. Proof of Riemann-Roch. Here are some things to think about over the "break".

Review of material from section II.6 (Cartier divisors and line bundles on curves) and section II.7 (complete linear systems); section IV.1 (Riemann-Roch). Look at these exercises.

Review of material from sections I.6 (abstract non-singular curves) and II.6 (Weil divisors on curves). Look at these exercises.