MATH 536B: Algebraic Geometry (Spring 2017)

Essential course information is contained in the syllabus.
A summary is below.

Instructor

Jack Hall
email: firstnamelastname at math dot arizona dot edu
office: ENR2 S349

Text

The Rising Sea: Foundations Of Algebraic Geometry Notes by Ravi Vakil.

Class

when: Tuesday and Thursday, 9:30am-10:45am
where: Math 514

Office hours

By appointment.

Topics covered

1/12: Recap of dimension and codimension from last semester, statement and applications of Krull's PID Theorem.
1/17: Crash course on artin rings and primary ideals, proof of Krull's PID Theorem, and proof of the algebraic Hartog's Theorem.
1/19: Associated points.
1/24: Associated points and primary decompositions.
1/26: Meromorphic functions.
1/31: Cartier divisors.
2/2: Weil divisors.
2/7: Applications of Weil divisors I.
2/9: Applications of Weil divisors II.
2/14: Right derived functors.
2/16: Long exact sequences, cohomology and higher pushforwards on a ringed space.
2/21: Flabby resolutions and applications.
2/23: Cohomology and higher pushforwards on schemes, Serre's Theorem.
2/28: Cohomology and colimits, higher pushforwards of quasi-coherents are quasi-coherent.
3/2: Cohomology of projective space, Hom and Ext, and Mayer-Vietoris.
3/7: Cohomological vanishing, coherence of higher pushforwards for projective morphisms.
3/9: Devissage, coherence of higher pushforwards for proper morphisms, Euler characteristics.
3/21: Cartier divisors on curves, Riemann's Theorem.
3/23: Hilbert polynomials, flatness.
3/28: Flat base change, flatness over DVRs, degenerations.
4/4: Smooth, unramified and etale morphisms; regularity.
4/6: Kahler differentials, derivations.
4/11: Exact sequences of Kahler differentials.
4/13: Unramified morphisms again.
4/18: Monomorphisms; Euler exact sequence.
4/20: Divisors on curves.
4/25: Numerical criteria for closed immersion of curves in projective space, curves of genus 0.
4/27: Riemann--Hurwitz, hyperelliptic curves, the canonical embedding.
5/2: Serre duality.