Curves and surfaces 

This course serves as an introduction to the concrete aspects of algebraic geometry in low dimensions, loosely following Hartshorne Chapters IV and V (with additional outside topics). The goal is to serve as a transition from the technical foundations of algebraic geometry to questions closer to actual research in the field. Topics will include the geometry of canonical curves, automorphisms of curves, equations for curves of low genus, birational geometry of surfaces with a view towards the Enriques classification, as well as more advanced topics dictated by student interest, for example the geometry of the moduli space of curves.

  • Riemann-Roch

  • Equations for low-genus curves, automorphisms of curves

  • Jacobians, the Torelli theorem

  • The canonical embedding

  • Curves in P^3

  • Intersection theory on surfaces, Riemann-Roch for surfaces

  • Rational and ruled surfaces, cubic surfaces

  • Enriques-Kodaira classification of surfaces and examples

  • Hodge index theorem for surfaces and the Weil conjectures for curves

  • Advanced topics

References

Homework

Bold problems are recommended but long. Some of these problems will be done in class.

  • Hartshorne IV.1, 1, 3, 5, 6, 7

  • Hartshorne IV.2, 2, 3, 5, 6, 7

  • Harthorne IV.3, 1, 5, 6, 9, 10

  • Hartshorne IV.5, 1, 2, 6

  • Hartshorne IV.6, 1, 2, 4, 7, 8, 9

office hours

1/16/2023: 3pm Tuesday, Huron 1028

Usually: 4pm Tuesdays, Huron 1028