Math 567: Topics in algebraic geometry
Rational curves on algebraic varieties

Prerequisites: Familiarity with basic algebraic geometry

Topics: The first part of the course will emphasize concrete examples with a classical flavor, to build up geometric intuition. The second part will develop the technical machinery needed to analyze how rational curves deform in families.

Grassmannians: Plücker embeddings, tangent spaces, universal sub- and quotient-bundles

Examples of varieties of lines: lines on cubic surfaces, tangent spaces, and smoothness results

Morphisms from P¹ to projective varieties

Hilbert schemes: sketch construction, universal properties, infinitesimal analysis

Chow varieties

Deformation theory of morphisms of curves and free curves

Cone of curves

Mori's bend and break technique

Smoothing morphisms of curves

Rationally connected and chain-connected varieties

Fano varieties are rationally connected

Reference books:

D. Mumford, Lectures on curves on an algebraic surface, Princeton 1966

J. Harris, Algebraic geometry, Spinger Verlag 1992

J. Kollár, Rational curves on algebraic varieties, Springer Verlag 1999

O. Debarre, Higher-dimensional algebraic geometry, Springer Verlag 2001

Assessment:
Weekly problem sets (60%): Due each Wednesday starting January 17. They should be turned in during class. Homework assignments are not pledged. You are strongly encouraged to work together, though each student should write up her/his own submission.

Presentation (40%): A public presentation of a fundamental research paper in the field.

Any student with a disability requiring accommodations in this course is encouraged to contact me after class or during office hours. Additionally, students should contact Disability Support Services in the Ley Student Center.

Brendan Hassett
Herman Brown 402
(713) 348-5261
hassett@math.rice.edu
http://www.math.rice.edu/~hassett