Math 567:
Topics in algebraic geometry
Rational curves on algebraic varieties
Lecture: MWF 3:00PM, HB 453
Review/question session: M 2:00 PM
Office hours: open door policy
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 quotientbundles
 Examples of varieties of lines: lines on cubic surfaces, tangent
spaces, and smoothness results
 Morphisms from P^{1} 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 chainconnected 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, Higherdimensional 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) 3485261
hassett@math.rice.edu
http://www.math.rice.edu/~hassett