Topics in algebraic geometry
Rational curves on algebraic varieties
Lecture: MWF 3:00PM, HB 453
Office hours: open door policy
Review/question session: M 2:00 PM
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,
- 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,
- 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
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.
Herman Brown 402