Colloquium Schedule for Fall 1997

Counting Rational Curves on Projective Varieties


Aaron Betram
University of Utah, Salt Lake City

Abstract: Even in the case of the complex projective plane, the result is stunning. Let Nd denote the number of rational curves of degree d passing through 3d-1 general points in ${\bf CP}^2$. Then the ``potential function'':

\begin{displaymath} \Phi(y_0,y_1,y_2) := \frac 12(y_0^2y_2 + y_0y_1^2) + \sum_{d \ge 1} N_d \frac{y_2^{3d-1}}{(3d-1)!} e^{dy_1}\end{displaymath}

satisfies the following partial differential equation:

\begin{displaymath} \Phi_{222} = \Phi_{112}^2 - \Phi_{111}\Phi_{122}\end{displaymath}

which determines a recursive formula for the Nd.


Such a statement, unimagined in the course of more than a hundred years of study of such numbers by enumerative algebraic geometers, is a typical result coming from the theory of quantum cohomology. In this talk, I will explain how the rational curves on a smooth projective variety determine a potential function and system of 3rd order partial differential equations in general. I'll explain why there is a commutative ring hiding here, and talk about some interesting examples.