## David Eisenbud

The free resolutions of powers of a homogeneous ideal behave in a a fairly random-seeming way; but the regularity of powers becomes a linear function. I'll explain the basic situation, and then talk about the question of how soon" the behavior of the regularity becomes regular. This is work-in-progress with Bernd Ulrich.

## 5:00PM: Hyperplane Sections of Determinental Varieties

Suppose that $X$ is the determinental variety defined by the maximal minors of a generic $m \times n$ matrix of variables. Much has been known classically about X - it is irreducible, Cohen Macaulay, and there's an easy formula for its codimension. If the matrix is not generic, however, the situation becomes much more subtle. In this talk I'll discuss a result of Merle and Giusti which states that if we look at coordinate hyperplane sections of $X$, (in other words, set some of the matrix entries to $0$), then all of the above properties can still be read off in the simplest way from the way the zeros are arranged in the matrix. We'll also see what lurks in the background if we take more general hyperplane sections.