Commutative Algebra Seminar

Title

Some recent results on Seshadri constants

Abstract

Let X be a projective variety, and let L be an ample line bundle on X. For a point x in X, the Seshadri constant of L at x is defined as the infimum, taken over all curves C passing through x, of the ratios \frac{L.C}{m}, where L.C denotes the intersection product of L and C, and m is the multiplicity of C at x. This concept was introduced by J.-P. Demailly in 1990, inspired by Seshadri’s ampleness criterion. Seshadri constants provide insights into both the local behavior of L at x and certain global properties of X.

The notion of Seshadri constants has been extended in various directions. Two important generalizations include replacing L with a vector bundle of arbitrary rank and replacing the point x with a finite collection of points. We will give an overview of the current research in this area and discuss some recent results.