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Computing in intersection theory and intersection rings of flag bundles and Grassmannians

Michael Stillman
(Cornell University)
Kollokvium
Onsdag, 18 august, 2010, at 16:15-17:15, in Aud. D4 (1531-219)
Abstrakt:
We describe some joint work with Dan Grayson, involving intersection rings of flag manifolds, and a package for Macaulay2, "Schubert 2", which is under development. Schubert 2 is roughly based on the Maple package Schubert, written by Katz and Stromme. Enumerative geometry is a beautiful and powerful subject in algebraic geometry. One of the key ingredients is the notion of intersection ring of a algebraic zero set. In order to make computations in this realm feasible, it is necessary to be able to compute effectively with these objects. One of the most important such zero-sets is the flag manifold. We start by briefly describing intersection theory and Grothendieck's theorem for the intersection ring of flag manifolds. We then show that, with respect to specific monomial orders, their defining ideals have simple to describe initial ideals, even over the integers. The resulting Groebner bases allow for fast computation in the intersection rings of flag bundles and Grassmannians. We then show some examples of computations in enumerative geometry using Schubert2 that take advantage of these methods. We will assume little or no algebraic geometry, making the talk accessible with those with only basic knowledge about polynomials and their zero-sets.