Allen Knutson

Abstract:   The intersection of the the x-axis and the parabola y=x^2 is a "nonreduced scheme", with two points sitting on top of one another at the origin. In many geometric examples coming from representation theory, this nonreducedness does not occur; for example, the intersection of any two Schubert varieties is always reduced, despite this intersection being very non-transverse. I will explain how reducedness questions naturally lead one to consider characteristic p computations, and prove the following:

Let f be a degree n polynomial in n variables. From the hypersurface {f=0} we can construct many other subschemes Y, by decomposing into components, intersecting those components, decomposing, intersecting, and so on. We can get even more if we take unions, too.

1. If f has coefficients in F_p, and #{v : f(v)=0} is not a multiple of p, then all these Y are reduced.

2. If f has rational coefficients, and f's leading term is the product of the variables, then each Y has a degeneration to a Stanley-Reisner scheme (a reduced union of coordinate subspaces).

Then I'll talk about an example studied by Lusztig and Postnikov, where the ambient space is a Grassmannian, and the subvarieties {Y} correspond to juggling patterns. This part is joint with Thomas Lam and David Speyer.