CAPITAL REGION ALGEBRA/NUMBER THEORY SEMINAR 


Speaker:       Mark Huibregtse, Skidmore College 

Time:          Wednesday, March 7, at 4:30pm, 

Place:         Bailey Hall 104, Union College, Schenectady  

Title:         A description of certain affine open subschemes 
               that form an open covering of the Hilbert scheme 
               of n points of the affine plane


Abstract: 

Let k be an algebraically closed field.  The Hilbert scheme H of
n points of the affine plane (over k) is a 2n-dimensional algebraic variety
whose points are in one-to-one correspondence with the ideals I of the
polynomial ring k[x,y] such that the quotient k[x,y] / I has finite
k-dimension n.  It is a celebrated fact (originally due to Fogarty) that H
is nonsingular and irreducible.  We will consider certain open subsets U of
H (introduced by M. Haiman) that have the structure of affine algebraic
varieties and whose union covers H.  We express the coordinate ring of U
explicitly as a quotient of a polynomial ring in two ways; as an
application, we obtain sufficient conditions for U to be isomorphic to
2n-dimensional affine space.