The cotangent space at a monomial ideal of the Hilbert scheme of points of an affine space

Details The cotangent space at a monomial ideal of the Hilbert scheme of points of an affine space The cotangent space at a monomial ideal of the Hilbert scheme of points of an affine space

2005-06-28

arxiv.org/abs/math/0506575v1

Mathematics

Algebraic Geometry

51 pages, 9 figures

Scientific paper

Let k be an algebraically closed field. We study the cotangent space of a point t corresponding to a monomial ideal I of k[x_1, ..., x_r] in the Hilbert scheme of n points of affine r-space (so the k-dimension of k[x_1, ..., x_r]/I = colength of I = n). Since t lies in the closure of the locus corresponding to subschemes supported at n distinct points of A^r_k, one knows that the k-dimension of the cotangent space is always >= r*n, and that t is nonsingular if and only if the dimension equals r*n. We construct an explicit linearly independent set S of cotangent vectors of size r*n, and then explore conditions on I under which S either is or is not a basis of the cotangent space. In particular, we give a condition on I sufficient for S to be a basis (equivalently, for t to be nonsingular) that holds for every monomial ideal in the case of r = 2 variables, and that characterizes such ideals when r = 3. We also give an easily-checked condition on I sufficient for S not to be a basis.

Affiliated with

Also associated with

No associations

Rating

The cotangent space at a monomial ideal of the Hilbert scheme of points of an affine space does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The cotangent space at a monomial ideal of the Hilbert scheme of points of an affine space, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The cotangent space at a monomial ideal of the Hilbert scheme of points of an affine space will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-434568