The cotangent space at a monomial ideal of the Hilbert scheme of points   of an affine space
  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. 
Mark E. Huibregtse