arXiv subjects

Mark E. Huibregtse

Publications and source records attributed to Mark E. Huibregtse.

4 recordsLinked to original sources

arXiv2022

More elementary components of the Hilbert scheme of points

Let $K$ be an algebraically closed field of characteristic $0$, and let $H^μ$ denote the Hilbert scheme of $μ$ points of the affine space $A^n$. An elementary component $E$ of $H^μ$ is an irreducible component such that every $K$-point $[I]$ $\in$ $E$ represents a length-$μ$ closed subscheme Spec$(K[x_1,\dots,x_n]/I)$ $\subseteq$ $A^n$ that is supported at one point. In a previous article we found some new examples of elementary components; in this article, we simplify the methods and extend the range of the previous paper to find several more examples. In addition, we present a "plausibility test" that suggests the existence of a vast number of similar examples.

math.AG↗

arXiv2016

Some elementary components of the Hilbert scheme of points

Let $K$ be an algebraically closed field of characteristic 0, and let $H$ denote the Hilbert scheme of $m$ points of affine n-space $A^n$. An elementary component $E$ of $H$ is an irreducible component such that every $K$-point $[I]$ in $E$ represents a length-$m$ closed subscheme Spec$(K[x_1,\dots,x_n]/I)$ of $A^n$ that is supported at one point. Iarrobino and Emsalem gave the first explicit examples (with $m > 1$) of elementary components in ["Some zero-dimensional generic singularities: Finite algebras having small tangent space", Comp. Math. 36 (1978), pp. 145-188]; in their examples, the ideals $I$ were homogeneous (up to a change of coordinates corresponding to a translation of $A^n$). We generalize their construction to obtain new examples of elementary components.

math.AG↗

arXiv2010

Some syzygies of the generators of the ideal of a border basis scheme

A border basis scheme is an affine scheme that can be viewed as an open subscheme of the Hilbert scheme of μpoints of affine n-space. We study syzygies of the generators of a border basis scheme's defining ideal. These generators arise as the entries of the commutators of certain matrices (the "generic multiplication matrices"). We consider two families of syzygies that are closely connected to these matrices: The first arises from the Jacobi identity, and the second from the fact that the trace of a commutator is 0. Several examples of both types of syzygy are presented, including a proof that the border basis schemes in case n = 2 are complete intersections.

math.AG↗

arXiv2005

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.

math.AG↗