A stratification of the Hilbert scheme of points in the projective plane (Q1098899)

Let k be an algebraically closed field of characteristic 0, \(d\in {\mathbb{N}}\), and H d the Hilbert scheme parametrizing ideals \({\mathcal I}\subset {\mathcal O}_{{\mathbb{P}}^ 2_ k}\) of \(colength\quad d.\) For \(p\in H\) d, let h(p) denote the Hilbert function of the ideal of \({\mathbb{P}}\) \(2\otimes k(p)\), which corresponds to the point p. If \(\phi: {\mathbb{N}}\to {\mathbb{N}}\) is any function, the subset \(H_{\phi}=\{p\in H\quad d| \quad h(p)=\phi \}\) is locally closed in H d (and possibly empty). The following result is proved: If \(H_{\phi}\) is provided with the reduced induced scheme structure, then \(H_{\phi}\) is connected and smooth over k. Moreover, in the appendix a formula is described, by means of which the dimension of \(H_{\phi}\) can be computed in terms of \(\phi\).