On the first infinitesimal neighborhood of a linear configuration of points in $\mathbb P^2$
| dc.creator | Geramita, A. V. | |
| dc.creator | Migliore, J. | |
| dc.creator | Sabourin, L. | |
| dc.date | 2004-11-19 | |
| dc.date.accessioned | 2026-07-07T05:14:31Z | |
| dc.date.available | 2026-07-07T05:14:31Z | |
| dc.description | We consider the following open questions. Fix a Hilbert function, $h$, that occurs for a reduced zero-dimensional subscheme of $\mathbb P^2$. Among all subschemes, $X$, with Hilbert function $h$, what are the possible Hilbert functions and graded Betti numbers for the first infinitesimal neighborhood, $Z$, of $X$ (i.e. the double point scheme supported on $X$)? Is there a minimum ($h^{\min}$) and maximum ($h^{\max}$) such function? The numerical information encoded in $h$ translates to a {\it type vector}, which allows us to find unions of points on lines, called {\it linear configurations}, with Hilbert function $h$. We give necessary and sufficient conditions for the Hilbert function and graded Betti numbers of the first infinitesimal neighborhoods of {\it all} such linear configurations to be the same. Even for those $h$ for which the Hilbert functions or graded Betti numbers of the resulting double point schemes are not uniquely determined, we give one (depending only on $h$) that does occur. We prove the existence of $h^{\max}$, in general, and discuss $h^{\min}$. Our methods include liaison techniques. | |
| dc.description | 46 pages | |
| dc.identifier | https://arxiv.org/abs/math/0411445 | |
| dc.identifier | http://arxiv.org/abs/math/0411445 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/73304 | |
| dc.subject | Commutative Algebra | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 13D40; 13D02; 13H15; 14M07 | |
| dc.title | On the first infinitesimal neighborhood of a linear configuration of points in $\mathbb P^2$ | |
| dc.type | text |