Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type
| dc.creator | Zanello, Fabrizio | |
| dc.date | 2005-02-22 | |
| dc.date | 2005-03-17 | |
| dc.date.accessioned | 2026-07-07T07:53:03Z | |
| dc.date.available | 2026-07-07T07:53:03Z | |
| dc.description | Given a vector space $V$ of homogeneous polynomials of the same degree over an infinite field, consider a generic subspace $W$ of $V$. The main result of this paper is a lower-bound (in general sharp) for the dimensions of the spaces spanned in each degree by the partial derivatives of the forms generating $W$, in terms of the dimensions of the spaces spanned by the partial derivatives of the forms generating the original space $V$. Rephrasing our result in the language of commutative algebra (where this result finds its most important applications), we have: let $A$ be a type $t$ artinian level algebra with $h$-vector $h=(1,h_1,h_2,...,h_e)$, and let, for $c=1,2,...,t-1$, $H^{c,gen}=(1,H_1^{c,gen},H_2^{c,gen},...,H_e^{c,gen})$ be the $h$-vector of the generic type $c$ level quotient of $A$ having the same socle degree $e$. Then we supply a lower-bound (in general sharp) for the $h$-vector $H^{c,gen}$. Explicitly, we will show that, for any $u\in \lbrace 1,...,e\rbrace $, $$H_u^{c,gen}\geq {1\over t^2-1}((t-c)h_{e-u}+(ct-1)h_u).$$ This result generalizes a recent theorem of Iarrobino (which treats the case $t=2$). Finally, we begin to obtain, as a consequence, some structure theorems for level $h$-vectors of type bigger than 2, which is, at this time, a very little explored topic. | |
| dc.description | 18 pages. To appear in the Trans. of the A.M.S | |
| dc.identifier | https://arxiv.org/abs/math/0502466 | |
| dc.identifier | http://arxiv.org/abs/math/0502466 | |
| dc.identifier | Trans. of the A.M.S. 359 (2007), No. 2, 2675-2686 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/126109 | |
| dc.subject | Commutative Algebra | |
| dc.subject | 13E10 (primary); 13H10 (secondary) | |
| dc.title | Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type | |
| dc.type | text |