Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type

Details Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type

2005-02-22

arxiv.org/abs/math/0502466v2

Trans. of the A.M.S. 359 (2007), No. 2, 2675-2686

Mathematics

Commutative Algebra

18 pages. To appear in the Trans. of the A.M.S

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-399054