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

Abstract:

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 $u\in \lbrace 1,...,e\rbrace$, \[ H_u^{c,gen}\geq {1\over t^2-1}\left ((t-c)h_{e-u}+(ct-1)h_u\right ).\] 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.