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

Author:
Fabrizio Zanello

Journal:
Trans. Amer. Math. Soc. **359** (2007), 2675-2686

MSC (2000):
Primary 13E10; Secondary 13H10

DOI:
https://doi.org/10.1090/S0002-9947-07-04015-9

Published electronically:
January 4, 2007

MathSciNet review:
2286051

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Abstract: Given a vector space of homogeneous polynomials of the same degree over an infinite field, consider a generic subspace of . 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 , in terms of the dimensions of the spaces spanned by the partial derivatives of the forms generating the original space .

Rephrasing our result in the language of commutative algebra (where this result finds its most important applications), we have: let be a type artinian level algebra with -vector , and let, for , be the -vector of the generic type level quotient of having the same socle degree . Then we supply a lower-bound (in general sharp) for the -vector . Explicitly, we will show that, for any ,

Finally, we begin to obtain, as a consequence, some structure theorems for level -vectors of type bigger than 2, which is, at this time, a very little explored topic.

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Additional Information

**Fabrizio Zanello**

Affiliation:
Dipartimento di Matematica, Università di Genova, Genova, Italy

Address at time of publication:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Email:
zanello@kth.se

DOI:
https://doi.org/10.1090/S0002-9947-07-04015-9

Keywords:
Artinian algebra,
level algebra,
$h$-vector,
generic quotient,
dimension,
partial derivatives.

Received by editor(s):
February 22, 2005

Received by editor(s) in revised form:
March 17, 2005

Published electronically:
January 4, 2007

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.