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), 26752686
MSC (2000):
Primary 13E10; Secondary 13H10
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 lowerbound (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 lowerbound (in general sharp) for the vector . Explicitly, we will show that, for any , This result generalizes a recent theorem of Iarrobino (which treats the case ). 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:
http://dx.doi.org/10.1090/S0002994707040159
PII:
S 00029947(07)040159
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.
