Partial derivatives of a generic subspace of a vector space of forms: Quotients of level algebras of arbitrary type
HTML articles powered by AMS MathViewer
- by Fabrizio Zanello PDF
- Trans. Amer. Math. Soc. 359 (2007), 2675-2686 Request permission
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.References
- Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- Anthony V. Geramita, Inverse systems of fat points: Waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals, The Curves Seminar at Queen’s, Vol. X (Kingston, ON, 1995) Queen’s Papers in Pure and Appl. Math., vol. 102, Queen’s Univ., Kingston, ON, 1996, pp. 2–114. MR 1381732
- A.V. Geramita, T. Harima, J. Migliore and Y.S. Shin: The Hilbert Function of a Level Algebra, Memoirs of the Amer. Math. Soc., to appear.
- Anthony Iarrobino, Hilbert functions of Gorenstein algebras associated to a pencil of forms, Projective varieties with unexpected properties, Walter de Gruyter, Berlin, 2005, pp. 273–286. MR 2202259
- Anthony Iarrobino, Compressed algebras: Artin algebras having given socle degrees and maximal length, Trans. Amer. Math. Soc. 285 (1984), no. 1, 337–378. MR 748843, DOI 10.1090/S0002-9947-1984-0748843-4
- Anthony Iarrobino and Vassil Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, vol. 1721, Springer-Verlag, Berlin, 1999. Appendix C by Iarrobino and Steven L. Kleiman. MR 1735271, DOI 10.1007/BFb0093426
- Noam Nisan and Avi Wigderson, Lower bounds on arithmetic circuits via partial derivatives, Comput. Complexity 6 (1996/97), no. 3, 217–234. MR 1486927, DOI 10.1007/BF01294256
- Fabrizio Zanello, Level algebras of type 2, Comm. Algebra 34 (2006), no. 2, 691–714. MR 2211949, DOI 10.1080/00927870500387986
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
- MR Author ID: 721303
- Email: zanello@kth.se
- Received by editor(s): February 22, 2005
- Received by editor(s) in revised form: March 17, 2005
- Published electronically: January 4, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2286051