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

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- 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