## Illinois Journal of Mathematics

### The strength of the Weak Lefschetz Property

#### Abstract

We study a number of conditions on the Hilbert function of a level Artinian algebra which imply the Weak Lefschetz Property (WLP). Possibly the most important open case is whether a codimension 3 SI-sequence forces the WLP for level algebras. In other words, does every codimension 3 Gorenstein algebra have the WLP? We give some new partial answers to this old question: we prove an affirmative answer when the initial degree is 2, or when the Hilbert function is relatively small. Then we give a complete answer to the question of what is the largest socle degree forcing the WLP.

#### Article information

Source
Illinois J. Math., Volume 52, Number 4 (2008), 1417-1433.

Dates
First available in Project Euclid: 18 November 2009

https://projecteuclid.org/euclid.ijm/1258554370

Digital Object Identifier
doi:10.1215/ijm/1258554370

Mathematical Reviews number (MathSciNet)
MR2595775

Zentralblatt MATH identifier
1178.13011

#### Citation

Migliore, Juan; Zanello, Fabrizio. The strength of the Weak Lefschetz Property. Illinois J. Math. 52 (2008), no. 4, 1417--1433. doi:10.1215/ijm/1258554370. https://projecteuclid.org/euclid.ijm/1258554370

#### References

• D. Bernstein and A. Iarrobino, A nonunimodal graded Gorenstein Artin algebra in codimension five, Comm. Algebra 20 (1992), 2323–2336.
• A. M. Bigatti and A. V. Geramita, Level algebras, Lex segments and minimal Hilbert functions, Comm. Algebra 31 (2003), 1427–1451.
• A. Bigatti, A. V. Geramita and J. Migliore, Geometric consequences of extremal behavior in a theorem of Macaulay, Trans. Amer. Math. Soc. 346 (1994), 203–235.
• M. Boij, Graded Gorenstein Artin algebras whose Hilbert functions have a large number of valleys, Comm. Algebra 23 (1995), 97–103.
• M. Boij, Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function, Pacific J. Math. 187 (1999), 1–11.
• M. Boij and D. Laksov, Nonunimodality of graded Gorenstein Artin algebras, Proc. Amer. Math. Soc. 120 (1994), 1083–1092.
• M. Boij and F. Zanello, Level algebras with bad properties, Proc. Amer. Math. Soc. 135 (2007), 2713–2722.
• H. Brenner and A. Kaid, Syzygy bundles on $\mathbb P^2$ and the weak Lefschetz property, Illinois J. Math. 51 (2007), 1299–1308.
• W. Bruns and J. Herzog, Cohen–Macaulay rings, revised ed., Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge Univ. Press, 1998.
• CoCoA, A system for doing computations in commutative algebra. Available at http://cocoa.dima.unige.it.
• E. D. Davis, Complete intersections of codimension 2 in $\mathbb P^{r}$: The Bezout–Jacobi–Segre theorem revisited, Rend. Sem. Mat. Univers. Politecn. Torino 43 (1985), 333–353.
• A. 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, Queen's Papers in Pure and Applied Mathematics no. 102.
• A. Geramita, T. Harima, J. Migliore and Y. Shin, The Hilbert function of a level algebra, Mem. Amer. Math. Soc. 186 (2007), no. 872.
• G. Gotzmann, Eine Bedingung für die Flachheit und das Hilbertpolynom cines graduierten Ringes, Math. Z. 158 (1978), 61–70.
• P. Gordan and M. Noether, Ueber die algebraischen Formen, deren Hessesche Determinante idensisch veschwindet, Math. Ann. 10 (1878).
• M. Green, Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann, Algebraic curves and projective geometry (1988), 76–86, Trento; Lecture Notes in Mathematics, vol. 1389, Springer, Berlin, 1989.
• T. Harima, Characterization of Hilbert functions of Gorenstein Artin algebras with the Weak Stanley property, Proc. Amer. Math. Soc. 123 (1995), 3631–3638.
• T. Harima, J. Migliore, U. Nagel and J. Watanabe, The weak and strong Lefschetz properties for Artinian $K$-algebras, J. Algebra 262 (2003), 99–126.
• A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, vol. 1721, Springer, Heidelberg, 1999.
• H. Ikeda, Results on Dilworth and Rees numbes of Artinian local rings, Japan J. Math. 22 (1996), 147–158.
• J. Migliore and R. Miró-Roig, Ideals of general forms and the ubiquity of the Weak Lefschetz property, J. Pure Appl. Algebra 182 (2003), 79–107.
• J. Migliore, R. Miró-Roig and U. Nagel, Almost complete intersections and the Weak Lefschetz property, avalable at arXiv:0811.1023.
• J. Migliore and U. Nagel, Reduced arithmetically Gorenstein schemes and Simplicial Polytopes with maximal Betti numbers, Adv. Math. 180 (2003), 1–63.
• J. Migliore, U. Nagel and F. Zanello, A characterization of Gorenstein Hilbert functions in codimension four with small initial degree, Math. Res. Lett. 15 (2008), 331–349.
• J. Migliore and F. Zanello, The Hilbert functions which force the Weak Lefschetz Property, J. Pure Appl. Algebra 210 (2007), 465–471.
• R. Stanley, Hilbert functions of graded algebras, Adv. Math. 28 (1978), 57–83.
• J. Watanabe, A remark on the Hessian of homogeneous polynomials, The Curves Seminar at Queen's, vol. XIII, 2000.
• A. Weiss, Some new non-unimodal level algebras, Ph.D. thesis, Tufts University, 2007, available at \href arXiv:0708.3354.
• F. Zanello, Stanley's theorem on codimension 3 Gorenstein $h$-vectors, Proc. Amer. Math. Soc. 134 (2006), 5–8.
• F. Zanello, A non-unimodal codimension 3 level $h$-vector, J. Algebra 305 (2006) 949–956.