Lefschetz properties for Artinian Gorenstein algebras presented by quadrics

Gondim, Rodrigo; Zappalà, Giuseppe

doi:10.1090/proc/13822

Lefschetz properties for Artinian Gorenstein algebras presented by quadrics
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by Rodrigo Gondim and Giuseppe Zappalà PDF

Proc. Amer. Math. Soc. 146 (2018), 993-1003 Request permission

Abstract:

We introduce a family of Artinian Gorenstein algebras, whose combinatorial structure characterizes the ones presented by quadrics. Under certain hypotheses these algebras have non-unimodal Hilbert vector. In particular we provide families of counterexamples to the conjecture that Artinian Gorenstein algebras presented by quadrics should satisfy the weak Lefschetz property.

References

Mats Boij and Dan Laksov, Nonunimodality of graded Gorenstein Artin algebras, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1083–1092. MR 1227512, DOI 10.1090/S0002-9939-1994-1227512-2
David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322, DOI 10.1090/gsm/124
Aldo Conca, Gröbner bases for spaces of quadrics of low codimension, Adv. in Appl. Math. 24 (2000), no. 2, 111–124. MR 1748965, DOI 10.1006/aama.1999.0676
Ciro Ciliberto, Francesco Russo, and Aron Simis, Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian, Adv. Math. 218 (2008), no. 6, 1759–1805. MR 2431661, DOI 10.1016/j.aim.2008.03.025
David Eisenbud, Mark Green, and Joe Harris, Cayley-Bacharach theorems and conjectures, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 3, 295–324. MR 1376653, DOI 10.1090/S0273-0979-96-00666-0
R. Gondim, On higher Hessians and the Lefschetz properties, arXiv:1506.06387.
Rodrigo Gondim and Francesco Russo, On cubic hypersurfaces with vanishing hessian, J. Pure Appl. Algebra 219 (2015), no. 4, 779–806. MR 3282110, DOI 10.1016/j.jpaa.2014.04.030
Tadahito Harima, Juan C. Migliore, Uwe Nagel, and Junzo Watanabe, The weak and strong Lefschetz properties for Artinian $K$-algebras, J. Algebra 262 (2003), no. 1, 99–126. MR 1970804, DOI 10.1016/S0021-8693(03)00038-3
Tadahito Harima, Toshiaki Maeno, Hideaki Morita, Yasuhide Numata, Akihito Wachi, and Junzo Watanabe, The Lefschetz properties, Lecture Notes in Mathematics, vol. 2080, Springer, Heidelberg, 2013. MR 3112920, DOI 10.1007/978-3-642-38206-2
Klaus Lamotke, The topology of complex projective varieties after S. Lefschetz, Topology 20 (1981), no. 1, 15–51. MR 592569, DOI 10.1016/0040-9383(81)90013-6
Juan C. Migliore, Rosa M. Miró-Roig, and Uwe Nagel, Monomial ideals, almost complete intersections and the weak Lefschetz property, Trans. Amer. Math. Soc. 363 (2011), no. 1, 229–257. MR 2719680, DOI 10.1090/S0002-9947-2010-05127-X
Juan Migliore and Uwe Nagel, Survey article: a tour of the weak and strong Lefschetz properties, J. Commut. Algebra 5 (2013), no. 3, 329–358. MR 3161738, DOI 10.1216/JCA-2013-5-3-329
Juan Migliore and Uwe Nagel, Gorenstein algebras presented by quadrics, Collect. Math. 64 (2013), no. 2, 211–233. MR 3041764, DOI 10.1007/s13348-012-0076-x
Toshiaki Maeno and Junzo Watanabe, Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials, Illinois J. Math. 53 (2009), no. 2, 591–603. MR 2594646
Francesco Russo, On the geometry of some special projective varieties, Lecture Notes of the Unione Matematica Italiana, vol. 18, Springer, Cham; Unione Matematica Italiana, Bologna, 2016. MR 3445582, DOI 10.1007/978-3-319-26765-4
Imre Simon, Recognizable sets with multiplicities in the tropical semiring, Mathematical foundations of computer science, 1988 (Carlsbad, 1988) Lecture Notes in Comput. Sci., vol. 324, Springer, Berlin, 1988, pp. 107–120. MR 1023416, DOI 10.1007/BFb0017135
Richard P. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, 168–184. MR 578321, DOI 10.1137/0601021
Richard P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), no. 1, 57–83. MR 485835, DOI 10.1016/0001-8708(78)90045-2
Richard P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph theory and its applications: East and West (Jinan, 1986) Ann. New York Acad. Sci., vol. 576, New York Acad. Sci., New York, 1989, pp. 500–535. MR 1110850, DOI 10.1111/j.1749-6632.1989.tb16434.x
Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949, DOI 10.1090/ulect/008
Paul Turán, Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok 48 (1941), 436–452 (Hungarian, with German summary). MR 18405
J. Watanabe, On the Theory of Gordan-Noether on Homogeneous Forms with Zero Hessian , Proc. Sch. Sci. TOKAI UNIV. 49 (2014), 1–21.