The weak Lefschetz property for quotients by quadratic monomials

  • Juan Migliore
  • Uwe Nagel
  • Hal Schenck

Abstract

Michałek and Miró-Roig, in J. Combin. Theory Ser. A 143 (2016), 66–87, give a beautiful geometric characterization of Artinian quotients by ideals generated by quadratic or cubic monomials, such that the multiplication map by a general linear form fails to be injective in the first nontrivial degree. Their work was motivated by conjectures of Ilardi and Mezzetti, Miró-Roig and Ottaviani, connecting the failure to Laplace equations and classical results of Togliatti on osculating planes. We study quotients by quadratic monomial ideals, explaining failure of the Weak Lefschetz Property for some cases not covered by Michałek and Miró-Roig.

References

Anick, D. J., Thin algebras of embedding dimension three, J. Algebra 100 (1986), no. 1, 235–259. https://doi.org/10.1016/0021-8693(86)90076-1

Boij, M., Migliore, J. C., Miró-Roig, R. M., Nagel, U., and Zanello, F., On the shape of a pure $O$-sequence, vol. 218, Mem. Amer. Math. Soc., no. 1024, American Mathematical Society, 2012. https://doi.org/10.1090/S0065-9266-2011-00647-7

Boij, M., Migliore, J. C., Miró-Roig, R. M., Nagel, U., and Zanello, F., On the weak Lefschetz property for Artinian Gorenstein algebras of codimension three, J. Algebra 403 (2014), 48–68. https://doi.org/10.1016/j.jalgebra.2014.01.003

Brenner, H. and Kaid, A., Syzygy bundles on $\Bbb P^2$ and the weak Lefschetz property, Illinois J. Math. 51 (2007), no. 4, 1299–1308.

Cook, II, D. and Nagel, U., The weak Lefschetz property for monomial ideals of small type, J. Algebra 462 (2016), 285–319. https://doi.org/10.1016/j.jalgebra.2016.06.004

Gondim, R. and Zappalà, G., Lefschetz properties for Artinian Gorenstein algebras presented by quadrics, Proc. Amer. Math. Soc. 146 (2018), no. 3, 993–1003. https://doi.org/10.1090/proc/13822

Harima, T., Maeno, T., Morita, H., Numata, Y., Wachi, A., and Watanabe, J., The Lefschetz properties, Lecture Notes in Mathematics, vol. 2080, Springer, Heidelberg, 2013. https://doi.org/10.1007/978-3-642-38206-2

Harima, T., Migliore, J. C., Nagel, U., and Watanabe, J., The weak and strong Lefschetz properties for Artinian $K$-algebras, J. Algebra 262 (2003), no. 1, 99–126. https://doi.org/10.1016/S0021-8693(03)00038-3

Hochster, M. and Laksov, D., The linear syzygies of generic forms, Comm. Algebra 15 (1987), no. 1-2, 227–239. https://doi.org/10.1080/00927872.1987.10487449

Hodge, W. V. D., The theory and applications of harmonic integrals, Cambridge, at the University Press, 1952, 2d ed.

Ilardi, G., Togliatti systems, Osaka J. Math. 43 (2006), no. 1, 1–12.

McMullen, P., The numbers of faces of simplicial polytopes, Israel J. Math. 9 (1971), 559–570. https://doi.org/10.1007/BF02771471

Mezzetti, E., Miró-Roig, R. M., and Ottaviani, G., Laplace equations and the weak Lefschetz property, Canad. J. Math. 65 (2013), no. 3, 634–654. https://doi.org/10.4153/CJM-2012-033-x

Michałek, M. and Miró-Roig, R. M., Smooth monomial Togliatti systems of cubics, J. Combin. Theory Ser. A 143 (2016), 66–87. https://doi.org/10.1016/j.jcta.2016.05.004

Migliore, J. C., Miró-Roig, R. M., and Nagel, U., Monomial ideals, almost complete intersections and the weak Lefschetz property, Trans. Amer. Math. Soc. 363 (2011), no. 1, 229–257. https://doi.org/10.1090/S0002-9947-2010-05127-X

Migliore, J. C., Miró-Roig, R. M., and Nagel, U., On the weak Lefschetz property for powers of linear forms, Algebra Number Theory 6 (2012), no. 3, 487–526. https://doi.org/10.2140/ant.2012.6.487

Migliore, J. C. and Nagel, U., Gorenstein algebras presented by quadrics, Collect. Math. 64 (2013), no. 2, 211–233. https://doi.org/10.1007/s13348-012-0076-x

Migliore, J. C. and Nagel, U., Survey article: a tour of the weak and strong Lefschetz properties, J. Commut. Algebra 5 (2013), no. 3, 329–358. https://doi.org/10.1216/JCA-2013-5-3-329

Migliore, J. C., Nagel, U., and Schenck, H., The weak Lefschetz property and products of linear forms, in preparation.

Perkinson, D., Inflections of toric varieties, vol. 48, 2000, Dedicated to William Fulton on the occasion of his 60th birthday, pp. 483–515. https://doi.org/10.1307/mmj/1030132730

Schenck, H., Computational algebraic geometry, London Mathematical Society Student Texts, vol. 58, Cambridge University Press, Cambridge, 2003. https://doi.org/10.1017/CBO9780511756320

Schenck, H. and Seceleanu, A., The weak Lefschetz property and powers of linear forms in $\Bbb K[x,y,z]$, Proc. Amer. Math. Soc. 138 (2010), no. 7, 2335–2339. https://doi.org/10.1090/S0002-9939-10-10288-3

Singh, A. K. and Walther, U., Bockstein homomorphisms in local cohomology, J. Reine Angew. Math. 655 (2011), 147–164. https://doi.org/10.1515/CRELLE.2011.039

Stanley, R. P., The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), no. 3, 236–238. https://doi.org/10.1016/0001-8708(80)90050-X

Stanley, R. P., Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, 168–184. https://doi.org/10.1137/0601021

Togliatti, E. G., Alcuni esempî di superficie algebriche degli iperspazî che rappresentano un' equazione di Laplace, Comment. Math. Helv. 1 (1929), no. 1, 255–272. https://doi.org/10.1007/BF01208366

Togliatti, E. G., Alcune osservazioni sulle superficie razionali che rappresentano equazioni di Laplace, Ann. Mat. Pura Appl. (4) 25 (1946), 325–339. https://doi.org/10.1007/BF02418089
Published
2020-03-29
How to Cite
Migliore, J., Nagel, U., & Schenck, H. (2020). The weak Lefschetz property for quotients by quadratic monomials. MATHEMATICA SCANDINAVICA, 126(1), 41-60. https://doi.org/10.7146/math.scand.a-116681
Section
Articles