On the strong Lefschetz problem for uniform powers of general linear forms in $k[x,y,z]$
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- by Juan C. Migliore and Rosa María Miró-Roig PDF
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Abstract:
Schenck and Seceleanu proved that if $R = k[x,y,z]$, where $k$ is an infinite field and $I$ is an ideal generated by any collection of powers of linear forms, then multiplication by a general linear form $L$ induces a homomorphism of maximal rank from any component of $R/I$ to the next. That is, $R/I$ has the weak Lefschetz property. Considering the more general strong Lefschetz problem of when $\times L^j$ has maximal rank for $j \geq 2$, we give the first systematic study of this problem. We assume that the linear forms are general and that the powers are all the same, i.e. that $I$ is generated by uniform powers of general linear forms. We prove that for any number of such generators, $\times L^2$ always has maximal rank. We then specialize to almost complete intersections, i.e. to four generators, and we show that for $j = 3,4,5$ the behavior depends on the uniform exponent and on $j$ in a way that we make precise. In particular, there is always at most one degree where $\times L^j$ fails maximal rank. Finally, we note that experimentally all higher powers of $L$ fail maximal rank in at least two degrees.References
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Additional Information
- Juan C. Migliore
- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
- MR Author ID: 124490
- ORCID: 0000-0001-5528-4520
- Email: migliore.1@nd.edu
- Rosa María Miró-Roig
- Affiliation: Departament de Matemàtiques i Informàtica, Universitat de Barcelona, 08007 Barce-lona, Spain
- MR Author ID: 125375
- ORCID: 0000-0003-1375-6547
- Email: miro@ub.edu
- Received by editor(s): November 18, 2016
- Received by editor(s) in revised form: March 28, 2017
- Published electronically: September 6, 2017
- Additional Notes: The first author was partially supported by Simons Foundation grant #309556
- Communicated by: Irena Peeva
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 507-523
- MSC (2010): Primary 13D40; Secondary 13D02, 13E10
- DOI: https://doi.org/10.1090/proc/13747
- MathSciNet review: 3731687