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On the strong Lefschetz problem for uniform powers of general linear forms in $ k[x,y,z]$


Authors: Juan C. Migliore and Rosa María Miró-Roig
Journal: Proc. Amer. Math. Soc. 146 (2018), 507-523
MSC (2010): Primary 13D40; Secondary 13D02, 13E10
DOI: https://doi.org/10.1090/proc/13747
Published electronically: September 6, 2017
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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.


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Additional Information

Juan C. Migliore
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: migliore.1@nd.edu

Rosa María Miró-Roig
Affiliation: Departament de Matemàtiques i Informàtica, Universitat de Barcelona, 08007 Barce-lona, Spain
Email: miro@ub.edu

DOI: https://doi.org/10.1090/proc/13747
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
Article copyright: © Copyright 2017 American Mathematical Society

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