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Fall 2021 The Lefschetz question for ideals generated by powers of linear forms in few variables
Juan Migliore, Uwe Nagel
J. Commut. Algebra 13(3): 381-405 (Fall 2021). DOI: 10.1216/jca.2021.13.381


The Lefschetz question asks if multiplication by a power of a general linear form, L, on a graded algebra has maximal rank (in every degree). We consider a quotient by an ideal that is generated by powers of linear forms. Then the Lefschetz question is, for example, related to the problem whether a set of fat points imposes the expected number of conditions on a linear system of hypersurfaces of fixed degree. Our starting point is a result that relates Lefschetz properties in different rings. It suggests to use induction on the number of variables, n. If n=3, then it is known that multiplication by L always has maximal rank. We show that the same is true for multiplication by L2 if all linear forms are general. Furthermore, we give a complete description of when multiplication by L3 has maximal rank (and its failure when it does not). As a consequence, for such ideals that contain a quadratic or cubic generator, we establish results on the so-called strong Lefschetz property for ideals in n=3 variables, and the weak Lefschetz property for ideals in n=4 variables.


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Juan Migliore. Uwe Nagel. "The Lefschetz question for ideals generated by powers of linear forms in few variables." J. Commut. Algebra 13 (3) 381 - 405, Fall 2021.


Received: 5 September 2017; Revised: 27 February 2019; Accepted: 4 March 2019; Published: Fall 2021
First available in Project Euclid: 18 January 2022

Digital Object Identifier: 10.1216/jca.2021.13.381

Primary: 13D40
Secondary: 13E10 , 14N05

Keywords: Cremona transformation , inverse system , maximal rank

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium


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Vol.13 • No. 3 • Fall 2021
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