Abstract
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 L 2 if all linear forms are general. Furthermore, we give a complete description of when multiplication by L 3 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.
Original language | English |
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Pages (from-to) | 381-405 |
Number of pages | 25 |
Journal | Journal of Commutative Algebra |
Volume | 13 |
Issue number | 3 |
DOIs | |
State | Published - 2021 |
Bibliographical note
Publisher Copyright:© 2021. Rocky Mountain Mathematics Consortium. All Rights Reserved.
Keywords
- Cremona transformation
- inverse system
- maximal rank
ASJC Scopus subject areas
- Algebra and Number Theory