The Geometry of the Weak Lefschetz Property and Level Sets of Points

Migliore, Juan

doi:10.4153/cjm-2008-019-2

Cited by 9 publications

(6 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As an immediate corollary, since monomial artinian ideals lift to ideals of points, we have another proof of the existence of reduced level sets of points in P r (for every r ≥ 3) whose artinian reductions are non-unimodal (this result was first shown in [Mi,Theorem 4.3]). …”

Section: Introductionmentioning

confidence: 82%

Level algebras with bad properties

Boij

Zanello

2007

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. This paper can be seen as a continuation of the works contained in the recent article (J. Alg., 305 (2006), 949-956) of the second author, and those of Juan Migliore (math. AC/0508067). Our results are: 1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property (WLP). In fact, for e 0, we will construct a codimension three, type two h-vector of socle degree e such that all the level algebras with that h-vector do not have the WLP. We will also describe the family of those algebras and compute its dimension, for each e 0. 2). There exist reduced level sets of points in P 3 of type two whose artinian reductions all fail to have the WLP. Indeed, the examples constructed here have the same h-vectors we mentioned in 1).3). For any integer r ≥ 3, there exist non-unimodal monomial artinian level algebras of codimension r. As an immediate consequence of this result, we obtain another proof of the fact (first shown by Migliore in the abovementioned preprint, Theorem 4.3) that, for any r ≥ 3, there exist reduced level sets of points in P r whose artinian reductions are non-unimodal.

show abstract

Section: Introductionmentioning

confidence: 82%

Level algebras with bad properties

Boij

Zanello

2007

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…In [53], an example is constructed of a level set of points in P 3 with the property that an Artinian reduction (hence any Artinian reduction) has a non-unimodal Hilbert function, but it is remarked that no example is known of more than two maxima. (By this we mean that there is at most one "valley" in the sense of [7].)…”

Section: Moreover If Equality Holds In Some Degree J Then It Holds mentioning

confidence: 99%

On the shape of a pure 𝑂-sequence

et al. 2012

Self Cite

View full text Add to dashboard Cite

“…In fact, he proved a stronger result than unimodality using the structure theorem of Buchsbaum and Eisenbud for the Gorenstein algebra of codimension 3 in [8]. Since then, the graded Artinian Gorenstein algebras of codimension 3 have been much studied (see [9,15,16,20,21,27,28,31,33]). In [3], Bernstein and Iarrobino showed how to construct non-unimodal graded Artinian Gorenstein algebras of codimension higher than or equal to 5.…”

Section: Introductionmentioning

confidence: 97%

Generic initial ideals and graded Artinian-level algebras not having the Weak-Lefschetz Property

Ahn

Shin

2007

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

We find a sufficient condition that H is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function, and also prove that any codimension 3 Artinian graded algebra A = R/I cannot be level if β 1,d+2 (Gin(I )) = β 2,d+2 (Gin(I )). In this case, the Hilbert function of A does not have to satisfy the conditionMoreover, we show that every codimension n graded Artinian level algebra having the Weak-Lefschetz Property has a strictly unimodal Hilbert function having a growth condition onIn particular, we show that if A is of codimension 3, then (h d−1 − h d ) < 2(h d − h d+1 ) for every θ < d < s and h s−1 ≤ 3h s , and prove that if A is a codimension 3 Artinian algebra with an h-vector (1, 3, h 2 , . . . , h s ) such that h d−1 − h d = 2(h d − h d+1 ) > 0 and soc(A) d−1 = 0 for some r 1 (A) < d < s, then (I ≤d+1 ) is (d + 1)-regular and dim k soc(A) d = h d − h d+1 .

show abstract

The Geometry of the Weak Lefschetz Property and Level Sets of Points

Cited by 9 publications

References 26 publications

Level algebras with bad properties

Level algebras with bad properties

On the shape of a pure 𝑂-sequence

Generic initial ideals and graded Artinian-level algebras not having the Weak-Lefschetz Property

Contact Info

Product

Resources

About