Enumerations deciding the weak Lefschetz property

Details Enumerations deciding the weak Lefschetz property Enumerations deciding the weak Lefschetz property

2011-05-30

arxiv.org/abs/1105.6062v2

Mathematics

Commutative Algebra

46 pages, 40 figures; multiple updates throughout--especially regarding prior results

Scientific paper

We introduce a natural correspondence between artinian monomial almost complete intersections in three variables and punctured hexagonal regions. We use this correspondence to investigate the algebras for the presence of the weak Lefschetz property. In particular, we relate the field characteristics in which such an algebra fails to have the weak Lefschetz property to the prime divisors of the enumeration of signed lozenge tilings of the associated punctured hexagonal region. On the one side this allows us to establish the weak Lefschetz property in many new cases. For numerous classes of punctured hexagonal regions we find closed formulae for the enumerations of signed lozenge tilings, and thus the field characteristics in which the associated algebras fail to the have the weak Lefschetz property. Further, we offer a conjecture for a closed formula for the enumerations of signed lozenge tilings of symmetric punctured hexagonal regions. These formulae are exploited to lend further evidence to a conjecture by Migliore, Mir\'o-Roig, and the second author that classifies the {\em level} artinian monomial almost complete intersections in three variables that have the weak Lefschetz property in characteristic zero. Moreover, the formulae are used to generate families of algebras which never, or always, have the weak Lefschetz property, regardless of field characteristic. Finally, we determine (in one case, depending on the presence of the weak Lefschetz property) the splitting type of the syzygy bundle of an artinian monomial almost complete intersection in three variables, when the characteristic of the base field is zero. Our results convey an intriguing interplay between problems in algebra, combinatorics, and algebraic geometry, which raises new questions and deserves further investigation.

Affiliated with

Also associated with

No associations

Rating

Enumerations deciding the weak Lefschetz property does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Enumerations deciding the weak Lefschetz property, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Enumerations deciding the weak Lefschetz property will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-678987