Trivariate monomial complete intersections and plane partitions

Chen, Charles; Guo, Alan; Jin, Xin; Liu, Gaku

doi:10.48550/arxiv.1008.1426

Cited by 2 publications

(8 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where 0 ≤ α < a, 0 ≤ β < b, and 0 ≤ γ < c. If α = β = γ = 0, then we define I a,b,c,0,0,0 to be (x a , y b , z c ) which is a complete intersection and is studied extensively in [22] and [6]. Assume at most one of α, β, and γ is zero.…”

Section: Almost Complete Intersectionsmentioning

confidence: 99%

“…(ii) The corollary extends [6, Theorem 1.2], where punctured hexagons with trivial puncture (i.e., M = 0) are considered. We further note that [18, Section 3.4] provides, independently, essentially the same proof as [6], and the proof of Lemma 5.1 builds on this technique.…”

Section: Proof Stepmentioning

confidence: 99%

“…Its Hilbert function is (1,3,6,6,3) and so it has multiplicity 19. It is worth noting that this ideal is extensively studied in [4, Example 3.1] and is the basis for an exploration of the subtlety of the Lefschetz properties in [9].…”

Section: Minimal Multiplicitymentioning

confidence: 99%

“…In this paper we make progress on the above conjecture and illustrate the depth of the problem by considering a larger class of algebras and relating the problem to a priori seemingly unrelated questions in combinatorics and algebraic geometry. This builds on the work of many authors (e.g., [4], [6], [8], [22], and [25]).…”

mentioning

confidence: 99%

“…In [8], the authors found a connection between certain families of level artinian monomial almost complete intersections and lozenge tilings of hexagons; independently, Li and Zanello [22] found a similar connection for artinian monomial complete intersections (see also Corollary 6.5). However, both were without combinatorial bijection until one was found by Chen, Guo, Jin, and Liu [6]; Boyle, Migliore, and Zanello [2] have pushed this connection further. Brenner and Kaid [5] also consider artinian monomial complete intersections in three variables with generators all of the same degree.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Enumerations deciding the weak Lefschetz property

Cook¹,

Nagel²

2011

Preprint

View full text Add to dashboard Cite

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. On the other side we can determine some of the prime divisors of the enumerations by means of an algebraic argument.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, Miró-Roig, and the second author that classifies the 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.

show abstract

Section: Almost Complete Intersectionsmentioning

confidence: 99%

Section: Proof Stepmentioning

confidence: 99%

Section: Minimal Multiplicitymentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Enumerations deciding the weak Lefschetz property

Cook¹,

Nagel²

2011

Preprint

View full text Add to dashboard Cite

show abstract

The Lefschetz properties of monomial complete intersections in positive characteristic

Cook

2011

Preprint

View full text Add to dashboard Cite

Stanley proved that, in characteristic zero, all artinian monomial complete intersections have the strong Lefschetz property. We provide a positive characteristic complement to Stanley's result in the case of artinian monomial complete intersections generated by monomials all of the same degree, and also for arbitrary artinian monomial complete intersections in characteristic two.To establish these results, we first prove an a priori lower bound on the characteristics that guarantee the Lefschetz properties. We then use a variety of techniques to complete the classifications.

show abstract

Trivariate monomial complete intersections and plane partitions

Cited by 2 publications

References 8 publications

Enumerations deciding the weak Lefschetz property

Enumerations deciding the weak Lefschetz property

The Lefschetz properties of monomial complete intersections in positive characteristic

Contact Info

Product

Resources

About