Return to search

LEFSCHETZ PROPERTIES AND ENUMERATIONS

An artinian standard graded algebra has the weak Lefschetz property if the multiplication by a general linear form induces maps of maximal rank between consecutive degree components. It has the strong Lefschetz property if the multiplication by powers of a general linear form also induce maps of maximal rank between the appropriate degree components. These properties are mainly studied for the constraints they place, when present, on the Hilbert series of the algebra. While the majority of research on the Lefschetz properties has focused on characteristic zero, we primarily consider the presence of the properties in positive characteristic. We study the Lefschetz properties by considering the prime divisors of determinants of critical maps.
First, we consider monomial complete intersections in a finite number of variables. We provide two complements to a result of Stanley. We next consider monomial almost complete intersections in three variables. We connect the characteristics in which the weak Lefschetz property fails with the prime divisors of the signed enumeration of lozenge tilings of a punctured hexagon. Last, we study how perturbations of a family of monomial algebras can change or preserve the presence of the Lefschetz properties. In particular, we introduce a new strategy for perturbations rooted in techniques from algebraic geometry.

Identiferoai:union.ndltd.org:uky.edu/oai:uknowledge.uky.edu:math_etds-1002
Date01 January 2012
CreatorsCook, David, II
PublisherUKnowledge
Source SetsUniversity of Kentucky
Detected LanguageEnglish
Typetext
Formatapplication/pdf
SourceTheses and Dissertations--Mathematics

Page generated in 0.0016 seconds