scholarly journals Geometric vertex decomposition and liaison

2021 ◽  
Vol 9 ◽  
Author(s):  
Patricia Klein ◽  
Jenna Rajchgot
Keyword(s):  
Lower Bound ◽  
Cluster Algebras ◽  
Algebraic Varieties ◽  
Determinantal Ideals ◽  
Vertex Decomposable

Abstract Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this paper, we establish an explicit connection between these approaches. In particular, we show that each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height $1$ to an ideal of indeterminates and, conversely, that every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes and defining ideals of graded lower bound cluster algebras.

scholarly journals Constructing Fano 3-folds from cluster varieties of rank 2

2020 ◽  
Vol 156 (9) ◽  
pp. 1873-1914
Author(s):  
Stephen Coughlan ◽  
Tom Ducat
Keyword(s):  
Cluster Algebras ◽  
Algebraic Varieties ◽  
Graded Ring ◽  
Gorenstein Rings ◽  
Main Application ◽  
Hilbert Polynomials ◽  
Rank 2 ◽  
Cluster Varieties

Cluster algebras give rise to a class of Gorenstein rings which enjoy a large amount of symmetry. Concentrating on the rank 2 cases, we show how cluster varieties can be used to construct many interesting projective algebraic varieties. Our main application is then to construct hundreds of families of Fano 3-folds in codimensions 4 and 5. In particular, for Fano 3-folds in codimension 4 we construct at least one family for 187 of the 206 possible Hilbert polynomials contained in the Graded Ring Database.

Algebraic varieties with simple singularities related to some reflection groups

2020 ◽  
pp. 2150070
Author(s):  
Juan García Escudero
Keyword(s):  
Lower Bound ◽  
Coxeter Group ◽  
Algebraic Varieties ◽  
Reflection Groups ◽  
Complex Reflection Groups ◽  
Hodge Numbers ◽  
Simple Singularities ◽  
Complex Projective ◽  
Double Coverings ◽  
Small Resolution

Related to a Coxeter group are certain sets of tangents of the deltoid with evenly distributed orientations forming simplicial line configurations. These configurations are used to construct curves and surfaces with [Formula: see text] singularities. Other surfaces associated with invariants of exceptional complex reflection groups are considered. A new lower bound for the maximal number of [Formula: see text] singularities in a sextic surface is obtained. Several Calabi–Yau threefolds defined as double coverings of the complex projective 3-space branched along nodal octic surfaces and Calabi–Yau quintic threefolds are analyzed. The Hodge numbers of a small resolution of all the nodes of the singular threefolds are obtained.

scholarly journals ON THE NUMBER OF POINTS OVER FINITE FIELDS ON VARIETIES RELATED TO CLUSTER ALGEBRAS

2010 ◽  
Vol 53 (1) ◽  
pp. 141-151
Author(s):  
F. CHAPOTON
Keyword(s):  
Finite Fields ◽  
Compact Support ◽  
Affine Space ◽  
Cluster Algebras ◽  
Algebraic Varieties ◽  
Dynkin Diagrams

AbstractWe start here the study of some algebraic varieties related to cluster algebras. These varieties are defined as the fibres of the projection map from the cluster variety to the affine space of coefficients. We compute the number of points over finite fields on these varieties, for all simply laced Dynkin diagrams. We also compute the cohomology with compact support in some cases.

scholarly journals Lower bound cluster algebras: presentations, Cohen–Macaulayness, and normality

2018 ◽  
Vol 1 (1) ◽  
pp. 95-114 ◽  
Author(s):  
Greg Muller ◽  
Jenna Rajchgot ◽  
Bradley Zykoski
Keyword(s):  
Lower Bound ◽  
Cluster Algebras

scholarly journals Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas

2017 ◽  
Vol 29 (4) ◽  
Author(s):  
Alberto Corso ◽  
Uwe Nagel ◽  
Sonja Petrović ◽  
Cornelia Yuen
Keyword(s):  
Blow Up ◽  
Symmetric Matrix ◽  
Monomial Ideals ◽  
Edge Ideals ◽  
Rees Algebras ◽  
Determinantal Ideals ◽  
Blowing Up ◽  
Initial Ideals ◽  
Vertex Decomposable ◽  
Liaison Theory

AbstractWe investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen–Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine their Hilbert functions and Castelnuovo–Mumford regularities. As a consequence, we find explicit minimal reductions for all Ferrers and many specialized Ferrers ideals, as well as their reduction numbers. These results can be viewed as extensions of the classical Dedekind–Mertens formula for the content of the product of two polynomials.

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Recent Work ◽  
Chow Groups ◽  
Ring Structure ◽  
Complete Intersections ◽  
Algebraic Varieties ◽  
Chow Ring ◽  
Hodge Structures ◽  
Chow Rings ◽  
Geometric Methods ◽  
The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

Sign in / Sign up

Export Citation Format

Share Document