Noncommutative deformations and flops
type: Item
instance of: scholarly article
title: Noncommutative deformations and flops
zbMATH Open document ID: 1346.14031
DOI: 10.1215/00127094-3449887
author: Will Donovan
author: Michael Wemyss
published in: Duke Mathematical Journal
publication date: 2016-07-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1309.0698
full work available at URL: https://projecteuclid.org/euclid.dmj/1458740997
full work available at URL: 130831.pdf
review text: The study of birational geometry of algebraic varieties via the minimal model program depends on the geometry of certain modifications of codimension 2 known as \textit{flips} and \textit{flops}. The geometry of these is very intricate, and is a contemporary object of study where a central problem is to classify flips and flops such that appropriate invariants can be constructed. This article gives a new invariant to every flipping or flopping curve in a 3-fold, using noncommutative deformation theory. In fact, the idea is to replace all the invariants by a new noncommutative geometric object represented by the \textit{noncommutative deformation algebra} \(A_{\mathrm{con}}\) associated to the curve. This algebra is finite dimensional, and can be associated to any contractible rational curve in any 3-fold, singular or not. A very nice property of this algebra is that it recovers the classical invariants in a natural way. The main difference from a separate set of invariants is that the fact that it is an algebra, makes it possible to give an intrinsic description of a derived autoequivalence associated to a general flopping curve. The authors give the background on 3-folds, the simplest example being the Atiyah flop. Then the normal bundle is \(\mathcal O(-1)\oplus\mathcal O(-1)\) and the curve is rigid (i.e. the orbit is finite under the action of the groups of projective transformations), and the flop can be factored as a blowup of the curve followed by a blowdown. For a general flopping irreducible rational curve \(C\) in a smooth 3-fold \(X\), the authors recall the classical invariants: The \textit{normal bundle} \((a,b):=\mathcal O(a)\oplus\mathcal O(b)\) which must be \((-1,-1),\;(-2,0)\), or \((-3,1)\), the \textit{width}, the \textit{Dynkin type}, the \textit{length}, and the \textit{Normal bundle sequence}: The flop \(f:X\dashrightarrow X^\prime\) factors into a sequence of blowups in centers \(C_1,\dots,C_n\), followed by blowdowns, and the normal bundles …
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14D15
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 6603535
zbMATH Keywords: noncommutative deformation theory
zbMATH Keywords: noncommutative deformation algebra
zbMATH Keywords: flips
zbMATH Keywords: flops
zbMATH Keywords: contraction algebra
zbMATH Keywords: analytic equivalence types
zbMATH Keywords: Dynkin type
zbMATH Keywords: derived autoequivalence
MaRDI profile type: Publication
OpenAlex ID: W1592798054
arXiv ID: 1309.0698
cites work: Orthogonally spherical objects and spherical fibrations
cites work: Quivers from matrix factorizations
cites work: Flops and derived categories.
cites work: The McKay correspondence as an equivalence of derived categories
cites work: Hochschild (co-)homology of schemes with tilting object
cites work: Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities.
cites work: Deformation theory of objects in homotopy and derived categories. I: General theory
cites work: Deformation theory of objects in homotopy and derived categories. II: Pro-representability of the deformation functor
cites work: Deformation theory of objects in homotopy and derived categories. III: Abelian categories
cites work: Homological Algebra on a Complete Intersection, with an Application to Group Representations
cites work: Q4317713
cites work: Q5710759
cites work: Computing Noncommutative Deformations of Presheaves and Sheaves of Modules
cites work: Q5439603
cites work: Fourier--Mukai transforms for Gorenstein schemes
cites work: Relative integral functors for singular fibrations and singular partners
cites work: Q5470192
cites work: Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras
cites work: The classification of special Cohen-Macaulay modules
cites work: Maximal modifications and Auslander-Reiten duality for non-isolated singularities
cites work: Singular derived categories of \(\mathbb{Q}\)-factorial terminalizations and maximal modification algebras
cites work: Q5287413
cites work: General hyperplane sections of nonsingular flops in dimension 3
cites work: Classification of Three-Dimensional Flips
cites work: Noncommutative deformations of modules
cites work: Deformation theory of abelian categories
cites work: The \(A_\infty\) deformation theory of a point and the derived categories of local Calabi-Yaus
cites work: Braid group actions on derived categories of coherent sheaves.
cites work: Growth, entropy and commutativity of algebras satisfying prescribed relations.
cites work: On a certain generalization of spherical twists
cites work: Non-commutative width and Gopakumar-Vafa invariants
cites work: Stability conditions and crepant small resolutions
cites work: Three-dimensional flops and noncommutative rings
description: scientific article