Noncommutative deformations and flops
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 of these curves form the \(\mathcal N\)-sequence. The authors give a table of relations between these invariants which proves that none of these classify all analytic equivalence types of flopping curves. The new invariant, i.e. the deformation algebra, of flopping and flipping rational curves \(C\) in a 3-fold \(X\) is constructed by noncommutatively deforming the associated sheaf \(E:=\mathcal O_C(-1)\). Infinitesimal deformations are controlled by \(\mathrm {Ext}^1_X(E,E)\), and its dimension is determined by the normal bundle \(\mathcal N_{C|X}.\) If \(\dim_{\mathbb C}\mathrm {Ext}^1_X(E,E)\leq 1\), then the curve \(C\) deforms over an Artinian base \(\mathbb C[x]/x^n\). When \(\mathrm {Ext}^1_X(E,E)\geq 2\) the commutative deformations of \(C\) does not induce enough invariants. The problem is solved by applying noncommutative deformation theory. The commutative deformation functor \(c\mathcal Def_E:\mathsf{CArt}\rightarrow\mathsf{Sets}\) is given by \(R\mapsto\{\mathrm{flat }R\)-families of coherent sheaves deforming \(E\)
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