The versal deformation space of a reflexive module on a rational cone
The versal deformation space \(R\) of a reflexive module on the cone over the rational normal curve of degree m is described. A resolution of each component of this space is given as the total space of a vector bundle on a Grassmannian. The vector bundle is a sum of copies of the cotangent bundle, the canonical sub-bundle, the dual of the canonical quotient bundle, and the trivial line bundle. Via an embedding in a trivial bundle, the components are obtained by projection. In particular equations for the minimal stratum in the Chern class filtration of the versal deformation space are given. The components are classified, and the deformation relations are described combinatorially. A formula for the number of components is given. The interesting case studied in this article is \(\text{rk}M>1\) so that \(M\) is decomposable. Let \(X\) be a rational surface singularity and let \(\pi:\widetilde{X}\rightarrow X\) be a minimal resolution. Then a reflexive module \(M\) on \(X\) corresponds to a full sheaf \(\widetilde{M}\) on \(\widetilde{X}\). For each \(d\in\text{Pic}(\widetilde{X})\), \textit{A. Ishii} [Math. Ann. 317, 239--262 (2000; Zbl 1016.14018)] defined a functor of families parametrised by schemes over \(R_{\text{red}}\), of semi-full sheaves \(\mathcal{E}\) on \(\widetilde{X}\) with an isomorphism. This functor is represented by a regular scheme \(F^d\) which is projective over \(R_{\text{red}}\). As \(d\) varies, a finite stratification \(\coprod S^d\) of \(R_{\text{red}}\) is obtained such that if the fibre of the versal family at \(t\in R\) is the reflexive module \(N\), then \(t\in S^d\) if and only if the full sheaf \(\widetilde{N}\) has Chern class \(d\). Each component in \(R_{\text{red}}\) is given as the closure of an \(S^d\). By the McKay correspondence this gives the stratification by isomorphism classes if \(X\) is a rational double point. When \(X\) is the cone over the rational normal curve of degree \(m\), there are \(m\) isomorphism classes of rank one reflexive modules and any reflexive module is a direct sum of these. The authors find \(F^d\) for all \(M\) and all \(d\). In theorem 1 an intrinsic description of \(F^d\) as the total space of a bundle \(\mathcal{E}_{\text{ A}}\) on a Grassmannian \(A\) is given. In theorem 2, \(\text{Ext}^1_{\widetilde{X}\times A/A}(\mathcal{E}_{\text{A}},\mathcal{E}_{\text{A}})\) is computed. Theorem 2 gives an embedding of the vector bundle in the trivial vector bundle \(\text{Ext}^1_X(M,M)\times A\) and the map to \(R\) is obtained. In corollary 1, equations describing the embedding of \(R^d\) in \(F^d\) lead to an explicit description of the image for all the minimal strata. This is corollary 2: \(R^d\) is the cone over a Segre embedding \(\times\) an incidence variety \(\times\) an affine space intersected with certain hyperplanes and quadratic hypersurfaces. Corollary 3 gives an ideal \(I_d\) of minors which gives \(T^d\) by blowing up \(R_d\). It gives \(T^d\) as the strict transform in resolutions of rank singularities, and the relation to the Chern class filtration is studied. Theorem 3 classifies the components of \(R_{\text{red}}\), and a formula for the number of components is given. The components corresponds to the geometrically rigid modules listed in corollary 4. The local deformation relations are given in corollaries 4--8. Finally, three explicit examples are found in the final section. This is a very nice application of local deformation theory to obtain interesting and deep results on this subject which however is hard to get acquainted with, so that reading the article forces some extra effort.
scientific article; zbMATH DE number 2104455
scientific article