Reflexive modules on normal Gorenstein Stein surfaces, their deformations and moduli (Q6605400)

The memoir generalizes \textit{M. Artin} and \textit{J. L. Verdier}'s [Math. Ann. 270, 79--82 (1985; Zbl 0553.14001)], \textit{H. Esnault}'s [J. Reine Angew. Math. 362, 63--71 (1985; Zbl 0553.14016)] and \textit{J. Wunram}'s [Math. Ann. 279, No. 4, 583--598 (1988; Zbl 0616.14001)] construction of McKay correspondence to Gorenstein surface singularities.\N\NThe McKay correspondence provides a one to one map between the set of (nontrivial) irreducible representations of finite subgroups of the special linear group \(\mathrm{SL}(2,\mathbb{C})\) and the irreducible components of the exceptional divisor of the minimal resolution of the corresponding quotient surface singularity. \textit{J. McKay} [Proc. Symp. Pure Math. 183--186 (1980; Zbl 0451.05026)] used the classification of finite subgroups of the special linear group.\N\NLater, \textit{G. Gonzalez-Sprinberg} and \textit{J. L. Verdier} [Ann. Sci. Éc. Norm. Supér. (4) 16, 409--449 (1983; Zbl 0538.14033)] and Artin and Verdier [loc. cit.] (respectively, Esnault [loc. cit.] and \textit{H. Knörrer} [Invent. Math. 88, 153--164 (1987; Zbl 0617.14033)]) achieved a geometric understanding of McKay correspondence at the level of vertices (respectively, edges) of the dual graph.\N\NThe paper concerns with the level of vertices. Setting \(\pi: \tilde{X} \rightarrow X\) the minimal resolution of the mentioned singularity, a (nontrivial) irreducible representation \(\rho\) is attached to a (nontrivial) indecomposable reflexive \(\mathcal{O}_X\)-module \(M\). \(\pi^*M/\)Torsion is locally free and the Poincaré dual of a curvette transversely intersecting a unique irreducible component of the exceptional divisor coincides with the Chern class \(c_1(\pi^*M/\)Torsion\()\). This component is exactly the image of \(\rho\) via McKay correspondence. Artin and Verdier [loc. cit.] showed that the Chern class determines \(M\) and \(\rho\). Also, fixed an irreducible component of the exceptional divisor, there is an \(M\) and \(\rho\) realizing it.\N\NSubsequently, Esnault [loc. cit.] extended the work by Artin and Verdier [loc. cit.] to arbitrary rational singularities. She observed that for quotient singularities given by finite subgroups of \(\mathrm{GL}(2,\mathbb{C})\) which are not subgroups of \(\mathrm{SL}(2,\mathbb{C})\), \(c_1(\pi^*M/\)Torsion\()\) and the rank of the module are not sufficient for determining the reflexive module. It was Wunram [loc. cit.] who provided a satisfactory McKay correspondence for arbitrary rational singularities.\N\NA complete study of reflexive modules in the case of minimally elliptic singularities is due to \textit{C. P. Kahn} [Math. Ann. 285, No. 1, 141--160 (1989; Zbl 0662.14022)] and reflexive modules on normal surface singularities were studied by \textit{O. Iyama} and \textit{M. Wemyss} [Math. Z. 265, No. 1, 41--83 (2010; Zbl 1192.13012)]. For this last class of singularities, reflexive modules coincide with maximal Cohen-Macaulay modules and a singularity has finite, tame or wild Cohen-Macaulay representation if the maximal dimension of the families of indecomposable maximal Cohen-Macaulay modules is \(0\), \(1\) or unbounded, respectively.\N\N\textit{Y. A. Drozd} et al. [Mosc. Math. J. 3, No. 2, 397--418 (2003; Zbl 1051.13006)] proved that log-canonical surface singularities have tame Cohen-Macaulay representation and conjectured that non-canonical surface singularities have wild Cohen-Macaulay representation.\N\NA very interesting result in this paper is a proof of this conjecture for the case of normal Gorenstein surface singularities.\N\NThe authors describe indecomposable maximal Cohen-Macaulay modules and characterize the irreducible components of the exceptional divisor in terms of reflexive modules for the case of normal Gorenstein surface singularities. Interesting results are provided. In particular, they\N\begin{itemize}\N\item[--] give a classification of special reflexive maximal Cohen-Macaulay modules in terms of divisorial valuations centered at the singularity. This can be seen as a generalization of McKay correspondence.\N\item[--] determine \(c_1\) at an adequate resolution of singularities.\N\item[--] construct moduli spaces for special reflexive modules.\N\item[--] give a classification of normal Gorenstein surface singularities in Cohen-Macaulay representation types.\N\item[--] study the deformation theory for maximal Cohen-Macaulay modules and its interaction with their pullbacks at resolutions.\N\end{itemize}\NThe authors also provide information about some possible applications for their results. Since the paper is long, the authors include a first section which describes the results, this makes reading the paper much easier.