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Weakly-exceptional singularities in higher dimensions. (English) Zbl 1305.14018

Weakly-exceptional singularities are a higher dimensional generalization of surface singularities of type \(\mathbb D\) and \(\mathbb E\) see Yu. G. Prokhorov [in: Algebra. Proceedings of the international algebraic conference on the occasion of the 90th birthday of A. G. Kurosh, Moscow, Russia, May 25–30, 1998. Berlin: Walter de Gruyter. 301–317 (2000; Zbl 1003.14005)]. If \((V\ni O)\) is a germ of a Kawamata log terminal singularity, a plt blow-up of \((V\ni O)\) is a birational morphism \(\pi: W\rightarrow V\) whose exceptional locus consists in a single irreducible divisor \(E\) such that \(O\in \pi(E)\), the log pair \((W, E)\) has purely log terminal singularities, and \(-E\) is a \(\pi\)-ample \(\mathbb Q\)-Cartier divisor. Then by definition \((V\ni O)\) is a weakly-exceptional singularitiy if it has a unique plt blow-up. This somehow generalizes the property of \(\mathbb D\) and \(\mathbb E\) surface singularities of having a \`\` fork” in the dual graph of a minimal resolution, i.e. a special curve intersecting three other exceptional curves in the graph.
The article under review aims at finding necessary and/or sufficient conditions for weakly-exceptional quotient singularities, and at giving some new classes of examples. The main results concern finite subgroups of \(\mathrm{GL}_n(\mathbb C)\) that do not contain reflections. If \(G\) is such a group, and \(\bar G\) denotes its projection in \(\mathrm{PGL}_n(\mathbb C)\), the authors prove that if \(\mathbb C^n/G\) is not weakly-exceptional, then there exists a \(\bar G\)-invariant, irreducible projectively normal subvariety \(V\subset \mathbb P^{n-1}\) of Fano type, such that \(\deg V\leq {{n-1}\choose \dim(V)}\), and satisfying certain cohomological conditions. As a corollary, a necessary condition for \(\mathbb C^n/G\) being weakly-exceptional is that, for every irreducible \(\bar G\)-invariant subvariety of \(\mathbb P^{n-1}\), there does not exist any hypersurface of degree \(\dim(V)+1\) containing \(V\).
For \(n=5, 6\) there are more precise results. If \(G\subset \mathrm{GL}_5(\mathbb C)\), then \(\mathbb C/G\) is weakly-exceptional if and only if \(G\) is transitive and does not have semi-invariants of degree \(\leq 4\). If \(G\subset \mathrm{GL}_6(\mathbb C)\), then \(\mathbb C/G\) is weakly-exceptional if five conditions are satisfied. The first two are the analogous to the previous ones, the others say that there is no irreducible \(\bar G\)-invariant variety of the following types: a smooth rational cubic scroll in \(\mathbb P^5\), a complete intersection of two quadrics, a normal projectively normal non-degenerate threefold in \(\mathbb P^5\) of Fano type, with at most rational singularities, of degree \(6\) and sectional genus \(3\), with \(h^0(\mathcal I_X(2))=0\) and \(h^0(\mathcal I_X(4))=4\).
The authors also construct infinite series of Gorenstein weakly-exceptional singularities in any dimension.
The proofs involve many beautiful geometrical constructions of classical flavour.

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
14E15 Global theory and resolution of singularities (algebro-geometric aspects)

Citations:

Zbl 1003.14005
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