On a series of Gorenstein cyclic quotient singularities admitting a unique projective crepant resolution
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In this paper we prove that the Gorenstein cyclic quotient singularities of type \frac 1l (1,..., 1,l-(r-1)) with $l\geq r\geq 2$, have a \textit{unique}torus-equivariant projective, crepant, partial resolution, which is ``full'' iff either $l\equiv 0$ mod $% (r-1) $ or $l\equiv 1$ mod $(r-1) $. As it turns out, if one of these two conditions is fulfilled, then the exceptional locus of the full desingularization consists of $\lfloor \frac{l}{r-1} \rfloor$ prime divisors, $\lfloor \frac{l}{r-1}\rfloor - 1$ of which are isomorphic to the total spaces of $\Bbb{P}_{\Bbb{C}}^1$-bundles over $\Bbb{P}_{\Bbb{C}%}^{r-2}$. Moreover, it is shown that intersection numbers are computable explicitly and that the resolution morphism can be viewed as a composite of successive (normalized) blow-ups. Obviously, the monoparametrized singularity-series of the above type contains (as its ``first member'') the well-known Gorenstein singularity defined by the origin of the affine cone which lies over the $r$-tuple Veronese embedding of $\Bbb{P}_{\Bbb{C}}^{r-1}$.
96 pages. 14 Figures. LaTeX 2e with AMS and epsfig macros. This is the revised version of the ZIB preprint SC-97-39. To appear in the DMV-Seminar-Volume "Combinatorial Convex Geometry and Toric Varieties", Birkhaeuser. (For correct placement of figures we would recommend the use of the LaTeX sources instead of the direct PostScript link)
96 pages. 14 Figures. LaTeX 2e with AMS and epsfig macros. This is the revised version of the ZIB preprint SC-97-39. To appear in the DMV-Seminar-Volume "Combinatorial Convex Geometry and Toric Varieties", Birkhaeuser. (For correct placement of figures we would recommend the use of the LaTeX sources instead of the direct PostScript link)