Mathematics – Algebraic Geometry
Scientific paper
1998-03-20
Mathematics
Algebraic Geometry
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
Scientific paper
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 $\Bbb{P}_{\Bbb{C}}^{r-1}$.
Dais Dimitrios I.
Henk Martin
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