##
**Weak simultaneous resolution for deformations of Gorenstein surface singularities.**
*(English)*
Zbl 0568.14008

Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 2, 1-29 (1983).

[For the entire collection see Zbl 0509.00008.]

This paper is a detailed and technical investigation of weak resolutions of deformations, culminating in the following theorem which provides a topological criterion for the existence of resolutions: Let \(f: V\to T\) be the germ of a flat deformation of the normal Gorenstein two dimensional singularity (V,p) with T a reduced analytic space. Then f has a weak simultaneous resolution if and only if each \(V_ t\) has a singularity \(p_ t\) with \((V_ t,p_ t)\) homeomorphic to (V,p). [This theorem essentially uses the result of W. Neumann, Trans. Am. Math. Soc. 268, 299-343 (1981; Zbl 0546.57002) which asserts that the oriented homotopy type of V-p determines the topology of the pair (M,A) where \(g: M\to V\) is the minimal (good) resolution and A the exceptional set.] - On the way to the main theorem, a number of other results appear. For (V,p) a purely two-dimensional singularity, \(g: M\to V\) a resolution, and K the canonical divisor on M, \(S_ m\) denotes g(\({\mathcal O}(mK))\). The blow-up X of V at M is (for \(m\geq 3)\) shown to be the rational double point resolution of V. Moreover the canonical map \(S_ m\otimes S_ n\to S_{m+n}\) is shown to be onto for \(m\geq 2\) and \(n\geq 3\) (these assertions extend work of Shephard-Barron and Reid). With (V,p) normal and Gorenstein and g minimal then K is supported in the exceptional set A in M. So \(K\cdot K\) is defined, and is constant in the deformation \(f: V\to T\) if and only if f has a ”very weak” simultaneous resolution; and if so then dim \(H^ 1(M_ t,{\mathcal O})\) is also constant.

This paper is a detailed and technical investigation of weak resolutions of deformations, culminating in the following theorem which provides a topological criterion for the existence of resolutions: Let \(f: V\to T\) be the germ of a flat deformation of the normal Gorenstein two dimensional singularity (V,p) with T a reduced analytic space. Then f has a weak simultaneous resolution if and only if each \(V_ t\) has a singularity \(p_ t\) with \((V_ t,p_ t)\) homeomorphic to (V,p). [This theorem essentially uses the result of W. Neumann, Trans. Am. Math. Soc. 268, 299-343 (1981; Zbl 0546.57002) which asserts that the oriented homotopy type of V-p determines the topology of the pair (M,A) where \(g: M\to V\) is the minimal (good) resolution and A the exceptional set.] - On the way to the main theorem, a number of other results appear. For (V,p) a purely two-dimensional singularity, \(g: M\to V\) a resolution, and K the canonical divisor on M, \(S_ m\) denotes g(\({\mathcal O}(mK))\). The blow-up X of V at M is (for \(m\geq 3)\) shown to be the rational double point resolution of V. Moreover the canonical map \(S_ m\otimes S_ n\to S_{m+n}\) is shown to be onto for \(m\geq 2\) and \(n\geq 3\) (these assertions extend work of Shephard-Barron and Reid). With (V,p) normal and Gorenstein and g minimal then K is supported in the exceptional set A in M. So \(K\cdot K\) is defined, and is constant in the deformation \(f: V\to T\) if and only if f has a ”very weak” simultaneous resolution; and if so then dim \(H^ 1(M_ t,{\mathcal O})\) is also constant.

Reviewer: A.R.Magid

### MSC:

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |

14B07 | Deformations of singularities |

14J17 | Singularities of surfaces or higher-dimensional varieties |

14B05 | Singularities in algebraic geometry |