## 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.
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

### Citations:

Zbl 0509.00008; Zbl 0546.57002