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The number of vanishing cycles for a quasihomogeneous mapping from $$\mathbb{C} ^ 2$$ to $$\mathbb{C} ^ 3$$. (English) Zbl 0764.32013
If the analytic map-germ $$f:(\mathbb{C}^ n,0)\to(\mathbb{C}^{n+1},0)$$ has finite $${\mathcal A}_ e$$-codimension, the image of a stable perturbation has the homotopy type of a wedge of $$n$$-spheres whose number $$\sigma$$ is an analytic invariant of $$f$$. In this paper this invariant (the “image Milnor number”) is computed, in terms of weights and degrees, when $$f$$ is a quasi-homogeneous map $$\mathbb{C}^ 2\to\mathbb{C}^ 3$$.
The computation is indirect, and depends on the formula $\sigma={1\over 2}\{\mu(D^ 2)-4T+C-1\}$ where $$D^ 2$$ is the closure of the set of points in $$\mathbb{C}^ 2$$ where $$f$$ is 2-to-1, and $$T$$ and $$C$$ are respectively, the number of triple points and the number of pinch points (cross-caps) in the stable image. We obtain $$C$$, $$\mu(D^ 2)$$ and $$T$$ separately; only the last presents any difficulty. It is calculated from the fact that $T=\dim_ \mathbb{C}({\mathcal O}_{\mathbb{C}^ 3,0}/{\mathcal F}_ 2(f_ *{\mathcal O}_{\mathbb{C}^ 2})_ 0),$ where $${\mathcal F}_ 2(f_ *{\mathcal O}_{\mathbb{C}^ 2})$$ is the second Fitting ideal sheaf of the $${\mathcal O}_{\mathbb{C}^ 3}$$-module $$f_ *{\mathcal O}_{\mathbb{C}^ 2}$$, using Jozefiak’s projective resolution of the quotient of a regular local ring $$R$$ by the ideal of submaximal minors of a symmetric matrix with entries in $$R$$.
The answer is: Theorem 1. Let $$f$$ be quasi-homogeneous with weights $$w_ 1$$ and $$w_ 2$$, and degrees $$d_ 1$$, $$d_ 2$$ and $$d_ 3$$, and suppose $$f$$ has finite $${\mathcal A}_ e$$-codimension.
Let $$\varepsilon=d_ 1+d_ 2+d_ 3-w_ 1-w_ 2$$, and let $$\delta={d_ 1d_ 2d_ 3\over w_ 1w_ 2}$$. Then $\sigma={1\over 6w_ 1w_ 2}\{(\delta-\varepsilon)(\delta+\varepsilon-3(w_ 1+w_ 2))=(d_ 2d_ 3+d_ 1d_{32} )+(w_ 1+w_ 2)\varepsilon+w_ 1w_ 2\}.$ Note that $${\mathcal A}_ e$$-codimension = image Milnor number for quasi-homogeneous maps $$\mathbb{C}^ 2\to\mathbb{C}^ 3$$.
Subsequent developments: other formula for invariants of quasi- homogeneous mappings in terms of weights and degrees:
(i) T. M. Cooper gives the Poincaré series for the $${\mathcal A}_ e$$-normal space $$(T^ 1)$$ for maps $$\mathbb{C}^ 2\to\mathbb{C}^ 3$$ [M. Sc. Diss. Univ. Warwick (1990)],
(ii) V. V. Goryunov and the author compute the image Milnor number of a map $$\mathbb{C}^ n\to\mathbb{C}^{n+1}$$ with corank 1 singularity at 0 [Vanishing cohomology of singularities of mappings (to appear in Compos. Math.)],
(iii) C. T. C. Wall computes the $${\mathcal A}_ e$$-codimension of maps $$\mathbb{C}^ n\to\mathbb{C}^ p(n\geq p)$$ [Weighted homogeneous complete intersections, Eur. Singularity Project, Preprint 4(1992)]. In view of the main theorem of J. Damon and the author, Invent. Math. 106, 217-242 (1991)], this is equal to the discriminant Milnor number provided $$(n,p)$$ are in Mather’s range of nice dimensions.
Reviewer: D.Mond

##### MSC:
 32S30 Deformations of complex singularities; vanishing cycles 14B05 Singularities in algebraic geometry
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