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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

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