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The Alexander polynomials of plane algebraic curves. II. (English. Russian original) Zbl 0882.14008

Proc. Steklov Inst. Math. 208, 198-211 (1995); translation from Tr. Mat. Inst. Steklova 208, 224-239 (1995).
In part I of this paper [Russ. Acad. Sci., Izv., Math. 42, No. 1, 67-89 (1994); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 57, No. 1, 76-101 (1993; Zbl 0811.14017)], V. S. Kulikov investigated the Alexander polynomial of an algebraic curve \(D \subset \mathbb{C}^2\), defined by an equation \(f(x,y) =0\). In the paper under review he generalizes this to the case \(\overline D \subset \mathbb{P}^2\) with irreducible decomposition \(\overline D= \overline D_1 \cap \cdots \cap \overline D_n\), \(D_i= \overline D_i \cap \mathbb{C}^2\), \(D_i\) defined by \(f_i (x,y) =0\). Then the author defines the Alexander \(\overline m\)-polynomial, \(\overline m= (m_1, \dots, m_n) \in \mathbb{N}^n\). For \(\overline m= (1, \dots, 1)\) the Alexander \(\overline m\)-polynomial is the Alexander polynomial of part I of this paper.
Also in part I the irregularity \(q(\overline X_k)\) of a nonsingular surface \(\overline X_k\), which is birationally isomorphic to the surface defined in \(\mathbb{C}^3\) by the equation \(z^k= \prod^n_{i=1} f_i(x,y)\), was calculated in the case of transversal intersections of curves \(D_i\).
The purpose of this paper is to extend the results of part I to the case of general \(\overline m\).
For the entire collection see [Zbl 0863.00012].
Reviewer: G.-E.Winkler

MSC:

14H30 Coverings of curves, fundamental group
14F45 Topological properties in algebraic geometry
14F35 Homotopy theory and fundamental groups in algebraic geometry

Citations:

Zbl 0811.14017
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