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Alexander polynomials of plane algebraic curves. (English. Russian original) Zbl 0811.14017

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).
Summary: The author studies the fundamental group of the complement of an algebraic curve \(D\subset \mathbb{C}^ 2\) defined by an equation \(f(x,y)=0\). Let \(F: X= \mathbb{C}^ 2\setminus D\to \mathbb{C}^*= \mathbb{C}\setminus \{0\}\) be the morphism defined by the equation \(z= f(x,y)\). The main result is that if the generic fiber \(Y= F^{-1} (z_ 0)\) is irreducible, then the kernel of the homomorphism \(F_ *: \pi_ 1(X)\to \pi_ 1(\mathbb{C}^*)\) is a finitely generated group. In particular, if \(D\) is an irreducible curve, then the commutator subgroup of \(\pi_ 1(X)\) is finitely generated.
The internal and external Alexander polynomials of a curve \(D\) (denoted by \(\Delta_{\text{in}} (t)\) and \(\Delta_{\text{ex}} (t)\) respectively) are introduced, and it is shown that the Alexander polynomial \(\Delta_ 1(t)\) of the curve \(D\) divides \(\Delta_{\text{in}}(t)\) and \(\Delta_{\text{ex}}(t)\) and is a reciprocal polynomial whose roots are roots of unity. Furthermore, if \(D\) is an irreducible curve, the Alexander polynomial \(\Delta_ 1(t)\) of the curve \(D\) satisfies the condition \(\Delta_ 1(1)= \pm1\). From this it follows that among the roots of the Alexander polynomial \(\Delta_ 1(t)\) of an irreducible curve there are no primitive roots of unity of degree \(p^ n\), where \(p\) is a prime number.

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
14F45 Topological properties in algebraic geometry
14H99 Curves in algebraic geometry
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