Alexander polynomials of plane algebraic curves

Vik. S. Kulikov

Izvestiya. Mathematics , Tome 42 (1994) no. 1, pp. 67-89

Résumé

The author studies the fundamental group of the complement of an algebraic curve $D\subset\mathbf C^2$ defined by an equation $f(x,y)=0$. Let $F\colon X=\mathbf C^2\setminus D\to\mathbf C^*=\mathbf C\setminus\{0\}$ 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 $F_*\colon\pi_1(X)\to\pi_1(\mathbf C^*)$ 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 $\Delta_{in}(t)$ and $\Delta_{ex}(t)$ respectively) are introduced, and it is shown that the Alexander polynomial $\Delta_1(t)$ of the curve $D$ divides $\Delta_{in}(t)$ and $\Delta_{ex}(t)$ 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.