On the fundamental groups of complements of toral curves

Vik. S. Kulikov

Izvestiya. Mathematics , Tome 61 (1997) no. 1, pp. 89-112

Résumé

We show that for almost all curves $D$ in $\mathbb C^2$ given by an equation of the form $g(x,y)^a+h(x,y)^b=0$, where $a>1$ and $b>1$ are coprime integers, the fundamental group of the complement of the curve has presentation $\pi_1(\mathbb C^2 \setminus D) \simeq (x_1,x_2\mid x_1^a=x_2^b)$, that is, it coincides with the group of the torus knot $K_{a,b}$. In the projective case, for almost every curve $\overline D$ in $\mathbb P^2$ which is the projective closure of a curve in $\mathbb C^2$ given by an equation of the form $g(x,y)^a+h(x,y)^b=0$, the fundamental group $\pi_1(\mathbb P^2\setminus\overline D)$ of the complement is a free product with amalgamated subgroup of two cyclic groups of finite order. In particular, for the general curve $\overline D\subset\mathbb P^2$ given by the equation $l_{bc}^a(z_0,z_1,z_2)+l_{ac}^b(z_0,z_1,z_2)=0$, where $l_q$ is a homogenous polynomial of degree $q$, we have $\pi_1(\mathbb P^2\setminus\overline D)\simeq\langle x_1,x_2\mid x_1^a=x_2^b,x_1^{ac}=1\rangle$.