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On the fundamental groups of complements of toral curves. (English. Russian original) Zbl 0907.14013

Izv. Math. 61, No. 1, 89-112 (1997); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 61, No. 1, 89-112 (1997).
Summary: 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 \(\pi_1(\mathbb{C}^2 \setminus D) \cong \langle x_1,x_2 \mid x_1^a =x_2^b\rangle\), 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 \(l^a_{bc} (z_0,z_1,z_2) +l^b_ {ac} (z_0,z_1, z_2)=0\), where \(l_q\) is a homogeneous polynomial of degree \(q\), we have \(\pi_1 (\mathbb{P}^2 \setminus \overline D) \cong \langle x_1,x_2 \mid x^a_1=x_2^b, x_1^{ac} =1\rangle\).

MSC:

14H30 Coverings of curves, fundamental group
14F35 Homotopy theory and fundamental groups in algebraic geometry
14H45 Special algebraic curves and curves of low genus
57M05 Fundamental group, presentations, free differential calculus
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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