A weighted version of Zariski’s hyperplane section theorem and fundamental groups of complements of plane curves

In this paper, we formulate and prove a weighted homogeneous version of Zariski’s hyperplane section theorem on the fundamental groups of the complements of hypersurfaces in a complex projective space, and apply it to the study of π1(P 2 \C), where C ⊂ P is a projective plane curve. The main application is to prove a comparison theorem as follows. Let φ : P → P be the composition of the Veronese embedding P →֒ P and the restriction of a general projection P · · → P. Our comparison theorem enables us to calculate π1(P 2 \ φ(C)) from π1(P 2 \ C). In [14] and [16], Zariski studied some projective plane curves with interesting properties. An example is sextic curves with 6 cusps. Zariski showed that the fundamental group of the complement depends on the placement of the 6 cusps. Another example is the 3-cuspidal quartic curve, whose complement has a non-abelian and finite fundamental group. This curve is the only known example with this property. Using the comparison theorem, we derive infinite series of curves with these interesting properties from the classical examples of Zariski. As another application, we shall discuss a relation between π1(P 2 \ C) and π1(P 2 \ (C ∪ L∞)), where L∞ is a line intersecting C transversely.