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- W2024972907 abstract "The classical hyperplane section theorem of Zariski about the fundamental groups of the complements to hypersurfaces in the complex projective space is generalized to the weighted homogeneous case. Using this result, we study how the fundamental group of the complement to a projective plane curve changes under covering of the projective plane by another projective plane. We also give an example of plane curves whose complements have nonabelian and finite fundamental groups. Few examples of such curves have been known. By the weighted Zariski's hyperplane section theorem, we can determine the group structure of these finite nonabelian groups. 0. Introduction. In the first part of this paper, we formulate and prove a weighted homogeneous version of Zariski's hyperplane section theorem on the fundamental groups of the complements to hypersurfaces in a complex projective space, and discuss some direct applications to the fundamental groups of com plements to projective plane curves. In the second part, we investigate a certain singular plane curve C(q9 k)9 which is a generalization of Zariski's three cuspidal quartics, and calculate the fundamental group ^(P2 C(q9k)) using the main result of the first part. This group turns out to be finite and nonabelian. Let jci, ... 9xn be variables with weights" @default.
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- W2024972907 date "1997-01-01" @default.
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- W2024972907 title "Fundamental groups of complements to singular plane curves" @default.
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- W2024972907 doi "https://doi.org/10.1353/ajm.1997.0006" @default.
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