Flex curves and their applications
This article is concerned with the topology of the complement of an algebraic curve in the complex projective plane. In previous work, the author produced an example of two irreducible complex plane projective curves \(C_1\) and \(C_2\), both of degree 12 and having 27 ordinary cusps, such that the fundamental groups \(\pi_1 (\mathbb{P}^2- C_1)\) and \(\pi_1 (\mathbb{P}^2- C_2)\) are (nonabelian and) not isomorphic. In the present article, he produces a further example of an irreducible plane curve \(C_3\), such that \(\pi_1 (\mathbb{P}^2- C_3)\) is abelian. Moreover, the Alexander polynomial \(\Delta_{C_3} (t)\) of \(C_3\) is not equal to that of \(C_1\). A consequence is the fact that the space parametrizing plane projective curves of degree 12 having 27 cusps has at least three connected components, something that was unknown before. The author's technique consists in using certain cyclic transformations \(\varphi: \mathbb{C}^2\to \mathbb{C}^2\). A curve \(C\) in the target plane, when pulled back to the domain via \(\varphi\), will acquire more singularities. So, starting from a suitable relatively simple curve, it is possible to construct new ones, with more interesting features. Care must be taken to see what happens ``at infinity'', i.e. when one considers the extension of \(\varphi\) to \(\mathbb{P}^2\), as well as to control the fundamental groups under consideration and the pertinent Alexander polynomials. In previous work the author considered transformations as above ramified along a line. In the present paper he introduces, given a plane curve \(C\) on a nonsingular point \(P\in C\), such that the tangent line to \(C\) at \(P\) is not vertical, the notion of flex curve of degree \(l\) at \(P\). For a fixed positive integer \(l\), this is the (unique) curve \({\mathcal F}^{(l)} (P)\) of degree \(\leq l\) whose multiplicity of intersection with \(C\) at \(P\) is \(>l\). Then the author considers certain mappings \(\varphi\) as above, ramified along \({\mathcal F}^{(l)} (P)\) (called flex coverings), which play an essential role in his new construction. The work involved in many proofs and in thr verification of the examples is very technical and delicate.
topology of the complement of an algebraic curve
scientific article; zbMATH DE number 1295084