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Topology of the complement of a real algebraic curve in \({\mathbb{C}}P^ 2\). (English. Russian original) Zbl 0542.14009
J. Sov. Math. 26, 1684-1689 (1984); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 122, 137-145 (1982).
The topology of the complement of any nonsingular real algebraic curve of a given degree in \({\mathbb{C}}{\mathbb{P}}^ 2\) is standard. The situation varies when we consider the set A of complex points of a real curve and the complement of \({\mathbb{R}}{\mathbb{P}}^ 2\cup A\) in \({\mathbb{C}}{\mathbb{P}}^ 2\). One of the principal reasons for the research is to find rigid homotopic invariants of curves that would be finer than the complex scheme that can be restored by the topology of \(A\cup {\mathbb{R}}{\mathbb{P}}^ 2\). The author constructs a two-dimensional CW-complex which is homotopically equivalent to \({\mathbb{C}}{\mathbb{P}}^ 2\backslash(A\cup {\mathbb{R}}{\mathbb{P}}^ 2)\) for some class of curves. It allows him to compute the fundamental group of the complement. In particular the fundamental group is computed for all nonsingular real curves of a degree \(\leq 5\). The question whether the homotopic type of the complement of \(A\cup {\mathbb{R}}{\mathbb{P}}^ 2\) is finer invariant than the complex scheme remains undecided. - All necessary constructions are given in details; the proofs are absent.
Reviewer: B.Bekker
MSC:
14F45 Topological properties in algebraic geometry
14E20 Coverings in algebraic geometry
14H45 Special algebraic curves and curves of low genus
14F35 Homotopy theory and fundamental groups in algebraic geometry
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