Complex algebraic plane curves via their links at infinity. (English) Zbl 0734.57011

Neumann, Walter D.

Invent. Math. 98, No. 3, 445-489 (1989).

An algebraic curve \(V=\{f(x,y)=0\}\subset {\mathbb{C}}^ 2\) is called regular if it is isotopic to the curves \(f(x,y)=c\) as c is close to 0. The assertion of the author consists in the fact that, for a regular curve, the topology of the pair \(({\mathbb{C}}^ 2,V)\) is determined by the link \((S^ 3,L)\) where \(S^ 3\) is a sphere in \({\mathbb{C}}^ 2\) of a sufficiently large radius with the center (0;0) and \(L=S^ 3\cap V\). In doing so, the Seifert surface of the link L in \(S^ 3\) is unique up to isotopy in \(S^ 3\) and V is renewed (up to isotopy in \({\mathbb{C}}^ 2)\) by fitting the part of V lying outside the sphere \(S^ 3\) with the Seifert manifold.

MSC:

57M25	Knots and links in the \(3\)-sphere (MSC2010)
14F45	Topological properties in algebraic geometry
14H45	Special algebraic curves and curves of low genus

Keywords:

links at infinity; regular algebraic curve; Seifert surface

References:

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