Complex algebraic plane curves via their links at infinity
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- Neumann, W.D. Invent Math (1989) 98: 445. doi:10.1007/BF01393832
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Considering that the study of plane cuves has an over 2000 year history and is the seed from which modern algebraic geometry grew, surprisingly little is known about the topology of affine algebraic plane curves. We topologically classify “regular” algebraic plane curves in complex affine 2-space using “splice diagrams:” certain decorated trees that code Puiseux data at infinity. (The regularity condition — that the curve be a “typical” fiber of its defining polynomial — can conjecturally be avoided.) We also show that the splice diagram determines such algebraic information as the minimal degree of the curve, even in the irregular case. Among other things, this enables algebraic classification of regular algebraic plane curves with given topology.