Counting real curves with passage/tangency conditions

Details Counting real curves with passage/tangency conditions Counting real curves with passage/tangency conditions

2010-11-07

arxiv.org/abs/1011.1686v2

Mathematics

Geometric Topology

23 pages, many figures; v2: significant changes, results generalized from cubics to rational curves of arbitrary degree

Scientific paper

We study the following question: given a set P of 3d-2 points and an immersed curve G in the real plane R^2, all in general position, how many real rational plane curves of degree d pass through these points and are tangent to this curve. We count each such curve with a certain sign, and present an explicit formula for their algebraic number. This number is preserved under small regular homotopies of a pair (P, G), but jumps (in a well-controlled way) when in the process of homotopy we pass a certain singular discriminant. We discuss the relation of such enumerative problems with finite type invariants. Our approach is based on maps of configuration spaces and the intersection theory in the spirit of classical algebraic topology.

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