Counting real curves with passage/tangency conditions

TL;DR

This paper develops a method to count real rational plane curves of a given degree passing through specified points and tangent to a curve, providing an explicit algebraic count that remains invariant under certain deformations.

Contribution

It introduces a new explicit formula for counting such curves with signs, linking enumerative geometry with topological invariants and intersection theory.

Findings

Provides an explicit algebraic count of real rational curves with tangency conditions.

Shows the count is invariant under small regular homotopies, with controlled jumps at singularities.

Connects enumerative problems to finite type invariants and classical algebraic topology.

Abstract

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.