Real Zeuthen numbers for two lines

Details Real Zeuthen numbers for two lines Real Zeuthen numbers for two lines

2007-10-04

arxiv.org/abs/0710.1095v1

Mathematics

Algebraic Geometry

6 pages, 3 figures

Scientific paper

Given three natural numbers $k,l,d$ such that $k+l=d(d+3)/2$, the Zeuthen number $N_{d}(l)$ is the number of nonsingular complex algebraic curves of degree $d$ passing through $k$ points and tangent to $l$ lines in $\PP^2$. It does not depend on the generic configuration $C$ of points and lines chosen. If the points and lines are real, the corresponding number $N_{d}^\RR(l,C)$ of real curves usually depends on the configuration chosen. We use Mikhalkin's tropical correspondence theorem to prove that for two lines the real Zeuthen problem is maximal: there exists a configuration $C$ such that $N_{d}^\RR(2,C)=N_{d}(2)$. The correspondence theorem reduces the computation to counting certain lattice paths with multiplicities.

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