Local geometry of smooth curves passing through rational double points.

David B. Jaffe

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We compute the number of irreducible rational curves of given degree with 1 tacnode in $¶^{2}$ or 1 node in $¶^{3}$ meeting an appropriate generic collection of points and lines. As a byproduct, we also compute the number of rational plane curves of degree $d$ passing through $3 d - 2$ given points and tangent to a given line. The method is ‘classical’, free of Quantum Cohomology.