Curvas de contato no espaço projetivo

Abstract

The odd dimensional projective space P2n-1 admits a contact structure arising from a non integrable distribution of hyperplanes determined by a symplectic form in C2n. Our object of interest is the set of rational curves of degree d which are tangent to that contact distribution in P3. Such curves are called contact curves or legendrian curves. To explore the geometry of contact curves, we construct the parameter space Ld using Kontsevichs stable maps, M0;0(P3; d), endowed with the structure of algebraic stack. The intersection theory on stacks allows us to define in that space the virtual invariant Nd, associated with the number of degree d contact curves incident to 2d + 1 lines. Using graph combinatorics and partitions originated from Botts localization formula, we determine a general formula for Nd. We explicitly calculate it for contact curves up to degree 4 - confirming the known cases of contact lines and conics and introducing the new numbers for cubics and quartics. Finally, we discussthe enumerative significance of these invariants, still conjectural for d > 4.