Résume | Rational Gromov-Witten and Welschinger invariants of the plane (or another del Pezo surface) count complex and real rational curves passing through appropriate configuration of points in general position and having nodes as their only singularities. One can count complex curves with non-nodal singularities (cusps etc.) which, under some regularity conditions, always leads to enumerative invariants. In 2006, Welschinger noticed that the count of real plane rational unicuspidal curves does depend on the point constraint, and he suggested a real enumerative invariant counting together the unicuspidal curves, reducible nodal curves, and irreducible curves matching some point and tangency conditions. We address the following question: Does there exist real enumerative invariants counting only rational curves with a given collection of non-nodal singularities and with weights depending only on the topology of the real point set?
Theorem 1. In degree 4, such invariants exist only for curves having either one singular point A_5, or one singular point D_4, or one singular point E_6, or three ordinary cusps, provided that, in the latter case, the constraint consists of four pairs of complex conjugate points. All these invariants are positive.
Theorem 2. For each degree d>4, there exists an enumerative invariant that counts real rational curves with one singularity of order d-1, combined of the transversal local branches of odd orders.
Joint work with O. Bojan. |