Buongiorno,

lunedi prossimo (attenzione alle 16:30 !) : see also http://www.dm.unipi.it/~bertrand/Seminari.html

Lunedì 15 dicembre, sala Seminari 16:30 Erwan Brugalle (Rennes - France)

Welschinger invariant and enumeration of real rational curves. (based on [IKS])

abstract

How many real rational curves of degree d pass through 3d-1 generic points in RP² ? This is an example of a problem in real enumerative geometry. It has been some time since the analogous problem in complex geometry has been solved. However, in the real case, it is still an open problem and even the existence of such curves has been proved only recently by I. Itenberg, V. Kharlamov and E. Shustin in [IKS]. The proof there combines Mikhalkin's (cf. [Mik]) tropical approach of real enumerative geometry and Welschinger invariant of real rational curves in RP2 (cf. [Wel]). After defining Welschinger invariant, I will explain how Mikhalkin counts real nodal curves. I will then give an idea of the proof of [IKS] which gives d!/2 as a lower bound for this number.

[Wel] J-Y Welschinger. Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, 2003, arXiv:math.AG/0303145 [IKS] I. Itenberg and V. Kharlamov and E. Shustin. Welschinger invariant and enumeration of real plane rational curves, arXiv:math.AG/0303378 [Mik] G. Mikhalkin. Counting curves via lattice paths in polygons. arXiv:math.AG/0209253, 2003.

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