[CvGmt News] [Notiziario] [Settimanale] seminario di geometria reale
bertrand at mail.dm.unipi.it
Fri Dec 12 14:12:32 CET 2003
lunedi prossimo (attenzione alle 16:30 !) :
Lunedì 15 dicembre, sala Seminari
Erwan Brugalle (Rennes - France)
Welschinger invariant and enumeration of real rational curves.
(based on [IKS])
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.
J-Y Welschinger. Invariants of real symplectic 4-manifolds and lower
bounds in real enumerative geometry, 2003,
I. Itenberg and V. Kharlamov and E. Shustin. Welschinger invariant and
enumeration of real plane rational curves,
G. Mikhalkin. Counting curves via lattice paths in polygons.
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