[CvGmt News] [Notiziario] [Settimanale] seminario di geometria reale

[Benoit BERTRAND]

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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