Organisateurs : Benoît Bertrand, Erwan Brugallé, Ilia Itenberg, Grigory Mikhalkin

Toulouse, 14 October 2013

Institut de mathématiques de Toulouse, Salle 207

10h:30 Coffee

11h Lionel Lang (Université de Genève)

From Harnack to multi-Harnack curves.

Harnack curves in the sense of Mikhalkin are real algebraic curves sitting in toric surfaces in the nicest possible way. They enjoy several beautiful and of course equivalent definitions. One of them (at least) can be naturally weakened to give rise to a wider class of curves: we choose to call them multi-Harnack curves. Indeed, they are closely related to Harnack curves and might possess several real structures. We will present some members of this family, from famous to more anonymous, and try to motivate interest about them.

14:00 Nermin Salepci (Université de Lyon 1)

Fillability of real open books by real Lefschetz fibrations.

We will discuss an example of a real open book which cannot be filled by a real Lefschetz fibration although it is filled by non-real Lefschetz fibrations. (joint with F. Öztürk.)

15h coffee

15h30 Séverine Fiedler Le Touzé (Toulouse)

Rational pencils of cubics and configurations of seven points in ℝℙ2

Consider m ⩽ 9 generic points in ℝℙ2, what do they determine? Two points determine a line, five points a conic, nine points a cubic. Seven points 1, . . . 7 determine seven rational cubics, with nodes at 1, . . . 7. Six points 1, . . . 6 determine six rational pencils of cubics with nodes at 1, . . . 6. Such a pencil has exactly five reducible cubics. A combinatorial cubic is a topological type (cubic, points on it) and a combinatorial rational pencil is a cyclic sequence of the five combinatorial reducible cubics. We classify such pencils: up to the action of the symmetric group S6 on {1, . . 6}, there are exactly four different lists of six combinatorial rational pencils of cubics, and up to the action of S7 on {1, . . 7}, there are exactly 14 different lists of seven rational combinatorial cubics.