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

Toulouse, 20 October 2011

This session is dedicated to the memory of Mikael Passare

Institut de mathématiques de Toulouse, Amphi Schwartz Bât. 1R3

10h:30 Coffee

11h On the curvature of the Real Amoeba.

Jean-Jacques Risler (Institut de mathématiques de Jussieu)

For a real smooth algebraic curve A ⊂ (*)2, the amoeba A2 is the image of A under the map Log : (x,y) → (log|x|, log| y |). We describe an universal bound for the total curvature of the real amoeba A and we prove that this bound is reached if and only if the curve A is a simple Harnack curve in the sense of Mikhalkin.

14:00 Refined curve counting on algebraic surfaces

Lothar Göttsche (ICTP Trieste)

Most of the results discussed in this talk are conjectural. Let L be ample line bundle on an a projective algebraic surface S. Let g be the genus of a smooth curve in the linear system |L|. If L is sufficiently ample with respect to d, the number of nL,d of d-nodal curves in a general d-dimensional sublinear system of |L| will be finite. Kool-Shende-Thomas use relative Hilbert schemes of points of the universal curve over |L| to define the numbers nL,d as BPS invariants and prove a conjecture of mine about their generating function. We use the generating function of the χy-genera of these relative Hilbert schemes to define and study refined curve counting invariants NL,g(y), which are now polynomials in y, with NL,d(1)=nL,d. If S is a K3 surface we relate these invariants to the Donaldson-Thomas invariants considered by Maulik-Pandharipande-Thomas. In the case of real toric surfaces we see that the refined invariants interpolate between the Gromov-Witten invariants (at y = 1) and the Welschinger invariants (which count real curves) at y = -1.

15h café

15h30 Tropical smoothness and the adjunction formula for curves in surfaces

Kristin Shaw (Université de Genève)

In classical algebraic geometry the adjunction formula relates the canonical class a divisor to the canonical class of the original variety. For smooth curves in surfaces this formula can be written using the genus of the curve, and in the case of singular curves it defines the arithmetic genus. Using Mikhalkin's definition of the canonical class of an abstract tropical surface together with tropical intersection theory we may consider a tropical version of this formula and give sufficient conditions for it to hold. However, examples show these conditions are not necessary.

It is equally interesting to consider situations where the tropical adjunction formula fails. In joint work with E. Brugallé, we use the adjunction formula and a correspondence between complex and tropical intersections of curves to obtain local obstructions to the approximation of tropical curves in surfaces. Applying these results to the pathological lines in smooth tropical surfaces discovered by Vigeland, we discovered a peculiar phenomenon in tropical geometry; smoothness as it is defined in the tropical world is not an intrinsic property of varieties.