Gurvan Mével soutiendra sa thèse à l'UFR des sciences et des techniques de Nantes université, bâtiment 11, salle 3 à 14h.
Titre de la thèse : Asymptotic properties of tropical refined invariants
Résumé :
In enumerative algebraic geometry, the number of curves on a surface behaves differently whether one fixes the number of nodes or the genus of the curves. It is polynomial in the first case, but grows more than exponentially fast in the second case. In the first case Göttsche conjecture, proven by Tzeng, gives a universal formula for the generating series of these numbers.
Tropical refined invariants were introduced by Block and Göttsche. They are polynomials that interpolates between real and complex enumerative questions. As expected, their coefficients behave polynomially when the number of nodes is fixed. However, Brugallé and Jaramillo-Puentes proved that some of their coefficients are also polynomial when the genus is fixed. In this thesis we prove some universal formulas for the first coefficients of the tropical refined invariants in genus 0 and 1, in the spirit of Göttsche conjecture.
The techniques we use fall under the scope of tropical geometry. Introduced by Brugallé and Mikhalkin, floor diagrams are a tool that turns the starting algebro-geometric question into a combinatorial problem. In this thesis, we precisely describe the floor diagrams that asymptotically take part in the computation of tropical refined invariants. This allows to write down universal formulas.
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