Lionel Lang: The vanishing cycles of curves in toric surfaces (joint work with Rémi Crétois)

Abstract: Take a generic curve C in a linear system L on a toric surface X. What are the simple closed curves in C that can be contracted along a degeneration to a nodal curve? This question can be rephrased in term of the image of the monodromy map given by the complement of the discriminant D ⊂ L into the mapping class group of C. Compared with degeneration in M̅_g  (g>1) where any cycles can be contracted, there are known obstructions to contract cycles in L, namely roots of the relative canonical bundle and hyperelliptic involution. We will review how we can detect them and show that there is no other obstructions. Indeed, we will show that the monodromy is surjective on the mapping class group MCG(C) when no such obstruction appears. 
There will be two main ingredients along the proof. First we will use explicit degenerations on a well studied class of curves: simple Harnack curves. Then we will construct explicit element of the monodromy by applying Mihkalkin's approximation Theorem to well chosen loops in some tropical compactification of the linear system L.