Monodromy of rational curves on toric surfaces

For an ample line bundle L on a complete toric surface X, we consider the subset VL⊂|L| of irreducible, nodal, rational curves contained in the smooth locus of X. We study the monodromy map from the fundamental group of VL to the permutation group on the set of nodes of a reference curve C∈VL. We identify a certain obstruction map ΨX defined on the set of nodes of C and show that the image of the monodromy is exactly the group of deck transformations of ΨX, provided that L is sufficiently big (in the sense we make precise below). Along the way, we construct a handy tool to compute the image of the monodromy for any pair (X,L). Eventually, we present a family of pairs (X,L) with small L and for which the image of the monodromy is strictly smaller than expected.