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Monodromy of rational curves on toric surfaces
Journal of Topology  (IF1.582),  Pub Date : 2020-10-23, DOI: 10.1112/topo.12171
Lionel Lang

For an ample line bundle L on a complete toric surface X , we consider the subset V L | L | of irreducible, nodal, rational curves contained in the smooth locus of X . We study the monodromy map from the fundamental group of V L to the permutation group on the set of nodes of a reference curve C V L . We identify a certain obstruction map Ψ X defined on the set of nodes of C and show that the image of the monodromy is exactly the group of deck transformations of Ψ X , 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.