Lionel Lang

For an ample line bundle $\mathcal{L}$ on a complete toric surface $X$, we consider the subset ${V}_{\mathcal{L}}\subset |\mathcal{L}|$ of irreducible, nodal, rational curves contained in the smooth locus of $X$. We study the monodromy map from the fundamental group of ${V}_{\mathcal{L}}$ to the permutation group on the set of nodes of a reference curve $C\in {V}_{\mathcal{L}}$. We identify a certain obstruction map ${\mathrm{\Psi}}_{X}$ defined on the set of nodes of $C$ and show that the image of the monodromy is exactly the group of deck transformations of ${\mathrm{\Psi}}_{X}$, provided that $\mathcal{L}$ 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 $(X,\mathcal{L})$. Eventually, we present a family of pairs $(X,\mathcal{L})$ with small $\mathcal{L}$ and for which the image of the monodromy is strictly smaller than expected.