Monodromy of rational curves on toric surfaces

Abstract

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 ${\Psi }_X$ defined on the set of nodes of $C$ and show that the image of the monodromy is exactly the group of deck transformations of ${\Psi }_X$, provided that $\mathcal{L}$ is sufficiently big (in a sense we precise below). Along the way, we provide 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.

28 pages, 1 figures. Final version. To appear in Journal of Topology