# Monodromy of rational curves on toric surfaces

## TL;DR

This paper investigates the monodromy of rational curves on toric surfaces, identifying the monodromy group as the deck transformations of an obstruction map and providing tools to compute it, with examples of smaller monodromy groups.

## Contribution

It introduces a method to determine the monodromy group of rational curves on toric surfaces and identifies conditions under which the monodromy is maximal or smaller.

## Key findings

- Monodromy group equals the deck transformations of the obstruction map.
- Provided a computational tool for monodromy in various cases.
- Constructed examples with smaller-than-expected monodromy groups.

## Abstract

For an ample line bundle $\mathcal{L}$ on a complete toric surface $X$, we consider the subset $V_{\mathcal{L}} \subset \vert \mathcal{L} \vert$ 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 $\varPsi_{X}$ defined on the set of nodes of $C$ and show that the image of the monodromy is exactly the group of deck transformations of $\varPsi_{X}$, 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.