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.

中文翻译：

---

复曲面上有理曲线的单峰

对于足够的线束 $\mathcal{大号}$ 在完整的复曲面上 $X$，我们考虑子集 $V_{\mathcal{大号}}\subset |\mathcal{大号}|$ 的平滑轨迹中包含的不可约的，节点的，有理曲线 $X$。我们从$V_{\mathcal{大号}}$ 到参考曲线节点集上的置换组 $C\in V_{\mathcal{大号}}$。我们确定某个障碍物图${\mathrm{\Psi }}_X$ 在以下节点的集合上定义 $C$ 并表明，单声道的图像恰好是 ${\mathrm{\Psi }}_X$，前提是 $\mathcal{大号}$足够大（在某种意义上，我们在下面进行精确说明）。一路上，我们构造了一个方便的工具来计算任何一对的单峰图像$（X，\mathcal{大号}）$。最终，我们提出了一个成对的家庭$（X，\mathcal{大号}）$ 小 $\mathcal{大号}$ 对于它，单声道的图像严格小于预期。