AI 中文总结
本文构造了热带态射的通用空间，证明其到热带模空间的覆盖度为Catalan数，并给出构造Catalan-多gonality见证映射的有效方法。
AI 中文摘要
在Brill-Noether理论的研究中，Eisenbud和Harris建立了曲线上线性级数通用参数空间的几何，证明了对于偶亏格$g$和度数$d = g/2 + 1$，到曲线模空间的投影是度为Catalan数$C_{g/2} = \frac{1}{g/2+1}\binom{g}{g/2}$的有限覆盖。在本文中，我们构造了这个通用族的热带对应物：一个多面体锥复形$\mathcal{G}_{g \to 0, d}^{\mathrm{trop}}$，参数化从亏格$g$的度量图到度量树的度-$d$热带态射。对于偶数的$g$和$d = g/2 + 1$，我们证明了遗忘投影$\Pi \colon \mathcal{G}_{g \to 0, d}^{\mathrm{trop}} \to \mathcal{M}_{g}^{\mathrm{trop}}$是一个度为$C_{g/2}$的分支覆盖，并配备了自然的行列式重数。我们通过证明在环的毛虫图上态射与选票序列一一对应来计算这个度，并通过跨余维数$1$墙的热带平衡条件建立了其在$\mathcal{M}_{g}^{\mathrm{trop}}$上的全局不变性。通过形变和路径提升，这为任何一般度量图构造Catalan-多的gonality见证映射提供了一种有效方法，确立了任何亏格$g$的度量图的树gonality至多为$\lceil g/2 \rceil + 1$。
英文摘要
In their work on Brill-Noether theory, Eisenbud and Harris established the geometry of the universal parameter space of linear series over curves, proving that for even genus $g$ and degree $d = g/2 + 1$, the projection to the moduli space of curves is a finite cover of degree equal to the Catalan number $C_{g/2} = \frac{1}{g/2+1}\binom{g}{g/2}$. In this paper, we construct the tropical counterpart of this universal family: a polyhedral cone complex $\mathcal{G}_{g \to 0, d}^{\mathrm{trop}}$ parametrizing degree-$d$ tropical morphisms from genus-$g$ metric graphs to metric trees. For even $g$ and $d = g/2 + 1$, we prove that the forgetful projection $Π\colon \mathcal{G}_{g \to 0, d}^{\mathrm{trop}} \to \mathcal{M}_{g}^{\mathrm{trop}}$ is a branched cover of degree $C_{g/2}$ equipped with natural determinantal multiplicities. We compute this degree by showing that on caterpillars of loops the morphisms are in bijection with ballot sequences, and we establish its global invariance across $\mathcal{M}_{g}^{\mathrm{trop}}$ via a tropical balancing condition across codimension-$1$ walls. Via deformation and path lifting, this yields an effective method to construct Catalan-many gonality-witnessing maps for any generic metric graph, establishing that the tree gonality of any genus-$g$ metric graph is at most $\lceil g/2 \rceil + 1$.
Comments38 pages, 5 figures. Comments are welcome!
