We prove a generalisation of the Brill-Noether theorem for the variety of special divisors $W^r_d(C)$ on a general curve C of prescribed gonality. Our main theorem gives a closed formula for the dimension of $W^r_d(C)$ . We build on previous work of Pflueger, who used an analysis of the tropical divisor theory of special chains of cycles to give upper bounds on the dimensions of Brill-Noether varieties on such curves. We prove his conjecture, that this upper bound is achieved for a general curve. Our methods introduce logarithmic stable maps as a systematic tool in Brill-Noether theory. A precise relation between the divisor theory on chains of cycles and the corresponding tropical maps theory is exploited to prove new regeneration theorems for linear series with negative Brill-Noether number. The strategy involves blending an analysis of obstruction theories for logarithmic stable maps with the geometry of Berkovich curves. To show the utility of these methods, we provide a short new derivation of lifting for special divisors on a chain of cycles with generic edge lengths, proved using different techniques by Cartwright, Jensen, and Payne. A crucial technical result is a new realisability theorem for tropical stable maps in obstructed geometries, generalising a well-known theorem of Speyer on genus $1$ curves to arbitrary genus.

中文翻译：

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固定边角曲线的 Brill-Noether 理论

我们证明了对各种特殊除数的 Brill-Noether 定理的推广 $W^r_d(C)$ 在一般曲线上C规定的角。我们的主要定理给出了维数的封闭公式 $W^r_d(C)$ . 我们以 Pflueger 先前的工作为基础，他使用对特殊循环链的热带因子理论的分析来给出此类曲线上 Brill-Noether 变体的维度的上限。我们证明了他的猜想，这个上限是针对一般曲线实现的。我们的方法引入了对数稳定映射作为 Brill-Noether 理论中的系统工具。利用循环链上的除数理论和相应的热带地图理论之间的精确关系来证明具有负布里尔-诺特数的线性级数的新再生定理。该策略涉及将对数稳定映射的障碍理论分析与 Berkovich 曲线的几何形状相结合。为了展示这些方法的实用性，我们为具有通用边长的循环链上的特殊除数提供了一个简短的新推导，Cartwright、Jensen 和 Payne 使用不同的技术证明了这一点。一个关键的技术成果是一个新的可实现性定理，适用于受阻几何中的热带稳定图，推广了著名的 Speyer 定理 $1$ 曲线到任意属。