We use Young tableaux to compute the dimension of $V^r$, the Prym-Brill-Noether locus of a folded chain of loops of any gonality. This tropical result yields a new upper bound on the dimensions of algebraic Prym-Brill-Noether loci. Moreover, we prove that $V^r$ is pure-dimensional and connected in codimension $1$ when $\dim V^r \geq 1$. We then compute the first Betti number of this locus for even gonality when the dimension is exactly $1$, and compute the cardinality when the locus is finite and the edge lengths are generic.

中文翻译：

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特殊曲线的 Prym-Brill-Noether 轨迹

我们使用 Young tableaux 来计算 $V^r$ 的维度，即任何多边形折叠链的 Prym-Brill-Noether 轨迹。这个热带结果在代数 Prym-Brill-Noether 轨迹的维度上产生了一个新的上限。此外，我们证明了当 $\dim V^r \geq 1$ 时，$V^r$ 是纯维的并且在余维 $1$ 上连通。然后，当维度正好为 $1$ 时，我们计算该轨迹的第一个 Betti 数，以计算偶数角性，并在轨迹有限且边长通用时计算基数。