用于枚举 $n$-标点曲线上线性级数的广义 RSK
A Generalized RSK for Enumerating Linear Series on $n$-pointed Curves
AI总结：
本文以推广 RSK 的 Young 表与多元序列双射，组合证明并解释了 $n$-标记曲线在指定入射条件下线性级数计数为 $(r+1)^g$（及 $r=1$ 重合点情形为 $2^g$）的几何结果。
摘要：
我们对 Farkas 和 Lian 最近关于具有给定关联条件的曲线上线性级数的几何结果给出了组合证明。该结果表明，对于一般亏格 $g$ 的 $n$-标记曲线 $C$，当 $d$ 足够大时，将 $C$ 上的标记点映到 $\mathbb{P}^r$ 中指定的一般点的 $d$ 次态射的期望数目等于 $(r+1)^g$。这一计算可以重新表述为 Grassmann 簇上的一个相交问题；借助经典的 Littlewood–Richardson 法则，它可用 Young 表给出自然的组合解释。我们构造了一个双射，将所讨论的表与长度为 $g$ 的 $(r+1)$ 元序列对应起来，该双射推广了著名的 RSK 对应；同时我们探究了该双射的组合性质。我们还应用类似方法，对如下事实给出组合解释与证明：在修改后的情形 $r=1$ 且若干标记点映到 $\mathbb{P}^1$ 中同一点时，当 $d$ 足够大，态射数目仍为 $2^g$。
英文摘要：
We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-$d$ morphisms from a general genus $g$, $n$-marked curve $C$ to $\mathbb{P}^r$, sending the marked points on $C$ to specified general points in $\mathbb{P}^r$, is equal to $(r+1)^g$ for sufficiently large $d$. This computation may be rephrased as an intersection problem on Grassmannians, which has a natural combinatorial interpretation in terms of Young tableaux by the classical Littlewood-Richardson rule. We give a bijection, generalizing the well-known RSK correspondence, between the tableaux in question and the $(r+1)$-ary sequences of length $g$, and we explore our bijection's combinatorial properties. We also apply similar methods to give a combinatorial interpretation and proof of the fact that, in the modified setting in which $r=1$ and several marked points map to the same point in $\mathbb{P}^1$, the number of morphisms is still $2^g$ for sufficiently large $d$.
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