论文标题
用于列举$ n $点曲线的线性系列的广义RSK
A Generalized RSK for Enumerating Linear Series on $n$-pointed Curves
论文作者
论文摘要
我们给出了Farkas和Lian在线性序列上的几何结果的组合证明，在有规定的发病率条件下。结果指出，从一般属$ g $，$ n $ g $，$ n $ c $ c $ c $ to $ \ mathbb {p}^r $中的预期$ d $形态的预期数，在$ \ c $上向特定的一般点发送标记点，以$ \ mathbb {p}^r $，等于$（r+1）^g $ d $ d $ d $。该计算可以作为格拉曼尼亚人的交叉问题来改编，格拉斯曼尼亚人在经典的利特尔伍德 - 里查森（Littlewood-Richardson）规则上就年轻的tableaux进行了自然的组合解释。我们进行了两次培训，概括了所讨论的tableaux和$（r+1）$ - $ g $的$（r+1）$ ary序列之间的概括，我们探索了两者的组合属性。 我们还采用了类似的方法来给出一个组合解释和证据，证明了以下事实：在修改的设置中，其中$ r = 1 $和几个标记的点映射到$ \ mathbb {p}^1 $中的同一点，形态的数量仍然是$ 2^g $，对于足够的大$ d $。
We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-$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 $d$.
