arXiv 1606.042812016-06-14math.AG

水平- $δ$极限线性级数

Level-$δ$ limit linear series

Eduardo Esteves, Antonio Nigro, Pedro Rizzo

摘要：

我们引入了水平- $δ$极限线性级数的概念，它描述了沿着退化为奇异曲线$X$的光滑曲线族的线性级数的极限。我们在这里只处理最简单的情况，其中$X$是两个光滑分量在$P$点横向相交的并集。整数$δ$代表在$P$处退化的总空间的奇异度。如果全空间是正则的，我们得到一级极限线性级数，这正是Osserman在2006年提出的。本文构造了一个射影模空间$G^r_{d,δ}(X)$，它参数化了$X$上秩$r$和度$d$的水平- $δ$极限线性级数，并证明了对于每个$δ$，它是Osserman精确极限线性级数的模空间的一个新的紧化，Osserman精确极限线性级数是Osserman已构造的空间$G^r_{d,1}(X)$的一个开子模式$G^{r,*}_{d,1}(X)$.最后，我们推广了Esteves和Osserman的工作相关联的每个确切的水平- $δ$限制线性系列$X$ $\mathfrak g$上的一个封闭的子计划$\mathbb P(\mathfrak g)\subseteq X^{(d)}$的$d$次对称产品的$X$，并表明，$\mathbb P(\mathfrak g)$是有限的因子空间的线性系列光滑曲线退化到$X$上的$\mathfrak g$，如果这种退化存在。特别是，我们完全描述了沿退化到这样的曲线$X$的因子的限制。

英文摘要：

We introduce the notion of level-$δ$ limit linear series, which describe limits of linear series along families of smooth curves degenerating to a singular curve $X$. We treat here only the simplest case where $X$ is the union of two smooth components meeting transversely at a point $P$. The integer $δ$ stands for the singularity degree of the total space of the degeneration at $P$. If the total space is regular, we get level-1 limit linear series, which are precisely those introduced by Osserman in 2006. We construct a projective moduli space $G^r_{d,δ}(X)$ parameterizing level-$δ$ limit linear series of rank $r$ and degree $d$ on $X$, and show that it is a new compactification, for each $δ$, of the moduli space of Osserman exact limit linear series, an open subscheme $G^{r,*}_{d,1}(X)$ of the space $G^r_{d,1}(X)$ already constructed by Osserman. Finally, we generalize work by Esteves and Osserman by associating to each exact level-$δ$ limit linear series $\mathfrak g$ on $X$ a closed subscheme $\mathbb P(\mathfrak g)\subseteq X^{(d)}$ of the $d$th symmetric product of $X$, and showing that $\mathbb P(\mathfrak g)$ is the limit of the spaces of divisors associated to linear series on smooth curves degenerating to $\mathfrak g$ on $X$, if such degenerations exist. In particular, we describe completely limits of divisors along degenerations to such a curve $X$.

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