标题： 完备射影几何中的退化与几何Tevelev度量$\mathbb{P}^r$ 显示英文标题

标题： Degenerations of complete collineations and geometric Tevelev degrees of $\mathbb{P}^r$

摘要： 我们考虑枚举从固定的一般亏格为$g$的曲线$C$到$\mathbb{P}^r$的度数为$d$的映射$f$，满足形式为$f(p_i)\in X_i$的相交条件，其中$p_i\in C$是一般点，$X_i\subset\mathbb{P}^r$是一般线性空间。 我们在点的场合给出了一个完整的答案，在这种场合下，$X_i$的计数，“$\mathbb{P}^r$的 Tevelev 度”，之前仅当$r=1$时已知，或者当$d$相对于$r,g$较大时已知，或者在 Gromov-Witten 理论中几乎已知。 我们还在任意入射条件下的$r=2$场合给出了一个完整的答案。 我们的主要方法研究了在各种退化下完全共线性的行为。 显示英文摘要

摘要： We consider the problem of enumerating maps $f$ of degree $d$ from a fixed general curve $C$ of genus $g$ to $\mathbb{P}^r$ satisfying incidence conditions of the form $f(p_i)\in X_i$, where $p_i\in C$ are general points and $X_i\subset\mathbb{P}^r$ are general linear spaces. We give a complete answer in the case where the $X_i$ are points, where the counts, the ``Tevelev degrees'' of $\mathbb{P}^r$, were previously known only when $r=1$, when $d$ is large compared to $r,g$, or virtually in Gromov-Witten theory. We also give a complete answer in the case $r=2$ with arbitrary incidence conditions. Our main approach studies the behavior of complete collineations under various degenerations.