发表机构
Instituto de Matemática Pura e Aplicada (IMPA); Stony Brook University; University of Pennsylvania（巴西应用数学研究所; 纽约州立大学石溪分校; 宾夕法尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究光滑射影簇上连接多个一般点的曲线族，在 Kodaira 维数非负时给出亏格线性下界，并刻画连接族 gonality 的极限值及高次超曲面的渐近行为。
AI 中文摘要
Kollár、Miyaoka 和 Mori 的一个定理指出，在有理连通簇上，任何有限点集都位于一条有理曲线上。受此启发，自然要问：对于任意光滑射影簇 X 上经过多个一般点的曲线族，我们能得出什么结论？当 X 的 Kodaira 维数非负时，我们建立了此类曲线亏格的尖锐线性下界，该下界依赖于点的数量和 X 的维数，推广了 Arapura 和 Archava 的一个定理。相比之下，连接族的最小可能的 gonality 随着点数的增长最终趋于稳定。我们用支配 X 且是有理连通簇的有限覆盖的簇来刻画其极限值。作为示例，我们研究了高次超曲面的这些不变量，特别确定了最小连接亏格随点数和次数变化的联合渐近行为。最后，我们简要考虑高维连接子簇，证明了其典范体积的线性界，并计算了超曲面的渐近结果。
英文摘要
A theorem of Kollár, Miyaoka, and Mori states that on a rationally connected variety, any finite collection of points lies on a rational curve. Motivated by this, it is natural to ask what one can say about families of curves passing through many general points of an arbitrary smooth projective variety X. When X has nonnegative Kodaira dimension, we establish a sharp linear lower bound for the genus of such curves in terms of the number of points and the dimension of X, generalizing a theorem of Arapura and Archava. By contrast, the least possible gonality of a connecting family eventually stabilizes as the number of points grows. We characterize its limiting value in terms of varieties dominating X that are generically finite covers of rationally connected varieties. As an illustration, we study these invariants for hypersurfaces of large degree, determining in particular the joint asymptotic behavior of the minimal connecting genus as the number of points and the degree vary. Finally, we briefly consider higher-dimensional connecting subvarieties, proving linear bounds for their canonical volumes and computing asymptotic results for hypersurfaces.
