Connecting families of curves
Abstract
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.
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BibTeX
@misc{chen2026connecting,
title = {{Connecting families of curves}},
author = {Nathan Chen and Robert Lazarsfeld and Federico Moretti},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20716},
eprint = {2609.20716},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20716},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Connecting families of curves
AU - Nathan Chen
AU - Robert Lazarsfeld
AU - Federico Moretti
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20716
UR - https://arxiv.org/abs/2609.20716
AB - 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.
ER -