PreprintarXiv licencearXiv
Finding large families of rational curves through Bend-and-Break
Abstract
We present a new construction that allows us to break off large-degree rational curves from families of higher genus curves. Our construction and results deepen the connection between rational curves and positivity of the anticanonical divisor. Specifically, we show that varieties with large Fujita invariant admit large families of rational curves. We also construct free rational curves on certain singular Fano varieties. As an explicit consequence of our results, we prove that for a general Fano hypersurface of index at least 3, all spaces of genus g curves of sufficiently large degree have the expected dimension.
Related
- Torsion points in families of elliptic curves
- Transverse rational curves and the LeBrun--Salamon conjecture
- Rational Points on Drinfeld Modular Curves of Prime Level
- On Free and Nearly Free Conjecture of Rational Unibranched Projective Plane Curves
- Hilbert schemes of low-degree rational curves on a prime Fano threefold of degree $22$
BibTeX
@misc{jovinelly2026finding,
title = {{Finding large families of rational curves through Bend-and-Break}},
author = {Eric Jovinelly and Brian Lehmann and Eric Riedl},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27696},
eprint = {2608.27696},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27696},
}
RIS
TY - UNPB
TI - Finding large families of rational curves through Bend-and-Break
AU - Eric Jovinelly
AU - Brian Lehmann
AU - Eric Riedl
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27696
UR - https://arxiv.org/abs/2608.27696
AB - We present a new construction that allows us to break off large-degree rational curves from families of higher genus curves. Our construction and results deepen the connection between rational curves and positivity of the anticanonical divisor. Specifically, we show that varieties with large Fujita invariant admit large families of rational curves. We also construct free rational curves on certain singular Fano varieties. As an explicit consequence of our results, we prove that for a general Fano hypersurface of index at least 3, all spaces of genus g curves of sufficiently large degree have the expected dimension.
ER -