Normal Bundle Splitting Strata of Rational Curves in Toric Varieties
Abstract
We study stratification by normal-bundle splitting type in families of unramified rational curves of fixed class in a smooth projective toric variety over an algebraically closed field {of arbitrary characteristic}. Using the Cox construction, we construct an explicit two-term resolution of the relative normal bundle by direct sums of line bundles on the universal curve over the fixed-source parameter space. Then we construct morphisms of vector bundles on the parameter space, which we call relative cohomology morphisms, from which we can investigate generic splitting type and jumping loci of the family. In particular, we obtain sufficient numerical criteria for unbalancedness of the normal bundle. We apply this method to blowups of projective space along linear subspaces, and show that the numerical criteria for unbalancedness are equivalent to those of Cela--Lian \cite{CelaLian2026}. We also obtain explicit formulas for the expected Chow classes of the jump loci.
Related
- Constructing Rational Curves via Jets on Projective Varieties with Non-Nef Canonical Bundle
- Quadratic hulls of osculating varieties of rational normal curves
- Stable Trace Formula for Newton strata of Shimura varieties
- Ekedahl-Oort strata with a superspecial factor in unitary Shimura varieties
- On the existence of smooth varieties of any dimension carrying no Ulrich bundles for some very ample line bundle
BibTeX
@misc{iwasaki2026normal,
title = {{Normal Bundle Splitting Strata of Rational Curves in Toric Varieties}},
author = {Hikari Iwasaki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39309},
eprint = {2609.39309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39309},
}
RIS
TY - UNPB
TI - Normal Bundle Splitting Strata of Rational Curves in Toric Varieties
AU - Hikari Iwasaki
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39309
UR - https://arxiv.org/abs/2609.39309
AB - We study stratification by normal-bundle splitting type in families of unramified rational curves of fixed class in a smooth projective toric variety over an algebraically closed field {of arbitrary characteristic}. Using the Cox construction, we construct an explicit two-term resolution of the relative normal bundle by direct sums of line bundles on the universal curve over the fixed-source parameter space. Then we construct morphisms of vector bundles on the parameter space, which we call relative cohomology morphisms, from which we can investigate generic splitting type and jumping loci of the family. In particular, we obtain sufficient numerical criteria for unbalancedness of the normal bundle. We apply this method to blowups of projective space along linear subspaces, and show that the numerical criteria for unbalancedness are equivalent to those of Cela--Lian \cite{CelaLian2026}. We also obtain explicit formulas for the expected Chow classes of the jump loci.
ER -