Geometry of m-direct-sum morphisms
Abstract
Let V be a complex vector space of dimension mk (m≥2). A morphism from a positive-dimensional projective variety f: X→Gr(k,V) is m-direct-sum if V=Ux1⊕⋯⊕ Uxm for every m pairwise distinct points x1,…,xm on X, where Ux⊂ V is the subspace represented by f(x). We show that X is necessarily a smooth rational curve. When the so-called drill d(f) of f is $1$ or when the rank k is at most $2$, we give complete classifications of f up to the automorphisms of Gr(k,V). As applications, we classify all so-called skew scrolls in ℙ3, which completely answers a question of Day. Then we show that if X⊂ ℙ(V) is a smooth, projective, nondegenerate subvariety whose higher-order Terracini locus is empty, then X is the rational normal curve, which strengthens the critical Terracini rigidity discovered by Ballico and Chiantini. By using the same machinery, we also prove a dual version of Terracini rigidity. Finally, we show that m-unisolvent systems form an abelian category of finite length and that $7$ is the least diagonal exponent of a simple rank-three $2$-unisolvent system.
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BibTeX
@misc{li2026geometry,
title = {{Geometry of $m$-direct-sum morphisms}},
author = {Duo Li and Yanjie Li and Francesco Russo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38184},
eprint = {2610.38184},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38184},
}
RIS
TY - UNPB
TI - Geometry of m-direct-sum morphisms
AU - Duo Li
AU - Yanjie Li
AU - Francesco Russo
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38184
UR - https://arxiv.org/abs/2610.38184
AB - Let V be a complex vector space of dimension mk (m≥2). A morphism from a positive-dimensional projective variety f: X→Gr(k,V) is m-direct-sum if V=U_x_1⊕⋯⊕ U_x_m for every m pairwise distinct points x_1,…,x_m on X, where U_x⊂ V is the subspace represented by f(x). We show that X is necessarily a smooth rational curve. When the so-called drill d(f) of f is $1$ or when the rank k is at most $2$, we give complete classifications of f up to the automorphisms of Gr(k,V). As applications, we classify all so-called skew scrolls in ℙ^3, which completely answers a question of Day. Then we show that if X⊂ ℙ(V) is a smooth, projective, nondegenerate subvariety whose higher-order Terracini locus is empty, then X is the rational normal curve, which strengthens the critical Terracini rigidity discovered by Ballico and Chiantini. By using the same machinery, we also prove a dual version of Terracini rigidity. Finally, we show that m-unisolvent systems form an abelian category of finite length and that $7$ is the least diagonal exponent of a simple rank-three $2$-unisolvent system.
ER -