2021/11/22 by María Inés de Frutos Fernández, Fernández, María Inés de Frutos, Sumita Garai +7
Computer Science · Mathematics · #Coding theory and cryptography #Limits and Structures in Graph Theory #Meromorphic and Entire Functions #math.AG
paper · pdf · doi:10.48550/arxiv.2111.10976
openalex publication_date 2021/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X ⊂ ℙn be a hypersurface of degree d defined over a finite field of characteristic p > 0. We prove that if n ≥ r + \binomd+rr+1, then X contains a rational r-plane. We prove better bounds when X is smooth and p is sufficiently large. We also present experimental data regarding the existence of rational lines on cubic threefolds over \mathbbF7, \mathbbF8, and \mathbbF9. In particular, we construct an example of a smooth cubic threefold over \mathbbF7 with exactly 8 rational lines. It remains an open question whether smooth cubic threefolds over \mathbbF7, \mathbbF8, and \mathbbF9 always contain a rational line.