TY - UNPB

T1 - Linear growth and moduli spaces of rational curves

AU - Glas, Jakob

N1 - 16 pages, comments welcome!

PY - 2025/5/12

Y1 - 2025/5/12

N2 - Working in positive characteristic, we show how one can use information about the dimension of moduli spaces of rational curves on a Fano variety $X$ over $\mathbb{F}_q$ to obtain strong estimates for the number of $\mathbb{F}_q(t)$-points of bounded height on $X$. Building on work of Beheshti, Lehmann, Riedl and Tanimoto~\cite{BeheshtiLehmannRiedlTanimoto.dP}, we apply our strategy to del Pezzo surfaces of degree at most 5. In addition, we also treat the case of smooth cubic hypersurfaces and smooth intersections of two quadrics of dimension at least 3 by showing that the moduli spaces of rational curves of fixed degree are of the expected dimension. For large but fixed $q$, the bounds obtained come arbitrarily close to the linear growth predicted by the Batyrev--Manin conjecture.

AB - Working in positive characteristic, we show how one can use information about the dimension of moduli spaces of rational curves on a Fano variety $X$ over $\mathbb{F}_q$ to obtain strong estimates for the number of $\mathbb{F}_q(t)$-points of bounded height on $X$. Building on work of Beheshti, Lehmann, Riedl and Tanimoto~\cite{BeheshtiLehmannRiedlTanimoto.dP}, we apply our strategy to del Pezzo surfaces of degree at most 5. In addition, we also treat the case of smooth cubic hypersurfaces and smooth intersections of two quadrics of dimension at least 3 by showing that the moduli spaces of rational curves of fixed degree are of the expected dimension. For large but fixed $q$, the bounds obtained come arbitrarily close to the linear growth predicted by the Batyrev--Manin conjecture.

KW - math.NT

KW - math.AG

KW - 14G05 (11D45, 11P55, 14H10, 14J70)

U2 - 10.48550/arXiv.2505.07645

DO - 10.48550/arXiv.2505.07645

M3 - Preprint

BT - Linear growth and moduli spaces of rational curves

ER -