Betti bounds for spaces of curves on varieties and Manin's conjecture for quartic del Pezzo surfaces
Abstract
We prove uniform exponential bounds for the compactly supported Betti numbers of spaces of morphisms from curves of fixed genus to projective varieties. For targets in a fixed projective space cut out by a prescribed number of equations of fixed degrees, the bound is exponential in the degree of the morphism and is independent of the ground field, the source curve, and the target. The proof constructs bounded-degree affine presentations involving only linearly many variables and equations, and then applies Katz's estimate. As an application, we establish a higher genus function field version of Manin's conjecture for split quartic del Pezzo surfaces, generalizing a recent theorem of Das--Lehmann--Tanimoto--Tosteson. The passage from ℙ1 to arbitrary source curves requires uniform bounds for compactly supported Betti numbers over all curves of fixed genus, together with new geometric results on higher genus morphism spaces, including irreducibility and dimension estimates for the relevant incidence strata. These results, combined with the necessary control of the configuration space contributions, allow the homological sieve and virtual height zeta function argument to yield, over sufficiently large finite fields and within a slightly shrunken nef cone, the predicted asymptotic with the expected leading constant.
Related
- Arithmetic jet spaces and the unramified Zilber--Pink conjecture for subvarieties of abelian varieties
- The irreducibility of Hurwitz spaces and Severi varieties on toric surfaces
- Du Bois cohomology of noncommutative spaces and the Hodge numbers invariance conjecture for singular varieties
- Universal torsors over quartic del Pezzo surfaces and stable rationality
- Quasi-Galois points for quartic surfaces
BibTeX
@misc{feng2026betti,
title = {{Betti bounds for spaces of curves on varieties and Manin's conjecture for quartic del Pezzo surfaces}},
author = {Enhao Feng and Matthew Hase-Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.28465},
eprint = {2608.28465},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28465},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Betti bounds for spaces of curves on varieties and Manin's conjecture for quartic del Pezzo surfaces
AU - Enhao Feng
AU - Matthew Hase-Liu
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2608.28465
UR - https://arxiv.org/abs/2608.28465
AB - We prove uniform exponential bounds for the compactly supported Betti numbers of spaces of morphisms from curves of fixed genus to projective varieties. For targets in a fixed projective space cut out by a prescribed number of equations of fixed degrees, the bound is exponential in the degree of the morphism and is independent of the ground field, the source curve, and the target. The proof constructs bounded-degree affine presentations involving only linearly many variables and equations, and then applies Katz's estimate. As an application, we establish a higher genus function field version of Manin's conjecture for split quartic del Pezzo surfaces, generalizing a recent theorem of Das--Lehmann--Tanimoto--Tosteson. The passage from ℙ^1 to arbitrary source curves requires uniform bounds for compactly supported Betti numbers over all curves of fixed genus, together with new geometric results on higher genus morphism spaces, including irreducibility and dimension estimates for the relevant incidence strata. These results, combined with the necessary control of the configuration space contributions, allow the homological sieve and virtual height zeta function argument to yield, over sufficiently large finite fields and within a slightly shrunken nef cone, the predicted asymptotic with the expected leading constant.
ER -