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. Over sufficiently large finite fields, and after restricting curve classes to a slightly shrunken nef cone, we obtain the predicted asymptotic with the expected leading constant. As in Das--Lehmann--Tanimoto--Tosteson's argument, we combine the uniform Betti bound with a higher genus homological sieve, a bar complex calculation, and a virtual height zeta function.