发表机构
University of Kansas; MIT（堪萨斯大学; 麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了射影簇上固定亏格曲线态射空间的紧支撑贝蒂数的一致指数界，并将其应用于建立分裂四次德尔佩佐曲面马宁猜想的高亏格函数域版本。
AI 中文摘要
我们证明了从固定亏格的曲线到射影簇的态射空间的紧支撑贝蒂数的一致指数界。对于由固定数量、固定次数的方程定义在固定射影空间中的目标，该界关于态射次数呈指数增长，且与基域、源曲线及目标无关。证明构造了仅含线性多个变量和方程的有界次数仿射表示，随后应用Katz的估计。作为应用，我们对分裂四次德尔佩佐曲面建立了马宁猜想的高亏格函数域版本，推广了Das–Lehmann–Tanimoto–Tosteson近期的定理。在足够大的有限域上，且将曲线类限制在略微收缩的内法锥后，我们得到了带有预期首项常数的渐近预测。与Das–Lehmann–Tanimoto–Tosteson的论证类似，我们将一致贝蒂界与高亏格同伦筛、bar复形计算及虚拟高度zeta函数相结合。
英文摘要
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 $\mathbb{P}^1$ 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.
Comments47 pages, v2: improved exposition
