A converse to geometric Manin's conjecture for general low degree hypersurfaces and Poincar\'e duality

TL;DR

This paper proves a converse to geometric Manin's conjecture for certain low-degree hypersurfaces, showing the absence of pathological accumulating maps and establishing a form of Poincaré duality in this context.

Contribution

It introduces a novel approach using positive characteristic and a higher genus circle method to confirm the absence of accumulating maps for specific hypersurfaces.

Findings

No accumulating maps for hypersurfaces with degree d ≤ n/4 + 3/2.

Establishment of a Poincaré duality version for moduli spaces of curves.

Application of positive characteristic techniques to geometric conjectures.

Abstract

Geometric Manin's conjecture predicts that components of the moduli space of curves on a Fano variety parametrizing non-free curves are pathological and arise from "accumulating" morphisms that increase the Fujita invariant. By passing to positive characteristic and employing a higher genus generalization of the circle method, we prove a converse to this conjecture for general hypersurfaces $X$ in $\mathbb{P}^{n}$ of degree $d\le n/4+3/2$, namely that there are no such accumulating maps to $X$. One consequence of this is a version of Poincar\'e duality for these moduli spaces in a range.