Manin’s conjecture for rational points over global function fields

Establish that, for a Fano variety X over a finite field and its base change X_K to the function field K of a smooth projective geometrically integral curve B, whenever X_K(K) is not thin, there exists an exceptional set Z such that the counting function of morphisms f:B\to X outside Z with anticanonical degree at most rd satisfies the predicted asymptotic c(B,-K_X,X)q^{rd}(rd)^{\rho(X)-1} as d\to\infty, with c(B,-K_X,X) equal to Peyre’s constant.

Background

The paper places its arithmetic application in the broader conjectural framework for counting rational points on Fano varieties over global function fields. A K-rational point on the base change X_K corresponds, via the valuative criterion, to a morphism from the base curve B to X, so the conjectural point-counting asymptotic can be expressed in terms of the finite-field points of spaces of morphisms and their anticanonical degrees.

The stated conjecture is not proved in this generality in the paper. The authors prove a higher-genus function-field analogue for split quartic del Pezzo surfaces after restricting curve classes to a slightly shrunken nef cone, thereby establishing a special case of the broader prediction.