A converse to geometric Manin's conjecture for general low degree hypersurfaces and Poincaré duality

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 Poincaré duality for these moduli spaces in a range.

💡 Research Summary

The paper addresses a converse to the geometric Manin conjecture for low‑degree hypersurfaces in projective space. The conjecture, originally formulated by Taniguchi, predicts that components of the moduli space of curves on a Fano variety that parametrize non‑free curves are “pathological” and arise from accumulating morphisms—maps f : Y → X for which the Fujita invariant satisfies a(Y, −f* K_X) ≥ 1. The author proves that for a general (or Fermat) hypersurface X ⊂ ℙⁿ of degree d with d ≥ 5 and n ≥ 4d − 6 (equivalently d ≤ n/4 + 3/2), no such accumulating maps exist.

The proof proceeds in two major steps. First, the author works in positive characteristic and adapts a higher‑genus version of the circle method, originally developed by Browning, Vishe, and Sawin, to count tuples of sections satisfying the hypersurface equation over finite fields. By imposing congruence conditions and using Katz’s bounds on exponential sums, the author obtains an expected‑dimension result for the incidence variety of maps from a fixed curve C to a very general hypersurface. This yields a linear bound n ≥ 4d − 6, improving on earlier exponential bounds. Consequently, any proper subvariety V ⊂ X of a very general hypersurface satisfies a(V, −K_X|_V) < 1.

Second, the author lifts this statement from “very general” to “general” hypersurfaces using the resolution of the Borisov–Alexeev–Borisov (BAB) conjecture. By the BAB theorem, adjoint‑rigid subvarieties with Fujita invariant at least 1 have bounded anticanonical degree, which forces them to lie in a proper closed subset of the parameter space of hypersurfaces. Hence a general hypersurface contains no such subvarieties. Extending from subvarieties to maps, any accumulating map would either have a branch divisor with Fujita invariant > 1 (impossible by the subvariety result) or would give a non‑trivial étale cover of X, contradicting the simple‑connectedness of smooth hypersurfaces.

With the non‑existence of accumulating maps established, the paper derives consequences for the moduli space Mor_e(C, X) of degree‑e maps from a fixed smooth projective curve C of genus g. Using deformation theory, the author shows that the locus of non‑free maps has codimension at least T_e for some T_e > 0, depending linearly on e. As a result, for any prime ℓ different from the characteristic, the ℓ‑adic cohomology of Mor_e(C, X) satisfies Poincaré duality in the range i < T_e:
\