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

Matthew Hase-Liu

TL;DR

This work addresses the existence of accumulating maps to Fano hypersurfaces in projective space, reframing the problem through the Fujita invariant and geometric Manin's conjecture. It proves a converse result in the regime $d\ge5$ and $n\ge4d-6$ for general hypersurfaces: there are no accumulating maps $f: Y\to X$ with $a(Y,-f^{*}K_X)\ge1$, by combining a higher-genus circle-method analysis in positive characteristic with a reduction from general to very general hypersurfaces via the BAB conjecture. The strategy yields an expected-dimension control for moduli spaces of high-genus curves mapping to hypersurfaces, and demonstrates that all proper subvarieties have $a(V,-K_X|_V)<1$, thereby ruling out accumulation phenomena and transferring the conclusion back to characteristic zero. The results improve known bounds relating the ambient dimension $n$ to the degree $d$ and illuminate the interplay between arithmetic circle-method techniques and modern birational geometry in controlling moduli spaces of curves on Fano varieties.

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 $X$.

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

TL;DR

This work addresses the existence of accumulating maps to Fano hypersurfaces in projective space, reframing the problem through the Fujita invariant and geometric Manin's conjecture. It proves a converse result in the regime and for general hypersurfaces: there are no accumulating maps with , by combining a higher-genus circle-method analysis in positive characteristic with a reduction from general to very general hypersurfaces via the BAB conjecture. The strategy yields an expected-dimension control for moduli spaces of high-genus curves mapping to hypersurfaces, and demonstrates that all proper subvarieties have , thereby ruling out accumulation phenomena and transferring the conclusion back to characteristic zero. The results improve known bounds relating the ambient dimension to the degree and illuminate the interplay between arithmetic circle-method techniques and modern birational geometry in controlling moduli spaces of curves on Fano varieties.

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 in of degree , namely that there are no such accumulating maps to .
Paper Structure (4 sections, 7 theorems, 42 equations)

This paper contains 4 sections, 7 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Reduction from general hypersurfaces to very general hypersurfaces
  3. Dimension of moduli spaces of curves on very general hypersurfaces
  4. From geometric Manin's conjecture in positive characteristic to Fujita invariants

Key Result

Theorem 3

Let $X\subset\mathbb{P}_{\mathbb{C}}^{n}$ be a general hypersurface of degree $d\ge5$ with $n\ge4d-6$. Then, there are no accumulating maps to $X$.

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Remark 4
  • Remark 5
  • Proposition 6
  • Corollary 7
  • Proposition 8
  • proof : Proof of Theorem \ref{['thm:smallfujitageneral']}
  • Remark 9
  • ...and 10 more