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A higher genus circle method and an application to geometric Manin's conjecture

Matthew Hase-Liu

TL;DR

The paper develops a geometric version of the Hardy–Littlewood circle method to study the moduli space of maps from a fixed genus $g$ curve to a smooth low-degree hypersurface $X$ in projective space. By viewing the circle as linear functionals on global sections $H^{0}(C,L^{\otimes d})$ and employing the Beauville–Laszlo descent together with Harder–Narasimhan theory, the authors prove a higher-genus Davenport shrinking lemma and obtain irreducibility and the expected dimension for ${\rm Mor}_{e}(C,X)$ under explicit numerical bounds. They spread the problem to finite characteristic and use Lang–Weil to compare point counts, while Weyl differencing and a geometric geometry-of-numbers framework control minor-arc contributions. These results yield a corollary on Fujita invariants, showing $a(V,-K_{X}|_{V})<1$ for all proper subvarieties, which supports geometric Manin-type conclusions and implies that all components of ${\rm Mor}(C,X)$ have the expected dimension. Overall, the work blends arithmetic circle-method techniques with vector-bundle geometry to advance the understanding of the distribution of curves on varieties and their birational geometry.

Abstract

Browning and Vishe used the Hardy-Littlewood circle method to show the moduli space of rational curves on smooth hypersurfaces of low degree is irreducible and of the expected dimension. We reinterpret the circle method geometrically and prove a generalization for higher genus smooth projective curves. In particular, we explain how the geometry of numbers can be understood via the Beauville-Laszlo theorem in terms of vector bundles on curves and their slopes, allowing us to prove a higher genus variant of Davenport's shrinking lemma. As a corollary, we apply this result to show the Fujita invariant of any proper subvariety of a smooth hypersurface of low degree is less than 1.

A higher genus circle method and an application to geometric Manin's conjecture

TL;DR

The paper develops a geometric version of the Hardy–Littlewood circle method to study the moduli space of maps from a fixed genus curve to a smooth low-degree hypersurface in projective space. By viewing the circle as linear functionals on global sections and employing the Beauville–Laszlo descent together with Harder–Narasimhan theory, the authors prove a higher-genus Davenport shrinking lemma and obtain irreducibility and the expected dimension for under explicit numerical bounds. They spread the problem to finite characteristic and use Lang–Weil to compare point counts, while Weyl differencing and a geometric geometry-of-numbers framework control minor-arc contributions. These results yield a corollary on Fujita invariants, showing for all proper subvarieties, which supports geometric Manin-type conclusions and implies that all components of have the expected dimension. Overall, the work blends arithmetic circle-method techniques with vector-bundle geometry to advance the understanding of the distribution of curves on varieties and their birational geometry.

Abstract

Browning and Vishe used the Hardy-Littlewood circle method to show the moduli space of rational curves on smooth hypersurfaces of low degree is irreducible and of the expected dimension. We reinterpret the circle method geometrically and prove a generalization for higher genus smooth projective curves. In particular, we explain how the geometry of numbers can be understood via the Beauville-Laszlo theorem in terms of vector bundles on curves and their slopes, allowing us to prove a higher genus variant of Davenport's shrinking lemma. As a corollary, we apply this result to show the Fujita invariant of any proper subvariety of a smooth hypersurface of low degree is less than 1.
Paper Structure (16 sections, 28 theorems, 196 equations)

This paper contains 16 sections, 28 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. A lower bound on $\dim\mathop{\mathrm{Mor}}\nolimits_{e}\left(C,X\right)$
  3. Spreading out and the Lang-Weil bound
  4. Setting up the circle method
  5. Weyl differencing
  6. Why is the circle linear functionals on $P_{de,C}$?
  7. The geometry of numbers via vector bundles
  8. Bounding $N_{s}\left(\alpha\right)$
  9. Putting everything together
  10. $d\ge4,g=1,e\ge2^{d}\left(d-1\right)^{2}+1,$ and $n\ge2^{d}\left(d-1\right)+1$
  11. $d\ge4,g\ge2,e\ge2^{d-1}\left(d-1\right)^{2}\left(3g+1\right)+d\left\lceil \frac{3g-1}{2}\right\rceil +\left\lfloor \frac{g-1}{2}\right\rfloor ,$ and $n\ge2^{d}\left(d-1\right)+1$
  12. $d=3,g=1,e\ge45,$ and $n\ge17$
  13. $d=3,g\ge2,e\ge170g/3+32/3,$ and $n\ge17$
  14. $d=2,g=1,e\ge14,$ and $n\ge5$
  15. $d=2,g\ge2,e\ge35g/2-15/2,$ and $n\ge5$
  16. ...and 1 more sections

Key Result

Theorem 1

Let $C$ be a smooth projective curve of genus $g\ge1$ and $X\subset\mathbb{P}^{n}$ a smooth hypersurface of degree $d$, both defined over $\mathbb{C}$. Suppose the following holds: Then, the mapping space $\mathop{\mathrm{Mor}}\nolimits_{e}\left(C,X\right)$ is irreducible and has the expected dimension $(n+1)(e+1-g)-de-2+2g$.

Theorems & Definitions (59)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Definition 4
  • Corollary 5
  • Remark 6
  • Example 7
  • Lemma 8
  • proof
  • proof
  • ...and 49 more