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Rational curves on Fano threefolds with Gorenstein terminal singularities

Fumiya Okamura

TL;DR

This work extends the study of rational curves to singular Fano geometry by focusing on $ ext{Q-factorial}$ Gorenstein terminal Fano threefolds. It develops a unified framework using $a$- and $b$-invariants to detect pathological components in spaces of maps, classifies subvarieties with elevated $a$-invariants, and proves Movable Bend-and-Break (MBB) in the terminal factorial setting. The authors provide partial classifications of $a$-covers by analyzing the Iitaka dimension $ ext{κ}ig(Y, K_Y-f^{*}K_Xig) ext{ in }ig\{0,1,2igig}$, and leverage MBB to advance Geometric Manin's Conjecture (GMC). Finally, they verify GMC for singular cubic and quartic del Pezzo threefolds with terminal factorial singularities, describing a two-component structure for low-degree curve spaces and obtaining precise asymptotics for the counts of Manin components. The results bridge the smooth and singular theories, offering a robust pathway to understanding rational curves on singular Fano threefolds and their moduli spaces.

Abstract

We study the spaces of rational curves on Fano threefolds with Gorenstein terminal singularities. We generalize the results regarding Geometric Manin's Conjecture for smooth Fano threefolds, including the classification of subvarieties with higher a-invariants and Movable Bend-and-Break lemma. We also show Geometric Manin's Conjecture for some singular del Pezzo threefolds.

Rational curves on Fano threefolds with Gorenstein terminal singularities

TL;DR

This work extends the study of rational curves to singular Fano geometry by focusing on Gorenstein terminal Fano threefolds. It develops a unified framework using - and -invariants to detect pathological components in spaces of maps, classifies subvarieties with elevated -invariants, and proves Movable Bend-and-Break (MBB) in the terminal factorial setting. The authors provide partial classifications of -covers by analyzing the Iitaka dimension , and leverage MBB to advance Geometric Manin's Conjecture (GMC). Finally, they verify GMC for singular cubic and quartic del Pezzo threefolds with terminal factorial singularities, describing a two-component structure for low-degree curve spaces and obtaining precise asymptotics for the counts of Manin components. The results bridge the smooth and singular theories, offering a robust pathway to understanding rational curves on singular Fano threefolds and their moduli spaces.

Abstract

We study the spaces of rational curves on Fano threefolds with Gorenstein terminal singularities. We generalize the results regarding Geometric Manin's Conjecture for smooth Fano threefolds, including the classification of subvarieties with higher a-invariants and Movable Bend-and-Break lemma. We also show Geometric Manin's Conjecture for some singular del Pezzo threefolds.
Paper Structure (16 sections, 25 theorems, 29 equations)

This paper contains 16 sections, 25 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Free rational curves
  4. Gorenstein terminal threefolds
  5. $a$, $b$-invariants
  6. Subvarieties with higher $a$-invariants
  7. $a$-covers
  8. Iitaka dimension $2$ case
  9. Iitaka dimension $1$ case
  10. Iitaka dimension $0$ case
  11. Movable Bend-and-Break
  12. MBB for multiple covers
  13. MBB for birational stable maps
  14. Geometric Manin's Conjecture
  15. Statement of Geometric Manin's Conjecture
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X$ be a Gorenstein terminal Fano threefold. Let $Y \subset X$ be a $2$-dimensional subvariety of $X$ satisfying $a(Y, -K_{X}) > 1$ and let $\phi \colon \tilde{Y} \rightarrow Y$ be a smooth resolution.

Theorems & Definitions (59)

  • Theorem 1.1: §.3
  • Theorem 1.2: §.4
  • Theorem 1.3: Movable Bend-and-Break, §.5
  • Theorem 1.4: §.6
  • Definition 2.1
  • Theorem 2.2: Kollar1996*II.3.11
  • Proposition 2.3: Kollar1996*II.1.3, Lehmann2023*Lemma 2.2
  • Lemma 2.4: Lehmann2023*Lemma 2.3
  • Lemma 2.5: Kawamata1988*Lemma 5.1
  • Theorem 2.6: Cutkosky1988
  • ...and 49 more