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Moduli spaces of rational curves on Artin-Mumford double solids

Fumiya Okamura

TL;DR

This work analyzes rational curves on general Artin-Mumford double solids, using their structure as del Pezzo-2 Fanos and the associated Brauer obstruction to show Geometric Manin's Conjecture with two Manin components per degree. It identifies two line components and four higher-degree components in Mor(P^1,X,d) (d≥2), with the latter splitting into double-cover and embedded-curve families, and proves a strong movable bend-and-break behavior that connects components via free curves. Central technical inputs include the conic-bundle geometry of degree-2 del Pezzo surfaces and the relationship with Reyé congruences on Enriques surfaces. Consequently, the AM solids provide the first known Fano example with multiple Manin components in the moduli space of rational curves for each degree, with Br_nr(K(X)/k) of order 2 governing the count of Manin components.

Abstract

We describe the irreducible components of the moduli spaces of rational curves on Artin-Mumford double solids. This provides the first example of Fano varieties that satisfy Geometric Manin's Conjecture with multiple Manin components in moduli space of rational curves for each degree.

Moduli spaces of rational curves on Artin-Mumford double solids

TL;DR

This work analyzes rational curves on general Artin-Mumford double solids, using their structure as del Pezzo-2 Fanos and the associated Brauer obstruction to show Geometric Manin's Conjecture with two Manin components per degree. It identifies two line components and four higher-degree components in Mor(P^1,X,d) (d≥2), with the latter splitting into double-cover and embedded-curve families, and proves a strong movable bend-and-break behavior that connects components via free curves. Central technical inputs include the conic-bundle geometry of degree-2 del Pezzo surfaces and the relationship with Reyé congruences on Enriques surfaces. Consequently, the AM solids provide the first known Fano example with multiple Manin components in the moduli space of rational curves for each degree, with Br_nr(K(X)/k) of order 2 governing the count of Manin components.

Abstract

We describe the irreducible components of the moduli spaces of rational curves on Artin-Mumford double solids. This provides the first example of Fano varieties that satisfy Geometric Manin's Conjecture with multiple Manin components in moduli space of rational curves for each degree.
Paper Structure (10 sections, 22 theorems, 29 equations)

This paper contains 10 sections, 22 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric Manin's Conjecture
  4. Conic bundles on degree $2$ del Pezzo surfaces
  5. Artin-Mumford double solids
  6. Construction of AM double solids
  7. Algebraic equivalence and numerical equivalence
  8. Rational curves on AM double solids
  9. Lines on AM double solids and Reye congruences
  10. Higher degree rational curves on AM double solids

Key Result

Theorem 1.1

Let $X$ be a general Artin-Mumford double solid and let $H$ be the ample generator of $\mathrm{Pic}(X)\cong \mathbb{Z}$. The same statements hold for the Kontsevich spaces $\overline{M}_{0,0}(X,d)$ parametrizing genus $0$ stable maps of $H$-degree $d\ge 1$, but the expected dimension of each component is less than by $3$.

Theorems & Definitions (47)

  • Theorem 1.1: = Theorem \ref{['thm: AM space of lines']} + Theorem \ref{['thm: AM spaces of higher deg curves']}
  • Theorem 1.2: = Theorem \ref{['thm: AM strong MBB']}
  • Corollary 1.3: = Corollary \ref{['cor: AM GMC']}
  • Definition 2.1
  • Lemma 2.2: Kollar1996, Theorem 1.3; LT2024dPfibI, Lemma 2.2
  • Lemma 2.3: cf. LT2024dPfibI, Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6: Okamura2024Gt, Proposition 3.1; cf. LT2019Compos, Proposition 4.2
  • ...and 37 more