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The desingularization of the theta divisor of a cubic threefold as a moduli space

Arend Bayer, Sjoerd Beentjes, Soheyla Feyzbakhsh, Georg Hein, Diletta Martinelli, Fatemeh Rezaee, Benjamin Schmidt

Abstract

We show that the moduli space $\overline{M}_X(v)$ of Gieseker stable sheaves on a smooth cubic threefold $X$ with Chern character $v = (3,-H,-H^2/2,H^3/6)$ is smooth and of dimension four. Moreover, the Abel-Jacobi map to the intermediate Jacobian of $X$ maps it birationally onto the theta divisor $Θ$, contracting only a copy of $X \subset \overline{M}_X(v)$ to the singular point $0 \in Θ$. We use this result to give a new proof of a categorical version of the Torelli theorem for cubic threefolds, which says that $X$ can be recovered from its Kuznetsov component $\operatorname{Ku}(X) \subset \mathrm{D}^{\mathrm{b}}(X)$. Similarly, this leads to a new proof of the description of the singularity of the theta divisor, and thus of the classical Torelli theorem for cubic threefolds, i.e., that $X$ can be recovered from its intermediate Jacobian.

The desingularization of the theta divisor of a cubic threefold as a moduli space

Abstract

We show that the moduli space of Gieseker stable sheaves on a smooth cubic threefold with Chern character is smooth and of dimension four. Moreover, the Abel-Jacobi map to the intermediate Jacobian of maps it birationally onto the theta divisor , contracting only a copy of to the singular point . We use this result to give a new proof of a categorical version of the Torelli theorem for cubic threefolds, which says that can be recovered from its Kuznetsov component . Similarly, this leads to a new proof of the description of the singularity of the theta divisor, and thus of the classical Torelli theorem for cubic threefolds, i.e., that can be recovered from its intermediate Jacobian.

Paper Structure

This paper contains 10 sections, 50 theorems, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Cubic threefolds and intermediate Jacobians
  3. Divisors on hyperplane sections
  4. Notions of stability
  5. Construction of sheaves
  6. Variation of stability
  7. Classification of some torsion sheaves
  8. Set-theoretic description of the moduli space
  9. Proof of the main theorem
  10. Kuznetsov component

Key Result

Theorem 1.1

Let $\sigma$ be an arbitrary Serre-invariant stability condition on $\mathop{\mathrm{Ku}}\nolimits(X)$. Then the moduli space $M_{\sigma}(v)$ is isomorphic to the moduli space $\overline{M}_X(v)$.

Figures (2)

  • Figure 1: Walls are nested semicircles or a unique vertical wall (Theorem \ref{['thm:nested_wall_thm']} (ii))
  • Figure 2: Walls are nested semicircles (Theorem \ref{['thm:nested_wall_thm']} (iii))

Theorems & Definitions (88)

  • Theorem 1.1: Theorem \ref{['thm:equalmodspace']} and Proposition \ref{['prop:unique']}
  • Theorem 1.2: Corollary \ref{['cor:classicalTorelli']} and Theorem \ref{['thm:categorical-Torelli']}
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 78 more