Explicit birational geometry of determinantal quartic 3-folds

Problem Statement
The paper investigates the explicit birational geometry of linear determinantal quartic threefolds in $\mathbb{P}^4$. While smooth cubic threefolds are well-understood via their intermediate Jacobians and associated plane curves, the intermediate Jacobian of a smooth quartic threefold is intricate. To circumvent this complexity, the authors restrict their study to the family $\mathcal{F}$ of linear determinantal quartics. A general member $Y \in \mathcal{F}$ is nodal (possessing 20 nodes), non-$\mathbb{Q}$-factorial, and rational. The central problem is to understand the birational geometry of these quartics and their moduli by analyzing the extrinsic geometry of a specific curve $C \subset \mathbb{P}^3$ of degree 10 and genus 11, which governs the structure of the quartic via a small resolution.

Methodology
The authors employ the Minimal Model Program (MMP) on the variety $X = \text{Bl}_C \mathbb{P}^3$, the blow-up of $\mathbb{P}^3$ along a smooth curve $C$ of degree 10 and genus 11. The strategy relies on the correspondence between the determinantal presentation of $Y$ and the geometry of $C$.

1. Classification of Curves: The authors analyze the Hilbert scheme $\mathcal{H}_{10,11}$ of curves of degree 10 and genus 11. They identify a unique irreducible component whose generic point parametrizes smooth, irreducible, arithmetically Cohen-Macaulay (ACM) curves. They further identify two distinguished divisors within this component:

   • $D_1$: The locus of semicanonical curves (satisfying $\omega_C \cong \mathcal{O}_C(2)$).
   • $D_2$: The locus of curves contained in a smooth cubic surface.
     Using the Hartshorne-Rao module and liaison theory, they prove that any smooth irreducible curve in this Hilbert scheme belongs to one of these three classes (ACM, semicanonical, or contained in a cubic).

2. MMP Execution: For each class of curves, the authors run the MMP on $X$. This involves:

   • Determining the nef, effective, and movable cones of $X$.
   • Computing the stable base locus decomposition (SBLD) of the effective cone.
   • Explicitly describing the birational models (flops, flips, divisorial contractions) induced by divisors in these chambers.
   • Utilizing the extrinsic geometry of $C$ (e.g., secant lines, containment in surfaces) to describe the exceptional loci of these maps.

3. Explicit Constructions: The paper utilizes Macaulay2 to verify specific geometric properties, such as the smoothness of ramification divisors and the generation of ideal sheaves, to ensure the general members of the families behave as predicted.

Key Contributions and Results

• Uniform Description of Birational Models: The paper establishes that the birational geometry of $X$ is governed by the extrinsic geometry of $C$. The effective cone of $X$ generally possesses four stable base locus chambers, leading to a specific sequence of birational models.

  • Case 1: General ACM Curves ($C \in \mathcal{H}^\circ_{10,11}$):

    • The anticanonical map $f: X \to Y$ contracts exactly twenty 4-secant lines of $C$ to 20 nodes, yielding a linear determinantal quartic $Y \subset \mathbb{P}^4$.
    • The variety $X$ is a Mori dream space.
    • The MMP proceeds via a flop of the 20 4-secant lines, followed by a divisorial contraction to $\mathbb{P}^3$. This forms a Sarkisov link connecting $X$ to $\mathbb{P}^3$ and $Y$.
    • The effective cone is spanned by the exceptional divisor $E$ and a rigid divisor class $40H - 11E$.

  • Case 2: Semicanonical Curves ($C \in D_1$):

    • The anticanonical map $f_0: X \to Y_0$ is a generically 2:1 morphism onto a quadric threefold $Y_0 \subset \mathbb{P}^4$.
    • The branch divisor is a degree 8 surface with 20 nodes.
    • The birational geometry involves a Galois involution (Galois flip) rather than a standard flop, reflecting the double cover structure.
    • This specialization corresponds to a rational non-$\mathbb{Q}$-factorial double quadric.

  • Case 3: Curves in a Cubic Surface ($C \in D_2$):

    • The geometry changes significantly: the anticanonical divisor is no longer nef.
    • The curve $C$ possesses two 5-secant lines (in addition to 4-secant lines).
    • The MMP involves an "inverse flip" of the two 5-secant lines. This leads to a singular terminal threefold $X^+$.
    • A subsequent divisorial contraction maps $X^+$ to a quartic $Y_0 \subset \mathbb{P}^4$ which is $\mathbb{Q}$-factorial but possesses an elliptic singularity (formed by the coalescence of ten nodes) and ten ordinary double points.

• Moduli Implications: The authors show that the family $\mathcal{F}$ of linear determinantal quartics contains rational $\mathbb{Q}$-factorial quartics (via the specialization to $D_2$) and rational non-$\mathbb{Q}$-factorial double quadrics (via specialization to $D_1$). They argue that the intermediate Jacobian of $X$ coincides with the Jacobian of $C$, implying that these specializations reflect variations in the moduli of the curve $C$ rather than just extrinsic degenerations.

Significance
The paper claims that its primary contribution is providing a detailed, explicit description of the birational geometry of determinantal quartics via the extrinsic geometry of the associated curve $C$. Specifically:

1. It generalizes the known Sarkisov link for general ACM curves to the cases where $C$ specializes to semicanonical curves or curves on a cubic surface.
2. It identifies two irreducible divisors in the moduli space of such quartics associated with the non-ACM smooth curves.
3. It demonstrates how the MMP can be used to explicitly describe the transition from a general determinantal quartic to singular specializations (double quadrics and $\mathbb{Q}$-factorial quartics with elliptic singularities).
4. The work suggests that the moduli space of these quartics can be studied through the geometry of the curve $C$, offering a novel approach to understanding the moduli of determinantal hypersurfaces, which are otherwise difficult to analyze due to the complexity of their intermediate Jacobians.

The authors note that while the generic case (ACM) was previously known, the detailed description of the non-ACM cases using the Hilbert scheme and Hartshorne-Rao modules is new. They explicitly state that a full analysis of the moduli space $M$ is postponed to a future paper to maintain the length of the current work.