On the Fano dimension of an Enriques surface

Technical Summary: On the Fano Dimension of an Enriques Surface

Problem Statement
The paper addresses the "Fano-visitor problem," a question in algebraic geometry and derived categories first posed by Bondal (2011). The problem asks whether, for any smooth projective variety $X$, there exists a smooth Fano variety $Y$ and a full and faithful functor $D(X) \hookrightarrow D(Y)$, where $D(X)$ denotes the bounded derived category of coherent sheaves. If such a $Y$ exists, $X$ is termed a "Fano-visitor" and $Y$ a "Fano-host." The "Fano dimension" of $X$ is defined as the minimal dimension of such Fano-hosts.

Prior to this work, Kuznetsov ([Kuz18]) established that a general Enriques surface $S$ is a Fano-visitor for a six-dimensional Fano host, while nodal Enriques surfaces admit a four-dimensional host. The paper seeks to lower the Fano dimension bound for a general Enriques surface from six to four.

Methodology
The author employs the theory of degeneracy loci of morphisms between vector bundles and the geometry of stratified Grassmann bundles. The core technical tools include:

1. Degeneracy Loci and Vector Bundles: The construction relies on a general morphism $\phi: E \to F$ of vector bundles on a smooth variety $X$, where $E^\vee \otimes F$ is globally generated. The paper utilizes the relationship between the degeneracy locus $D_r(\phi)$ and the zero loci of associated sections on projective bundles.
2. Stratified Projective Bundles: A key lemma (Lemma 2.3) establishes that for a general morphism $\phi: E \to F$ with ranks $e > f$, the zero locus $Y = V(\phi_E)$ inside the projective bundle $\mathbb{P}_X(E)$ is birational to $X$. Crucially, if $f = e-1$, $Y$ is isomorphic to the blow-up of $X$ along the degeneracy locus $D_{f-1}(\phi)$.
3. Derived Category Embeddings: The paper applies Orlov's blow-up formula and results on semiorthogonal decompositions (specifically [Kuz07, Theorem 8.8]) to establish embeddings of derived categories. If $Y$ is a blow-up of $X$ along a center $Z$, and $Y$ is Fano, then $D(Y)$ contains $D(Z)$ as a semiorthogonal component.
4. Proposition 2.4: The author proves a general result stating that under specific conditions (vanishing of lower degeneracy loci and an anti-ampleness condition on a specific line bundle), the zero locus $Y$ of a section on a projective bundle is a Fano host for the degeneracy locus $Z$.

Key Contributions and Results

• Construction of a Four-Dimensional Fano Host: The primary result (Theorem 3.1) constructs a specific Fano fourfold $Y$ that serves as a host for a general Enriques surface $S$.

  • Geometric Realization: Let $X = \mathbb{P}^2 \times \mathbb{P}^2$. The author considers vector bundles $E = \mathcal{O}_X \otimes W_3$ and $F = \mathcal{O}_X(2,0) \oplus \mathcal{O}_X(0,2)$. A general morphism $\phi: E \to F$ defines a general Enriques surface $S$ as the first degeneracy locus $D_1(\phi)$.
  • The Host Variety: The variety $Y$ is defined as the zero locus of a section of the bundle $\mathcal{O}(1,0,2) \oplus \mathcal{O}(1,2,0)$ inside $\mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^2$.
  • Isomorphism: The paper proves that $Y \cong \text{Bl}_S(\mathbb{P}^2 \times \mathbb{P}^2)$, the blow-up of $\mathbb{P}^2 \times \mathbb{P}^2$ along the Enriques surface $S$.
  • Fano Property: Using the adjunction formula, $Y$ is shown to be a Fano variety (its anticanonical bundle is ample).

• Derived Category Decomposition: Corollary 3.3 establishes the semiorthogonal decomposition of the derived category of $Y$:
  $$D(Y) = \langle D(S), E_1, \dots, E_9 \rangle$$
  where $E_1, \dots, E_9$ are exceptional bundles arising from the exceptional collection of $\mathbb{P}^2 \times \mathbb{P}^2$. This confirms $S$ is a Fano-visitor for $Y$.

• Invariants and Torsion:

  • The Hodge diamond of $Y$ is diagonal.
  • The integral cohomology $H^4(Y, \mathbb{Z})$ contains a summand isomorphic to the rank-ten lattice $\Lambda$ of the Enriques surface plus a $\mathbb{Z}/2\mathbb{Z}$ torsion component.
  • Consequently, $K_0(Y)$ contains a 2-torsion class, implying that $D(Y)$ does not admit a full exceptional collection.

• Lower Bound Argument: The paper argues that the Fano dimension of a general Enriques surface is exactly 4. Since $S$ cannot be embedded into a Fano curve or surface (due to dimension and Hodge theoretic constraints), the author rules out Fano threefolds. If a Fano threefold $Y'$ were a host, its Picard rank would need to be at least 5 to accommodate the Hochschild homology dimensions. However, all known Fano threefolds with Picard rank $\ge 5$ admit full exceptional collections, implying their Grothendieck groups are torsion-free, which contradicts the presence of 2-torsion induced by $S$.

Significance
The paper claims to improve the known bound for the Fano dimension of a general Enriques surface from 6 to 4. By explicitly constructing a Fano fourfold $Y$ containing $D(S)$ as a semiorthogonal component, the work provides a concrete solution to the Fano-visitor problem for this class of surfaces. The construction highlights the deep interplay between degeneracy loci, blow-ups of products of projective spaces, and derived categories. The result also reinforces the connection between the torsion in the Grothendieck group of the host and the existence of full exceptional collections.