On the Fano dimension of an Enriques surface

We construct a family of Fano fourfolds with the derived category of coherent sheaves of a general Enriques surface as semiorthogonal component. This improves a result of Kuznetsov, lowering the Fano dimension of a general Enriques surface from six to four.

💡 Research Summary

The paper addresses the “Fano‑visitor” problem, which asks whether for a given smooth projective variety X there exists a smooth Fano variety Y whose derived category D(Y) contains D(X) as a full, faithful subcategory; the minimal possible dimension of such a Y is called the Fano dimension of X. Earlier work by Kuznetsov showed that a general Enriques surface has Fano dimension at most 6, realized by a specific six‑dimensional Fano host.

The author improves this bound to 4. The construction starts with three‑dimensional vector spaces V, V′, W and the product X = P(V) × P(V′). Two vector bundles on X are defined: E = O_X ⊗ W (rank 3) and F = O_X(2,0) ⊕ O_X(0,2) (rank 2). A general morphism ϕ : E → F is taken; its first degeneracy locus D₁(ϕ) is a smooth surface, which is shown to be isomorphic to a general Enriques surface S.

Two Fano hosts are then considered. The first, already known, is a six‑dimensional variety T obtained as the zero‑locus of a section of O_F∨(1) ⊠ O_{P(W)}(1) on P_X(F∨) × P(W). This yields a semi‑orthogonal decomposition D(T)=⟨D(S), D(P_X(F∨)), D(P_X(F∨))⟩.

The new contribution is a four‑dimensional Fano variety Y defined as the zero‑locus of a section of the rank‑2 bundle O(1,0,2) ⊕ O(1,2,0) on P(W) × P(V) × P(V′). Using Lemma 2.3, the author shows that Y is birational to X and, more precisely, that Y ≅ Bl_S(P² × P²). The conditions of Proposition 2.4 are verified (the second degeneracy locus is empty and an anti‑ampleness condition holds), guaranteeing that Y is indeed Fano.

Finally, the derived category of Y admits a semi‑orthogonal decomposition D(Y)=⟨D(S), E₁,…,E₉⟩ where the E_i are nine exceptional bundles. The Hodge diamond of Y is diagonal, its K‑group contains a 2‑torsion class, and D(Y) does not possess a full exceptional collection.

Thus the paper establishes that the Fano dimension of a general Enriques surface is exactly 4, improving the previous bound of 6, and provides a concrete geometric construction that may be adaptable to other varieties via degeneracy‑locus techniques.