Fano Visitor Varieties

• Fano visitor is defined as a smooth projective variety (or stack) whose bounded derived category embeds fully faithfully into that of a smooth Fano host.
• Constructions employ techniques like Cayley’s trick and semiorthogonal decompositions to connect categorical structures with birational geometry.
• The framework introduces invariants such as the Fano dimension, spurring studies on orbifold, weak Fano, and noncommutative variants in algebraic geometry.

A Fano visitor is a smooth projective variety (or more generally, an algebraic stack) whose bounded derived category of coherent sheaves can be embedded fully faithfully into the derived category of a smooth projective Fano variety, called a Fano host. Originating from Bondal's question on the universality of Fano-derived categories, the theory connects semiorthogonal decompositions, birational geometry, and the construction of new categorical and geometric invariants. Central to the subject is the interplay between derived categories, homological invariants, and geometric properties of Fano and related varieties, with significant applications and open questions in algebraic geometry.

1. Definitions and Framework

Let $Y$ be a smooth projective variety over $\mathbb C$ and $D^b(Y)$ its bounded derived category of coherent sheaves. $Y$ is a Fano visitor if there exists a smooth projective Fano variety $X$ (the Fano host) and a fully faithful exact functor

$\Phi : D^b(Y) \hookrightarrow D^b(X)$

such that $D^b(X)$ admits a semiorthogonal decomposition

$$D^b(X) = \langle A_1, \dotsc, A_{k-1}, \Phi(D^b(Y)) \rangle$$

(Kiem et al., 2015). This positions $D^b(Y)$ as an admissible subcategory within $D^b(X)$. The classical Fano dimension of $Y$, denoted $\fandim(Y)$, is the minimal $\dim X$ among all Fano hosts $X$; if none exists, $\fandim(Y) = +\infty$ (Kiem et al., 2015).

Variants include:

• Weak Fano visitor: If the host $X$ is only weak Fano (i.e., $-K_X$ is big and nef) (Aravena, 9 Oct 2024).
• Fano orbifold host: If $X$ is a smooth Deligne-Mumford stack whose coarse moduli is Fano, with a derived embedding $D^b(Y) \hookrightarrow D^b(X)$ (Kiem et al., 2015).

2. Existence and Constructions of Fano Hosts

Complete Intersections

All smooth complete intersections in projective space are Fano visitors (Kiem et al., 2015, Kiem et al., 2015). Explicitly, for a smooth $Y \subset \PP^n$ defined by $c$ equations of degrees $d_1 \geq \cdots \geq d_c$, set

• $S = \PP^{n+r}$ (for $r \gg 0$)
• $E = \oplus_i \cO_S(d_i) \oplus \cO_S(1)^{\oplus r-c}$ so that $Y = s^{-1}(0)$ for a regular section $s \in H^0(S,E)$, $\operatorname{rank} E = c+r$. Form the projective bundle $q : \PP(E^\vee) \to S$, with tautological $w \in H^0(\PP(E^\vee),\cO(1))$, and let $X = \{w=0\} \subset \PP(E^\vee)$. $X$ is smooth Fano, and Orlov's theorem yields a semiorthogonal decomposition with $D^b(Y)$ as the last block.

General Criteria and Cayley’s Trick

A more general construction via Cayley's trick (Kiem et al., 2015) applies when $Y = s^{-1}(0) \subset S$ is a regular zero locus of a vector bundle $E \to S$: the zero locus in $\PP(E^\vee)$ determines a hypersurface $X$ that, subject to certain positivity hypotheses (e.g., $E$ is ample, $K_S \otimes \det E$ nef), is Fano and thus a host for $Y$.

Fano Visitors Among K3 and Hyperkähler Varieties

For K3 surfaces of Picard number $1$ and genus $g \not\equiv 3 \pmod 4$, $X$ is a Fano visitor; if $g \equiv 3 \pmod 4$, only weak Fano hosts are constructed (Aravena, 9 Oct 2024). The construction proceeds via wall-crossing in the moduli spaces of semistable sheaves on $X$, using sequences of Mukai flops and antisymplectic involutions to terminate at (weak) Fano varieties.

The Hilbert scheme of $d$ points $\Hilb^d(X)$ for such $X$ (with $0 \le d \le (g+1)/4$) is also a (weak) Fano visitor.

Higher Dimensional and Special Varieties

The Hilbert square of a cubic hypersurface $Y \subset \PP^{n+1}$, $\Hilb^2(Y)$, is a Fano variety for $n \ge 3$, and the Fano variety of lines $F(Y)$ is a Fano visitor via a fully faithful embedding into $D^b(\Hilb^2(Y))$ (Belmans et al., 2020). This construction leverages standard flips in birational geometry and Orlov's theory of semiorthogonal decompositions.

3. Semiorthogonal Decompositions and Embedding Functors

A central methodological tool is the use of semiorthogonal decompositions (SODs) of derived categories. Given a Fano host $X$, there exists a decomposition of the form

$$D^b(X) = \langle \text{exceptional blocks},\,\Phi(D^b(Y)) \rangle$$

where $\Phi$ is typically realized as a pull-push functor via the incidence correspondence between $Y$ and $X$ (e.g., $F \mapsto i_* p^* F$ for proper maps $p$, $i$ from $Y$ to $X$) (Kiem et al., 2015). SODs also arise in the categorical study of flips, allowing embeddings of $D^b(X)$ into the post-flip variety $D^b(X')$ (Belmans et al., 2020, Aravena, 9 Oct 2024). Bondal-Orlov's theorem guarantees fully faithful functors in standard flip situations, often providing the bridge between a variety and its Fano host.

4. Invariants Arising From Fano Embeddings

Given that many varieties are Fano visitors, associated invariants capture the "distance" from the Fano world (Kiem et al., 2015, Kiem et al., 2015):

• Fano dimension $\fandim(Y)$: minimal dimension of a Fano host.
• Fano number: count of deformation classes of minimal-dimensional hosts.
• Fano function $K_Y(n)$: count of deformation classes of Fano hosts of $Y$ in dimension $n$.
• Toric Fano function $T_Y(n)$: refined to hosts that are hypersurfaces in toric varieties.

For complete intersection Calabi–Yau varieties of dimension $n$, $\fandim(Y) = n+2$ (Kiem et al., 2015). For plane curves of degree $d$, quartic K3 surfaces, and quintic threefolds, explicit Fano dimensions have been computed.

5. Orbifold Generalizations and Noncommutative Visitors

The notion of Fano visitor extends naturally to Fano orbifolds (smooth DM stacks with Fano coarse moduli) (Kiem et al., 2015). Any quasi-smooth complete intersection in a weighted projective space admits an orbifold Fano host of dimension one higher than its ambient space. Embeddings can also be constructed for derived categories of various stacks, and in many instances, orbifold Fano hosts contain subcategories with exotic properties (e.g., quasi-phantom or phantom categories seen in Godeaux surface constructions).

There is further interest in determining which "noncommutative K3 components" in derived categories of higher-dimensional varieties can be realized as full subcategories of Fano orbifold derived categories.

6. Examples and Applications

A selection of significant cases includes:

• Plane curves and elliptic curves: Realized as complete intersections, with Fano hosts constructed explicitly in projective bundle frameworks (Kiem et al., 2015).
• K3 and Enriques surfaces: Embeddings via Mukai moduli spaces, flips, and involutions produce concrete Fano hosts or orbifold analogues for surfaces with complex structure, e.g., quartic K3s, Enriques, Kummer, and bielliptic surfaces (Aravena, 9 Oct 2024, Kiem et al., 2015).
• Fano varieties of lines: $F(Y)$ for cubic $Y$ is a Fano visitor with host $\Hilb^2(Y)$; the embedding is realized via projective bundle formulae and categorical flips (Belmans et al., 2020).
• Holomorphic symplectic and Calabi-Yau varieties: Certain Hilbert schemes, moduli spaces, and special varieties arising from homogeneous spaces are confirmed Fano or weak Fano visitors through these constructions.

7. Open Problems and Future Directions

The theory raises several categorical and geometric questions (Kiem et al., 2015, Kiem et al., 2015):

• Characterization of Fano dimensions and Fano numbers, especially for classically significant varieties (e.g., del Pezzo, quartic K3, quintic threefolds).
• Whether the Fano function or related invariants uniquely characterize geometric families such as del Pezzo surfaces.
• Existence of smooth Fano varieties whose derived categories contain (quasi-)phantom subcategories.
• Variational behavior of Fano dimension under deformations and birational modifications.
• Extension to noncommutative and stack-theoretic settings; identifying which triangulated categories or noncommutative schemes arise as summands in Fano hosts.
• Combinatorial constructions for toric and spherical varieties and their Fano dimensions.

These avenues situate the Fano visitor problem at the crossroads of categorical algebraic geometry, birational geometry, and moduli theory, with deep connections to derived categories, $K$-theory, Hodge theory, and homological mirror symmetry.