Reducible Fake Quadrics

• The paper identifies reducible fake quadrics as smooth minimal surfaces with pg=0 and K²=8 that are uniformized by H×H and arise via mixed-type quotients of product curves.
• It details a two-dimensional Néron–Severi group generated by divisors D₁ and D₂ with a hyperbolic intersection form, confirming their status as Mori dream surfaces with finitely generated Cox rings.
• The study classifies these surfaces into five distinct families based on specific group actions, branch data, and Cox ring relations, providing insights into their moduli and birational geometry.

A reducible fake quadric is a smooth minimal complex surface of general type, characterized by the Hodge numbers $p_g = h^{2,0} = 0$, $q = h^{1,0} = 0$, and self-intersection number $K_S^2 = 8$, that admits an unramified cover biregular to a product of two algebraic curves—i.e., it is isogenous to a product. Such surfaces are uniformized by $\mathbb{H} \times \mathbb{H}$ (where $\mathbb{H}$ denotes the upper half-plane) and, up to finite étale cover, split as quotients of products of curves by a freely acting finite group. Reducible fake quadrics are distinguished from their irreducible counterparts, which do not admit such a splitting. The geometry of reducible fake quadrics is realized through the explicit construction of mixed-type surfaces isogenous to a product, and their algebraic and birational invariants place them as rare and structurally significant examples within the moduli of minimal surfaces with $p_g = 0$, $K^2 = 8$ (Frapporti et al., 2019).

1. Structure and Classification of Reducible Fake Quadrics

Surfaces isogenous to a product are defined as $S = (C_1 \times C_2)/G$, where $C_1$, $C_2$ are smooth projective curves of genus at least 2, and $G$ is a finite group acting freely on the product. If an element of $G$ exchanges the two factors, the quotient is said to be of mixed type; otherwise, it is of unmixed type. Reducible fake quadrics arise exactly as the mixed-type surfaces with $\chi(\mathcal{O}_S) = 1$, which yields $p_g = 0$, $K_S^2 = 8$.

Bauer–Catanese–Grunewald and Frapporti established that there exist precisely five connected families of mixed-type surfaces with $\chi = 1$, each giving a distinct irreducible component in the moduli space of minimal surfaces of general type with $p_g = 0$ and $K^2 = 8$ (Frapporti et al., 2019). The invariants—including the group $G$, the subgroup $G^0$ of index 2, the ramification type of the covering $C \to C/G^0$, the first homology $H_1(S, \mathbb{Z})$, and the dimension—are summarized as follows:

Family	$G$	$G^0$	Branch Type	$H_1(S, \mathbb{Z})$	Dim
(1)	SmallGroup(64,92)	SmallGroup(32,46)	[0;2,2,2,2,2]	$(\mathbb{Z}/2)^4$	4
(2)	G(256,3679)	G(128,36)	[0;4,3,3]	$\mathbb{Z}/8 \times \mathbb{Z}/2$	3
(3)	G(256,3678)	G(128,36)	[0;4,3,3]	$(\mathbb{Z}/2)^2$	3
(4)	G(256,3678)	G(128,36)	[0;4,3,3]	$\mathbb{Z}/4 \times \mathbb{Z}$	3
(5)	G(768,1085341)	G(384,4)	[0;4,3,3]	$\mathbb{Z}/3$	3

The branch data notation $[0; m_1, \ldots, m_r]$ refers to a degree-$r$ covering of $\mathbb{P}^1$ branched at $r$ points with respective ramification indices $m_i$.

2. Néron–Severi Group, Intersection Form, and Divisors

For any reducible fake quadric $S$, the Picard number is $\rho(S) = \operatorname{rank} NS(S) = 2$ due to $p_g = 0, q = 0, \chi = 1$ and $b_2(S) = 2$. The real Néron–Severi space $NS(S)_{\mathbb{R}} \cong \mathbb{R}^2$ is generated by two G-orbit divisors, $D_1$ and $D_2$, arising from the orbits of specific automorphisms of $C$.

In this basis, the intersection form is the hyperbolic matrix: $$(D_i \cdot D_j)_{i,j=1,2} = \begin{pmatrix} 0 & m \ m & 0 \end{pmatrix}$$ where $m = 4$ for family (1) and $m = 16$ for families (2)-(5). The canonical class is a positive combination $K_S \equiv a D_1 + b D_2$ satisfying $K_S^2 = 2 m a b = 8$.

3. Cones of Divisors: Effective, Nef, and Semiample

The geometry of divisor classes on reducible fake quadrics is particularly rigid. The closed convex cone of effective divisors $\mathrm{Eff}(S)$, the nef cone $\mathrm{Nef}(S)$, and the semiample cone $\mathrm{SAmp}(S)$ all coincide and are generated by the classes $D_1$, $D_2$, which are both effective, nef, and semiample: $$\mathrm{Eff}(S) = \mathrm{Nef}(S) = \mathrm{SAmp}(S) = \{ x D_1 + y D_2 \mid x, y \geq 0 \} \subset NS(S)_{\mathbb{R}}$$ There are no other irreducible effective curve classes in $NS(S)_{\mathbb{R}}$: $D_1$ and $D_2$ generate the two extremal rays, and all other effective divisors are positive linear combinations.

The intersection pairing, being of hyperbolic signature $(1,1)$, enforces non-negativity in the nef cone precisely when $x, y \geq 0$.

4. Mori Dream Surface Structure and Cox Rings

Each reducible fake quadric is a Mori dream surface. A surface $S$ is Mori dream if its Cox ring,

$$\operatorname{Cox}(S) = \bigoplus_{(a,b) \in \mathbb{Z}^2} H^0(S, \mathcal{O}_S(aD_1 + bD_2))$$

is finitely generated—a property confirmed by verifying that all extremal rays are semiample and the cones are polyhedral.

The two divisors $D_1$, $D_2$ are base-point-free and determine sections in degrees $(1,0)$ and $(0,1)$, respectively, which suffice to generate the Cox ring. Relations arise from $2 \times 2$ minors reflecting the intersection structure. The effective, nef, and semiample cones coincide, meaning the Mori–Mukai program is particularly simple: every effective divisor is movable, and the chamber decomposition of the movable cone is finite and easily described (Frapporti et al., 2019).

5. Examples and Moduli

A prototypical example from family (1) has $G \cong$SmallGroup$(64,92)$, $G^0 \cong$SmallGroup$(32,46)$, and the curve $C \rightarrow \mathbb{P}^1$ branched at 5 points with ramification index 2, leading to genus $g(C)=17$. The two generators of $NS(S)$ arise from $G^0$-orbits of involutions in $\mathrm{Aut}(C)$.

In families (2)-(5), a larger group $H$ of order 768 acts on $C$, with $G^0$ sitting as an index 6 subgroup. Non-conjugate elements of $H$ yield divisors whose descendants generate $NS(S)$. The intersection numbers and structure of $NS(S)$ in these families are explicit and uniform within each family.

All reducible fake quadrics are accounted for by the five mixed-type families, in addition to four infinite families of unmixed-type isogenous surfaces. The existence and structure of the so-called irreducible, or "quaternionic," fake quadrics represent an outstanding open problem; their Cox ring finiteness and Mori dream status remain unresolved (Frapporti et al., 2019).

6. Consequences, Applications, and Open Problems

The identification of reducible fake quadrics as Mori dream surfaces yields rare instances of general-type surfaces with positive Kodaira dimension and finitely generated Cox rings. Embeddings into toric ambient spaces via their Cox rings and explicit classification of triggered Mori programs are direct consequences.

Key open problems include:

• Determining whether all irreducible fake quadrics are Mori dream surfaces; no counterexamples are known.
• Whether every minimal smooth surface of general type with $p_g = 0$ admits a Mori dream structure—higher $K^2$ complicates the geometry.
• Uniform Cox ring presentations for all reducible fake quadrics, possibly via toric degenerations.
• Understanding how the chamber structure of the movable cone reflects the underlying group-theoretical data in surfaces isogenous to a product.

The study of Cox rings and Mori dream structures thus serves as a pathway to deeper insights into moduli and birational geometry of surfaces with $p_g = 0$, extending the landscape of explicit classification in the theory of algebraic surfaces (Frapporti et al., 2019).