[Paper Review] The classification of surfaces with p_g = q = 0 isogenous to a product of curves

Ingrid Bauer, Fabrizio Catanese+1 more|ArXiv.org|Oct 8, 2006

Algebraic Geometry and Number Theory10 references19 citations

TL;DR

This paper classifies smooth projective surfaces isogenous to a product of curves with geometric genus $p_g = 0$ and irregularity $q = 0$, proving that such surfaces arise as quotients $(C_1 \times C_2)/G$ where $G$ is a finite group acting freely and faithfully on the product of two curves of genus $\geq 2$. The key contribution is a complete classification of all such groups $G$ and their corresponding ramification structures, yielding a finite list of possible groups and surface invariants.

ABSTRACT

We classify all the surfaces with p_g = q = 0 which admit an unramified covering which is isomorphic to a product of curves. Beyond the trivial case \PP^1 x \PP^1 we find 17 families which we explicitly describe. We reduce the problem to a combinatorial description of certain generating systems for finite groups which we solve using also MAGMA's library of groups of small order.

Motivation & Objective

• To classify all smooth projective surfaces isogenous to a product of curves with $p_g = q = 0$, which are of general type and not rational or ruled.
• To determine the finite groups $G$ that can act freely on a product of two curves $C_1 \times C_2$ (with $g(C_1), g(C_2) \geq 2$) such that the quotient $S = (C_1 \times C_2)/G$ satisfies $p_g(S) = q(S) = 0$.
• To distinguish between the unmixed and mixed cases based on whether $G$ preserves or exchanges the factors of the product $C_1 \times C_2$.
• To provide explicit ramification structures and group presentations for all such surfaces, including non-abelian groups of order up to 256.
• To construct concrete models for each group and show that the moduli spaces of such surfaces have finitely many irreducible components of equal dimension.

Proposed method

• Use the theory of surfaces isogenous to a higher product, where $S \cong (C_1 \times C_2)/G$ with $G$ acting freely and preserving the product structure.
• Classify groups $G$ via their action on the curves $C_1$ and $C_2$, distinguishing between the unmixed case (where $G$ acts diagonally) and the mixed case (where $G$ contains elements swapping $C_1$ and $C_2$).
• Apply combinatorial group theory to classify tuples of group elements corresponding to ramification data, using the signature of the quotient map and the Riemann-Hurwitz formula.
• Construct explicit group presentations for $G$ as semidirect products $\mathbb{Z}_2^n \rtimes \mathbb{Z}_2^k$, with homomorphisms $\Phi$ and bilinear maps $\Theta$ encoding the action and commutator relations.
• Verify compatibility of ramification structures via the conditions in Definition 1.2, ensuring that the triple of tuples forms a valid mixed or unmixed ramification structure.
• Use the moduli space theory to show that for fixed $G$ and genera $g_1, g_2$, the surfaces form finitely many irreducible components of equal dimension $D$.

Experimental results

Research questions

• RQ1 Which finite groups $G$ can act freely on a product $C_1 \times C_2$ of two curves of genus $\geq 2$ such that the quotient surface $S = (C_1 \times C_2)/G$ satisfies $p_g(S) = q(S) = 0$?
• RQ2 What are the complete lists of groups $G$ and corresponding curve genera $g_1, g_2$ that yield surfaces isogenous to a product with $p_g = q = 0$?
• RQ3 How do the ramification structures (tuples of group elements) classify the possible quotients in both the unmixed and mixed cases?
• RQ4 What are the explicit group presentations and realizations (e.g., as semidirect products) for the groups $G$ that yield such surfaces?
• RQ5 How many irreducible components do the moduli spaces of such surfaces have, and what is their dimension?

Key findings

• The only surfaces isogenous to a product with $p_g = q = 0$ are either $\mathbb{P}^1 \times \mathbb{P}^1$ or quotients $(C_1 \times C_2)/G$ with $g(C_1), g(C_2) \geq 2$ and $G$ acting freely.
• All such groups $G$ are classified into 11 isomorphism types: $\mathfrak{A}_5$, $\mathfrak{S}_4$, $\mathfrak{D}_4 \times \mathbb{Z}_2$, $\mathfrak{S}_4 \times \mathbb{Z}_2$, $G(16)$, $G(32)$, $G(256,1)$, $G(256,2)$, and others from the families $\mathcal{N}_3, \mathcal{N}_4, \mathcal{N}_5, \mathcal{N}_6, \mathcal{M}_3, \mathcal{M}_4, \mathcal{M}_5, \mathcal{M}_6, \mathcal{M}_8$.
• For each group $G$, the paper provides explicit ramification structures: for example, $G(32)$ admits an unmixed ramification structure of type $([2,2,2,4]_8, [2,2,4,4]_4)$, and $G(256,1)$ admits three mixed ramification structures of type $[4,4,4]_{16}$.
• The group $G(256,1)$ is realized as a semidirect product $\mathbb{Z}_2^5 \rtimes_\Phi \mathbb{Z}_2^3$ with specific matrices $\Phi_{e_1}, \Phi_{e_2}, \Phi_{e_3}$ and a bilinear map $\Theta$ defined on 6 non-zero pairs.
• The group $G(256,2)$ is isomorphic to $\mathbb{Z}_2^5 \rtimes_\Phi \mathbb{Z}_2^3$ with a different $\Phi$ and $\Theta$, and admits a single mixed ramification structure of type $[4,4,4]_{16}$.
• For each such $G$, the moduli space of surfaces $S = (C_1 \times C_2)/G$ has finitely many irreducible components, all of the same dimension $D$, and the number of components $N$ is finite for each $G$.