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Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland)

This paper introduces a computational method to detect highly singular algebraic varieties, which is used to construct an explicit fake quadric as a $\mathbb{Z}/2 \times \mathbb{Z}/4$ abelian cover of a singular Godeaux surface, marking the first example of such a surface not arising as a quotient of a product of curves.

Original authors: Carlos Rito, Christian Gleissner, Noah Ruhland

Published 2026-07-13

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CC BY 4.0

Original authors: Carlos Rito, Christian Gleissner, Noah Ruhland

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to find a very specific, rare shape hidden inside a massive, shifting cloud of possibilities. That's essentially what mathematicians Carlos Rito and his team did when they hunted for a "fake quadric."

The Mystery: What is a "Fake Quadric"?
Think of a standard quadric surface (like a hyperboloid or a saddle shape) as a perfect, smooth ball of clay. It has very specific measurements: it has a "curvature score" of 8, and it has zero "holes" in certain directions (mathematicians call these $p_g$ and $q$ , and they are both 0).

A "fake quadric" is a shape that looks exactly like this perfect ball in terms of its measurements ( $K^2 = 8$ , $p_g = 0$ , $q = 0$ ), but it's actually a weird, twisted alien shape underneath. It's a "wolf in sheep's clothing." For a long time, mathematicians knew these shapes existed, but they were like ghosts: everyone knew they were there, but no one could write down the exact recipe (the equations) to build one.

The Old Way vs. The New Trick
Usually, to find these shapes, mathematicians tried to build them by gluing together two loops of string (products of curves) and then cutting them up. But this paper argues that the specific fake quadric they found cannot be made that way. It's not a simple remix of two loops; it's something entirely new.

To find it, Rito invented a new computational "metal detector."

The Search: Imagine a giant field where every spot represents a different shape. Most spots are smooth, but some have "knots" or "cracks" (singularities).
The Metal Detector: Instead of checking the whole field, the team used a clever trick. They looked at the field through a "finite field" lens (like checking the ground with a grid of tiny, discrete stepping stones). They found spots with knots, wrote down the coordinates, and then used a mathematical interpolation (like connecting the dots) to draw the exact line where the knots live.
The Climb: They didn't stop at one knot. They climbed up the "knot ladder," finding spots with two knots, then four, then six. It was like finding a treasure map that led deeper and deeper into a cave of complexity.

The Big Discovery
After running these calculations over and over again on more than 600 different prime numbers (to make sure the answer wasn't just a fluke), they finally found a surface with a very specific, messy scar pattern: two simple cracks ( $A_1$ ) and two complex, three-part cracks ( $A_3$ ).

This surface is a "Godeaux surface" (a specific type of mathematical shape). But here is the magic trick:

They took this cracked surface and built a "cover" over it. Think of it like wrapping a gift.
They wrapped it with a specific pattern (an abelian cover of type $Z/2 \times Z/4$ ).
When they smoothed out the cracks in this new wrapped version, the result was a smooth, perfect fake quadric.

The "Not a Product" Rule
This is the most critical part of the discovery. The paper explicitly rules out the idea that this shape came from the old "gluing loops" method.

The Argument: A separate team (Gleissner and Ruhland) proved a rule: if a shape is made by gluing loops, any symmetry it has must come from the loops themselves.
The Proof: The fake quadric Rito found has symmetries that cannot come from gluing loops. Therefore, the paper proves with certainty that this fake quadric is not a quotient of a product of curves. It is a "quaternionic" fake quadric, a rigid, unique creature that doesn't fit into the old categories.

How Sure Are They?
The paper is not just guessing or simulating. They have:

Explicit Equations: They wrote down the exact mathematical formulas for the surface.
Verification: They used computer software (Magma) to check that the equations actually work and that the surface has the right properties ( $K^2 = 8$ , $p_g = 0$ ).
Rigorous Proof: They mathematically proved that this shape cannot be built from products of curves.

The One Open Question
There is one thing they don't know yet. They know this shape is a fake quadric, but they don't know exactly what kind of "universe" (universal cover) it lives in.

Some fake quadrics live in a "bidisk" universe (like a double donut).
Others might live in a different universe entirely.
The paper suggests that if this new shape lives in the bidisk, it would be the first explicit example of a "quaternionic fake quadric." But until they prove which universe it belongs to, that part remains a mystery.

In Summary
The team used a high-tech "knot detector" to find a cracked surface, smoothed it out to reveal a perfect fake quadric, and proved it's a unique shape that can't be built from simple loops. It's the first time anyone has written down the exact recipe for this specific type of mathematical ghost.

Technical Summary: Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric

Problem Statement
The paper addresses the classification and explicit construction of "fake quadrics"—smooth minimal complex algebraic surfaces of general type sharing the invariants of the quadric surface in $\mathbb{P}^3$ ( $K^2=8, p_g=q=0$ ). While fake projective planes are known to be uniformized by the complex 2-ball, the universal cover of fake quadrics remains an open question. Existing examples are uniformized by the bidisk $\mathbb{H} \times \mathbb{H}$ and fall into two classes: those isogenous to a product of curves (which admit moduli) and "quaternionic" fake quadrics (which are rigid).

A central challenge in the field is the explicit construction of fake quadrics that are not quotients of products of curves. Previous work by Džambić and Roulleau suggested that if a quaternionic fake quadric admits an automorphism group isomorphic to $\mathbb{Z}/2 \times \mathbb{Z}/4$ , its quotient would be a numerical Godeaux surface with a specific singular locus: $2A_1 + 2A_3$ . However, no explicit examples of such Godeaux surfaces had been constructed, and it was unknown whether such configurations could arise from products of curves.

Methodology
The authors employ a computational strategy to detect highly singular members within families of algebraic varieties, specifically targeting numerical Godeaux surfaces. The method proceeds as follows:

Interpolation over Finite Fields: The authors define a family of varieties by explicit equations depending on parameters. To find members with a high number of singularities (nodes or higher-order singularities), they work over finite fields $\mathbb{F}_q$ .
Random Sampling and Locus Detection: By randomly sampling parameters, they identify surfaces with at least one node. The locus of such surfaces forms a hypersurface in the parameter space.
Iterative Interpolation: Using the LinearSystem function in Magma, they interpolate polynomials defining these loci. They iteratively restrict to subfamilies with increasing numbers of nodes (e.g., from 1 to 4, then to 6).
Lifting to Characteristic Zero: Once defining equations are determined modulo various primes, the authors use the Chinese Remainder Theorem and RationalReconstruction in Magma to lift these equations to characteristic zero (rational numbers). This process required computations over more than 600 distinct primes to handle the large rational coefficients involved.
Divisibility and Covering Construction: For a specific Godeaux surface $X$ with singularities $2A_1 + 2A_3$ , the authors compute its minimal resolution $X'$ . They determine the existence of specific divisors $C'$ and $D'$ on $X'$ satisfying numerical divisibility relations required to construct a $\mathbb{Z}/2 \times \mathbb{Z}/4$ abelian cover. This involves solving for linear equivalences involving the exceptional curves of the resolution.
Verification of Non-Product Origin: To ensure the resulting fake quadric is not a quotient of a product of curves, the authors rely on a result established in an Appendix by Gleissner and Ruhland. This result states that every automorphism of a variety isogenous to a product of curves lifts to an automorphism of the covering curves.

Key Contributions and Results

Explicit Construction of a Singular Godeaux Surface: The authors successfully constructed a 2-parameter family of $\mathbb{Z}/2$ -Godeaux surfaces with 6 nodes. Within this family, they identified a specific surface with a singular locus consisting of two $A_1$ and two $A_3$ singularities ( $2A_1 + 2A_3$ ). This surface is defined over the rational numbers.
Construction of a New Fake Quadric: They demonstrated that this specific Godeaux surface admits a $\mathbb{Z}/2 \times \mathbb{Z}/4$ abelian cover. The resulting surface $S$ is smooth, minimal, and of general type with invariants $K^2=8$ and $p_g=q=0$ , thus constituting a fake quadric.
Proof of Non-Product Origin: By combining their construction with the Appendix's result, the authors prove that their fake quadric $S$ cannot be isogenous to a product of curves. Specifically, the fundamental group of the quotient $X$ is $\mathbb{Z}/2$ , which contradicts the classification of mixed product-quotient surfaces with the required singularity basket (which would require a fundamental group of $\mathbb{Z}/4$ ).
Computational Methodology: The paper introduces and validates a robust computational framework for detecting highly singular members in algebraic families, overcoming difficulties related to large rational coefficients and high-dimensional parameter spaces.

Significance and Claims
The paper claims to provide the first explicit construction (via equations) of a fake quadric that does not arise as a quotient of a product of curves. This fills a gap in the geography of surfaces of general type, offering a candidate for a "quaternionic" fake quadric.

The authors note that while the surface is explicitly constructed, it remains an open question whether this specific surface is uniformized by the bidisk $\mathbb{H} \times \mathbb{H}$ or possesses a different universal cover. If it is uniformized by the bidisk, it would represent the first explicit example of a quaternionic fake quadric. The work highlights the difficulty of such constructions, noting that unlike fake projective planes, where the geometry of examples guided the construction, quaternionic fake quadrics lack a complete classification, making the discovery of such examples via computational algebraic geometry a significant step forward.

The computations were performed using Magma, and the defining equations, which involve relatively small rational coefficients but required extensive lifting procedures, are provided as ancillary files.

Technical Summary: Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric

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