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Explicit Bicanonical Models of Eight Fake Quadrics

This paper presents the first explicit projective models of eight rigid, pairwise non-isomorphic fake quadrics that are not isogenous to a product of curves, defining them over $\mathbb{Q}$ as $\mathbb{Z}/2\times\mathbb{Z}/4$ -covers of singular $\mathbb{Z}/2$ -Godeaux surfaces.

Original authors: Lev Borisov, Carlos Rito

Published 2026-08-13

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CC BY 4.0

Original authors: Lev Borisov, Carlos Rito

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 the universe of shapes not as the smooth balls and cubes we see in the playground, but as a vast, invisible landscape of complex, multi-dimensional surfaces. In this world, mathematicians are like cartographers trying to map out every possible type of "terrain" that follows specific, strict rules. One particularly tricky type of terrain is called a "fake quadric." Think of a real quadric as a perfect, smooth sphere or a saddle shape you can draw on a piece of paper. A "fake" quadric is a shape that looks exactly like that perfect sphere from a distance—having the same number of holes, curves, and bumps—but is actually made of a completely different material. It's like finding a rock that looks and feels exactly like a diamond, but is actually made of glass.

For a long time, mathematicians knew these "fake" shapes existed, but they were like ghosts: we knew they were there, but we couldn't see them clearly or write down their blueprints. Most of the ones we knew about were built by gluing together simpler shapes, like stitching two long ribbons together to make a tube. But there was a nagging question: could there be a fake quadric that wasn't just a stitched-together ribbon? Could there be a shape that was a single, unique, indivisible masterpiece? This paper steps into that mystery, using the tools of algebra and geometry to finally catch these ghosts and write down their exact addresses.

The Great Shape Hunt: Catching Eight New Ghosts

In this paper, two mathematicians, Lev Borisov and Carlos Rito, have done something remarkable: they have found and built explicit mathematical models for eight new "fake quadrics." Until now, these shapes were known to exist only in theory, like a recipe for a cake that no one had ever baked. The authors didn't just prove they exist; they wrote down the exact equations needed to construct them, turning these abstract ideas into concrete geometric objects that can be studied in a computer.

The Recipe for a Fake Quadric
To understand what they did, imagine you are trying to build a complex sculpture. You start with a rough, bumpy base that has some cracks and holes in it. In math-speak, this base is a "singular Godeaux surface." It's a bit messy, with specific types of cracks (called $2A_1$ and $2A_3$ singularities). The authors started with two different versions of these bumpy bases.

Next, they used a magical "unfolding" technique. Imagine taking a piece of paper with a pattern on it and folding it over itself in a very specific way, following a set of rules (a group called $\mathbb{Z}/2 \times \mathbb{Z}/4$ ). When you unfold it, the messy cracks on the base paper get smoothed out, and you end up with a brand new, perfectly smooth surface. This new surface is the "fake quadric." The authors calculated exactly how to do this unfolding for their two base shapes, and because there were different ways to fold the paper, they ended up with eight distinct, smooth surfaces.

The Blueprint: Equations in 8D
The most exciting part of their work is that they didn't just say, "Here is a shape." They gave the shape an address. They found the specific mathematical equations that define these eight surfaces. These equations describe the surfaces as living in a space with 9 dimensions (specifically, a projective space called $\mathbb{P}^8$ ).

Think of it like this: if you wanted to describe a 3D object to a blind person, you might give them a list of coordinates. The authors did the same thing, but for objects that are so complex they live in 8-dimensional space. They wrote down the "homogeneous ideals" (a fancy way of saying the master list of rules) that these eight surfaces must follow. All eight of these blueprints are defined using simple rational numbers (fractions and whole numbers), meaning they are "real" in the most fundamental sense, not just theoretical possibilities.

Are They All the Same?
You might wonder: if they all come from similar recipes, are they just copies of each other? The authors checked this carefully. They proved that all eight surfaces are different from one another. They are like eight siblings who look similar but have different fingerprints. They did this by looking at the "pencil of quadrics" inside the surfaces—a fancy way of looking at the family of simpler shapes hidden inside the complex ones. By comparing the "degeneracy loci" (the points where these inner shapes break down), they showed that no two surfaces can be transformed into each other. They are unique.

The "Not a Product" Discovery
One of the biggest questions in this field was whether these fake quadrics were just "stitched together" shapes (isogenous to a product of curves). For a long time, every known fake quadric was just a combination of two simpler curves. The authors confirmed that these eight new surfaces are not stitched together. They are indivisible, unique entities. This is a big deal because it proves that there is a whole new class of these shapes that we didn't have explicit models for before.

Rigid and Unchanging
Finally, the authors showed that these surfaces are "rigid." In the world of shapes, "rigid" means they are frozen in place. You can't wiggle them, stretch them, or deform them into a slightly different shape without breaking them. They are locked into their exact form. This was proven by checking the "tangent bundle" (a mathematical way of measuring how much a shape can wiggle) and finding that there is zero room for movement.

What About the Big Mystery?
While the authors have successfully built these eight surfaces and proved they are unique and rigid, one giant question remains open. We know these shapes exist and we have their blueprints, but we don't yet know if they are "uniformized by the bidisk." In simple terms, this asks if these shapes are built from a specific type of infinite, double-hyperbolic grid. The paper doesn't solve this; it leaves that door open for future explorers.

The Bottom Line
This paper is a triumph of explicit construction. It takes a theoretical existence proof and turns it into a set of concrete equations. The authors have found eight new, rigid, unique fake quadrics that are not just combinations of simpler shapes. They have provided the first explicit projective models of these surfaces, allowing mathematicians to finally hold these "ghosts" in their hands and study them in detail. The journey from "we think they exist" to "here are their exact equations" is a massive leap forward in understanding the hidden geometry of our universe.

=== DRAFT ===
Problem Statement
The paper addresses the construction of explicit projective models for "fake quadrics," which are smooth minimal complex surfaces of general type sharing the numerical invariants of a smooth quadric in $\mathbb{P}^3$ ( $K_S^2 = 8$ , $p_g(S) = q(S) = 0$ , and $c_2(S) = 4$ ). While known fake quadrics are generally uniformized by the bidisk $\mathbb{H} \times \mathbb{H}$ , they fall into two categories: those isogenous to a product of curves (reducible lattices) and those arising from quaternion algebras (irreducible lattices, or "quaternionic fake quadrics").

Prior to this work, obtaining explicit projective equations for irreducible (quaternionic) fake quadrics was noted as extremely difficult, analogous to the decade-long gap between the arithmetic classification of fake projective planes and the first explicit equations obtained by the first author and Keum. The specific problem tackled here is the construction of explicit defining equations for fake quadrics that are not isogenous to a product of curves. The existence of such surfaces was previously established theoretically by the second author in [12] via abelian covers of singular Godeaux surfaces, but no projective models were provided. Notably, while these surfaces are not isogenous to a product, the paper explicitly states that whether they are uniformized by the bidisk (and thus classified as quaternionic) remains an open problem.

Methodology
The authors employ a constructive approach starting from two specific singular $\mathbb{Z}/2$ -Godeaux surfaces ( $X$ ) identified in earlier work [12]. These surfaces possess singular sets of type $2A_1 + 2A_3$ . The methodology proceeds in three main stages:

Abelian Cover Construction:
The authors utilize Pardini's theory of abelian covers to construct smooth surfaces $S$ as $G$ -covers of the resolution $X'$ of the singular Godeaux surface $X$ , where $G \cong \mathbb{Z}/2 \times \mathbb{Z}/4$ . The cover is determined by specific divisibility relations involving canonical divisors and exceptional curves on the resolution. The lifting theorem of Gleissner and Ruhland is used to argue that the resulting surfaces are not isogenous to a product.
Explicit Equation Computation:
To obtain explicit equations, the authors compute the bicanonical models of the covering surfaces in $\mathbb{P}^8$ .
- Step 1: They analyze the universal cover $Y$ of the Godeaux surface and the intermediate cover $Y'$ . By computing character eigenspaces of the bicanonical system $H^0(Y', 2K_{Y'})$ , they derive polynomial relations. This involves lifting sections from the base to the cover and solving systems of equations to find the homogeneous ideals.
- Step 2: They recover the double cover $Z \to Y'$ (where $Z$ is the universal cover of the fake quadric) by analyzing the order-4 character spaces. This step involves computing square roots of sections and determining compatible lifts to form a basis for $H^0(Z, 2K_Z)$ .
- Step 3: The construction yields four distinct fake quadrics for each of the two starting Godeaux surfaces, resulting in a total of eight surfaces ( $S_1, \dots, S_8$ ). The authors ensure these models are defined over $\mathbb{Q}$ .
Verification and Analysis:
All computations are performed using computer algebra systems (Mathematica and Magma). The authors verify the geometric properties of the resulting models, including smoothness, dimension, and cohomological invariants, using reduction modulo a prime ( $p=97$ ) and upper semicontinuity arguments.

Key Contributions and Results

Explicit Models: The paper provides the first explicit defining equations for fake quadrics which are not isogenous to a product of curves. All eight models are defined over the rational numbers $\mathbb{Q}$ .
Bicanonical Embeddings: The authors prove that the computed projective models in $\mathbb{P}^8$ are indeed the bicanonical models of the surfaces. They verify that $K_S^2 = 8$ , $p_g(S) = q(S) = 0$ , and that the embedding is given by the complete bicanonical system ( $h^0(S, 2K_S) = 9$ ).
Rigidity: The authors prove that all eight surfaces are rigid ( $H^1(S, T_S) = 0$ ) by computing the cohomology of the tangent bundle via reduction modulo $p$ .
Non-Isomorphism: The eight surfaces are proven to be pairwise non-isomorphic. This is established by analyzing the pencil of quadrics contained in their bicanonical ideals. The authors compute the degeneracy divisors (determinants of linear combinations of the defining quadrics) and show that no projective equivalence exists between the degeneracy loci of any pair of the eight surfaces.
Non-Product Property: The paper confirms that these surfaces are not isogenous to a product of curves, relying on the classification of product-quotient Godeaux surfaces and the specific fundamental group constraints of the base surfaces.

Significance
The paper claims to provide the first explicit projective models of fake quadrics which are not isogenous to a product of curves. This bridges the gap between the theoretical existence of such surfaces (established in [12]) and concrete geometric realization. By providing explicit equations, the authors make the geometry of these rigid surfaces accessible for further study. The work parallels the historical progression in the study of fake projective planes, moving from arithmetic classification to explicit equation-based models. The authors note that while the surfaces are constructed, the question of whether they are uniformized by the bidisk (and thus whether they fall into the "quaternionic" class) remains an open problem.

The Great Shape Hunt: Catching Eight New Ghosts

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