Quartic surface, its bitangents and rational points

This paper establishes that a smooth quartic surface over a number field contains only finitely many rational bitangents by leveraging the geometry of its associated quartic double solid, while simultaneously demonstrating that its set of quadratic algebraic points is Zariski-dense.

The Gist

Imagine you are standing in front of a perfectly smooth, four-dimensional sculpture (a "quartic surface") floating in a mathematical universe. This sculpture is defined by a specific set of rules, and we are interested in two things: lines that touch it in a special way and points on it that can be described using simple numbers (rational points).

This paper by Pietro Corvaja and Francesco Zucconi is like a detective story solving two mysteries about this sculpture.

The Main Characters

1. The Quartic Surface ($X$): Think of this as a complex, bumpy 3D shape (like a weirdly shaped potato) floating in space. It's defined over a "number field," which is just a fancy way of saying the rules for its shape use specific types of numbers (like fractions or integers).
2. The Bitangents: Imagine a straight stick (a line) floating near the sculpture. Usually, a stick might poke through the surface once or twice. A bitangent is a special stick that just grazes the surface at two different spots, touching it gently without cutting through, like a feather landing on a pillow.
3. The "Map" ($S$): The authors imagine a magical map where every single possible bitangent stick is represented by a single dot. If you have a million different bitangents, you have a million dots on this map. This map itself is a surface.

Mystery #1: Are there infinite "Special" Points? (Theorem A)

The Question: If you look at the sculpture, can you find an infinite number of points that are "quadratic" (a specific type of simple number relationship) that are spread out all over the surface?

The Answer: Yes.

The Analogy:
Imagine the sculpture is covered in a net. The authors prove that you can weave a net of "hyperelliptic curves" (think of these as special, twisted loops) over the entire sculpture.

• These loops are like train tracks.
• On any single track, there are infinitely many stations (points) that are easy to find.
• Because you can cover the whole sculpture with these tracks, you can find infinitely many easy-to-find points everywhere on the surface.

Why is this cool? It shows that even though the surface is complex, it's not "empty" of simple points. You can always find more if you look in the right way.

Mystery #2: Are there infinite "Special" Sticks? (Theorem B)

The Question: If we restrict ourselves to a specific set of numbers (like the rational numbers we use in daily life), are there infinitely many bitangent sticks that can be drawn using only those numbers?

The Answer: No (with one exception).

The Analogy:
Think of the "Map of Bitangents" ($S$) mentioned earlier.

• The authors prove that if the sculpture is "generic" (meaning it doesn't have any straight lines running through it), this map is a very "rigid" and "sparse" place.
• It's like a desert. You can't find an infinite number of oases (rational points) in this desert. There are only a finite number of them.
• The Exception: If the sculpture is a "special" one (like Schur's Quartic, which is full of straight lines running through it), then the map breaks down, and you can find infinite sticks. But for almost all sculptures, the answer is "finite."

Why is this cool? It confirms a famous mathematical guess (the Bombieri–Lang conjecture) for this specific type of surface. It says that for "complex" shapes, the "simple" solutions are rare and finite.

The Secret Weapon: The "Double Solid"

How did they solve Mystery #2? They didn't just look at the sculpture; they looked at a shadow or a double version of it.

• Imagine taking the sculpture and creating a "double solid" (a 3D object that covers the sculpture twice).
• They looked at the lines inside this double solid.
• They discovered that the "Map of Bitangents" is actually a cover of the "Map of Lines" in this double solid.
• Using deep math (Faltings' Theorem), they showed that the "Map of Lines" is so complex and "curvy" that it cannot hold an infinite number of simple points.
• Since the "Map of Bitangents" is tied to this, it also cannot hold infinite simple points.

The "No Curves" Rule (Theorem C)

There is a third, deeper result. The authors prove that the "Map of Bitangents" is so rigid that it doesn't even contain any simple loops (curves with a specific mathematical property called "genus 1").

• Analogy: Imagine a landscape that is so craggy and full of sharp peaks that you cannot draw a single smooth circle on it without it breaking. This proves the surface is "hyperbolic"—it repels simple shapes.

Summary

• Good News: You can find infinitely many simple points on the sculpture if you look at it through the right "lens" (Theorem A).
• Bad News: You cannot find infinitely many simple "grazing sticks" (bitangents) on a standard sculpture (Theorem B).
• The Reason: The geometry of the sculpture is so complex that it forces simple solutions to be rare, unless the sculpture is a very specific, "broken" type.

The paper uses advanced geometry to show that while these mathematical shapes are rich and full of life, they are also disciplined: they don't allow for infinite chaos in their simple structures.

Technical Summary

1. Problem Statement

The paper addresses two distinct but related arithmetic-geometric problems concerning smooth quartic surfaces $X \subset \mathbb{P}^3$ defined over a number field $\kappa$:

1. Finiteness of Rational Bitangents: While it is generally believed that the set of rational points on a smooth quartic surface is potentially dense (i.e., dense over some finite extension), the authors investigate the specific subset of points arising from bitangent lines (lines tangent to the surface at two distinct points). The central question is whether there can be infinitely many bitangents defined over the base field $\kappa$.
2. Density of Quadratic Points: Conversely, the authors seek to establish the existence of a large set of algebraic points of degree 2 (quadratic points) on $X$ over a suitable finite extension of $\kappa$.

The paper specifically addresses the case where $X$ contains no lines. The presence of lines is a critical obstruction, as demonstrated by the authors' counter-example (Schur's quartic), which contains 64 lines and admits infinitely many rational bitangents.

2. Methodology

The authors employ a sophisticated blend of algebraic geometry, Hodge theory, and arithmetic geometry. The methodology relies on the interplay between the quartic surface $X$, the surface of bitangents $S$, and the associated quartic double solid $Q$.

A. Geometric Framework

• The Surface of Bitangents ($S$): The set of bitangents to $X$ forms an algebraic surface $S$. If $X$ contains no lines, $S$ is smooth.
• The Quartic Double Solid ($Q$): Let $\pi_Q: Q \to \mathbb{P}^3$ be the double cover of $\mathbb{P}^3$ branched along $X$. $Q$ is a Fano threefold.
• The Hilbert Scheme of Lines ($S_X$): The authors utilize the Hilbert scheme of lines on $Q$. A key result from previous work (Welters, Tikhomirov) establishes that $S_X$ is a 2-to-1 étale cover of the bitangent surface $S$.
• Contact Points: The geometry of the "surface of contact points" $Y$ (points where bitangents touch $X$) is analyzed to understand the structure of $S$.

B. Arithmetic Tools

• Chevalley–Weil Theorem: Used to transfer the problem of rational points from $S$ to its étale cover $S_X$. If rational points on $S_X$ are finite, so are those on $S$.
• Faltings' Theorem (Mordell Conjecture): Applied to the surface $S_X$. Since $S_X$ has a high irregularity, the authors use Faltings' theorem to show that rational points on $S_X$ are degenerate (contained in a finite union of curves).
• Abelian Varieties and Albanese Maps: The authors analyze the Albanese variety of $S_X$. They utilize the fact that the Albanese map $\alpha: S_X \to \text{Alb}(S_X)$ is an embedding (injective differential) and that $S_X$ is related to the intermediate Jacobian of $Q$.

3. Key Contributions and Results

Theorem A: Density of Quadratic Points

Statement: For any smooth quartic surface $X$ over a number field $\kappa$, there exists a finite extension $\kappa'$ such that the set of algebraic points in $X(\bar{\kappa})$ which are quadratic over $\kappa'$ is Zariski-dense.

Method of Proof:

• The authors show that $X$ contains a 1-dimensional family of curves of geometric genus 1 (elliptic curves).
• By choosing a point $p$ on such a curve defined over $\kappa'$, the tangent plane section $X_p$ is a singular quartic with a node at $p$.
• This singular curve admits a degree 2 morphism to $\mathbb{P}^1$. Since $\mathbb{P}^1$ has infinitely many rational points, their pre-images provide infinitely many quadratic points on $X_p$.
• Varying $p$ over the infinite set of rational points on the genus 1 curve yields a Zariski-dense set of quadratic points on $X$.
• Note: This result holds even if $X$ contains lines, unlike Theorem B.

Theorem B: Finiteness of Rational Bitangents

Statement: Let $X \subset \mathbb{P}^3$ be a smooth quartic surface defined over a number field $\kappa$ that contains no lines. Then there are only finitely many bitangent lines to $X$ defined over $\kappa$.

Method of Proof:

1. Reduction to $S_X$: By the Chevalley–Weil theorem, proving finiteness for $S$ reduces to proving it for the étale double cover $S_X$ (the Hilbert scheme of lines on $Q$).
2. Degeneracy: The irregularity of $S_X$ is $q(S_X) = 10$, which exceeds its dimension ($\dim S_X = 2$). By a theorem of Faltings, rational points on $S_X$ are degenerate; they lie on a finite union of curves of geometric genus $\le 1$.
3. Exclusion of Curves:
   • Genus 0: $S_X$ contains no rational curves because the Albanese map is an embedding, and Abelian varieties contain no rational curves.
   • Genus 1: The authors prove (Theorem C) that $S_X$ (and thus $S$) contains no curves of geometric genus 1.
4. Conclusion: Since there are no curves of genus $\le 1$ to support infinite families of rational points, the set of rational points on $S_X$ (and consequently $S$) must be finite.

Theorem C: Absence of Low-Genus Curves

Statement: Let $X$ be a smooth quartic containing no lines, and let $S$ be the surface of bitangents. Then the surface $S$ (and its cover $S_X$) contains no curve of geometric genus $\le 1$.

Method of Proof:

• The proof relies on the geometry of the divisor $D_l$ on $S_X$ (parametrizing lines intersecting a fixed line $l$).
• Using the identification of the Albanese variety with the Picard variety and the properties of the Abel-Jacobi map, the authors show that the existence of an elliptic curve in $S_X$ would imply the existence of a non-constant morphism from $S$ to an Abelian variety (specifically the Jacobian of the elliptic curve).
• However, the irregularity of $S$ is $q(S) = 0$ (since $S$ is a surface of general type with specific cohomological properties derived from $X$). A surface with $q=0$ cannot map non-trivially to an Abelian variety. This contradiction proves the non-existence of such curves.

4. Significance and Implications

• Bombieri–Lang Conjecture: Theorem B provides a strong instance of the Bombieri–Lang conjecture for surfaces of general type. It confirms that for a specific class of surfaces (bitangent surfaces of quartics without lines), the set of rational points is not just potentially dense, but actually finite over any number field.
• Hyperbolicity: The results imply that the surface $S$ is hyperbolic in the sense of Brody (no entire curves) and Kobayashi (pseudo-distance is a metric).
• Bogomolov's Conjecture: Theorem C proves Bogomolov's conjecture (that surfaces of general type contain only finitely many curves of genus $\le 1$) in the strong form for these specific surfaces: they contain none.
• Arithmetic Geometry: The paper highlights the delicate balance between the geometry of the surface and its arithmetic properties. The presence of lines (Picard number 20 in the counter-example) completely changes the arithmetic behavior, allowing for infinite rational bitangents, whereas the "generic" case (Picard number 1, no lines) forces finiteness.
• Quadratic Points: Theorem A clarifies that while rational points might be sparse or finite in specific contexts, quadratic points are abundant and dense, providing a different perspective on the "potential density" of points on K3 surfaces.

In summary, the paper successfully bridges the geometry of the quartic double solid and the arithmetic of rational points, using deep results on Abelian varieties and Hodge theory to prove finiteness theorems that were previously conjectural.