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Rank jumps for Jacobians of Hyperelliptic curves on K3 surfaces

This paper investigates Mordell-Weil rank jumps in families of Jacobians associated with a pencil of genus-2 curves on a K3 surface over a number field, demonstrating that such jumps occur infinitely often over a finite extension and identifying geometric conditions that ensure these jumps happen on a non-thin set of fibers.

Original authors: Ander Arriola Corpion, Cecília Salgado

Published 2026-04-07

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CC BY 4.0

Original authors: Ander Arriola Corpion, Cecília Salgado

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 an architect designing a massive, infinite library. This library isn't made of books, but of mathematical shapes called Jacobian varieties. Each "room" in this library corresponds to a specific curve (a twisted loop) on a special type of surface known as a K3 surface.

The authors of this paper, Arriola and Salgado, are investigating a very specific question about this library: Can we find rooms where the "complexity" of the math suddenly spikes?

In mathematical terms, they are looking for Rank Jumps.

The Core Concept: The "Rank" of a Room

Think of the "rank" of a room as the number of independent keys you need to unlock all the treasures inside it.

Generic Rank: Most rooms in the library have a standard number of keys (say, 2). This is the "expected" complexity.
Rank Jump: Occasionally, you walk into a room and discover it has more keys than expected (say, 5). This is a "Rank Jump." It means that specific room is much richer and more complex than the average.

The big question in number theory is: Are these "rich" rooms rare anomalies, or are there infinitely many of them?

The Setting: The K3 Surface and the Pencil

The authors are looking at a specific type of library built on a K3 surface.

The Surface: Imagine a K3 surface as a magical, two-dimensional sheet of fabric that folds over itself.
The Pencil: They draw a family of loops (genus-2 curves) across this fabric. Think of this as a "pencil" of lines, but instead of straight lines, they are these complex loops.
The Family: As you move your finger along the fabric, you trace different loops. Each loop has its own "room" (Jacobian) with its own number of keys (rank).

The Problem: Finding the Spikes

Usually, the number of keys stays the same or drops slightly as you move along the fabric. The authors want to prove that if you look hard enough (and maybe change your perspective slightly by extending the number system you're using), you will find infinitely many spots where the number of keys jumps up.

How They Did It: The "Flashlight" and the "Rigid Path"

To find these spikes, the authors used a clever strategy involving multisections.

The Analogy of the Flashlight:
Imagine shining a flashlight beam across the library floor.

The Beam (Multisection): The beam represents a path that cuts across many different rooms at once.
The Twist (Salient Ramification): The authors found a special kind of path that twists in a very specific, "rigid" way as it crosses the rooms. It's not just a straight line; it's a path that gets "stuck" or "twisted" at certain points on the surface.
The Result: Because this path is twisted in just the right way, it forces the rooms it passes through to suddenly gain extra keys.

They found two main ways to create these twisting paths:

The Tangent Line (Rigid): If the surface has a specific curved edge, they draw a line that just barely touches (is tangent to) that edge. This creates a path that is "stuck" in a specific spot, forcing a rank jump.
The Singularity (Moving): If the surface has a "kink" or a sharp point (a singularity), they draw lines radiating from that kink. These lines create a whole family of paths that move around, but they all share the property of causing rank jumps.

The Main Discoveries

The paper proves two major things:

Infinite Richness (Theorem 1.1): If the surface has a curved edge that isn't just a straight line, there is a way to extend the number system (like adding a new type of number) so that infinitely many rooms have a rank jump. It's not just one lucky room; it's an endless corridor of them.
The "Not Thin" Result (Theorem 1.2): If the surface has a "kink" (a singularity), the set of rich rooms is so dense that you can't avoid them. In math-speak, the set of these rooms is "not thin." Imagine trying to throw a dart at a wall covered in gold coins; if the set is "not thin," you are almost guaranteed to hit a coin.

Why This Matters

Before this paper, we knew that rank jumps could happen, but proving they happen infinitely often in these specific geometric settings was very hard. Previous methods required the surface to have very special, complex properties (like having multiple ways to slice it into loops).

The authors showed that you don't need those fancy conditions. Even with a "simpler" surface that might only have one way to be sliced, you can still find these infinite spikes in complexity.

The Takeaway

Think of the K3 surface as a landscape. The authors have discovered that if you look for specific features (like a curved edge or a sharp kink), you can find a "treasure map" (a multisection) that leads you to an infinite number of treasure chests (fibers with rank jumps).

They didn't just find one chest; they proved the map leads to a whole forest of them, changing our understanding of how complexity behaves in these mathematical landscapes.

1. Problem Statement

The paper investigates the Mordell–Weil rank jump phenomenon in families of abelian varieties defined over a number field $k$ . Specifically, it focuses on families of Jacobians associated with a pencil of genus-2 hyperelliptic curves on a K3 surface $X$ .

Context: Let $\eta: A \to S$ be a family of abelian varieties over a curve $S$ (here $S \cong \mathbb{P}^1$ ). By the Lang–Néron theorem, the generic fiber $A$ has a finitely generated Mordell–Weil group $A(k(S))$ with rank $r(\eta, k)$ .
Specialization: Silverman's specialization theorem states that for all but finitely many points $s \in S(k)$ , the rank of the fiber $A_s(k)$ satisfies $r_s \geq r(\eta, k)$ .
The Goal: The authors study the set of rank jumps, defined as $R(\eta, k) = \{ s \in S(k) \mid r_s > r(\eta, k) \}$ . The central question is: Under what geometric conditions on the K3 surface is this set infinite? Furthermore, is it "large" in the sense of being non-thin (i.e., not contained in a finite union of proper closed subvarieties or images of generically finite maps of degree $\geq 2$ )?

2. Methodology

The authors employ a geometric approach centered on the construction of saliently ramified multisections. This technique builds on work by Bogomolov, Tschinkel, Pasten, and Salgado.

Geometric Setup: The K3 surface $X$ is realized as a double cover $f: X \to \mathbb{P}^2_k$ branched along a plane sextic curve $B$ . A pencil of lines in $\mathbb{P}^2$ passing through a point $P$ pulls back to a pencil of genus-2 curves on $X$ (after resolving singularities).
Saliently Ramified Multisections: A multisection $M$ $M$ of the fibration is called saliently ramified if the map from its normalization to the base curve is ramified over the locus of smooth fibers.
- Key Lemma (2.4): If a fibration admits a saliently ramified multisection, the Mordell–Weil rank of the Jacobian over the function field of the multisection is strictly greater than the generic rank of the original fibration.
- Proposition 2.5: If such a multisection $M$ has infinitely many rational points (over a suitable extension), then the set of rank jumps is infinite.
Construction Strategies: The authors construct these multisections by analyzing the geometry of the branch curve $B$ $B$ :
1. Rigid Case: Pulling back tangent lines to a non-linear irreducible component of $B$ .
2. Moving Case: Pulling back lines passing through a singular point of $B$ , which induces a genus-1 fibration on $X$ . The fibers of this genus-1 fibration act as multisections for the genus-2 pencil.
Hilbert Property: To prove the set of rank jumps is non-thin (rather than just infinite), the authors utilize the Hilbert property and linear disjointness arguments, ensuring the multisections are not contained in the image of any generically finite map of degree $\geq 2$ .

3. Key Contributions and Results

Main Theorems

Theorem 1.1 (Infinite Rank Jumps):
If the branch sextic $B$ contains a geometrically irreducible non-linear component defined over $k$ , there exists a field extension $l/k$ of degree at most 12 such that the set of rank jumps $R(\eta, l)$ is infinite.
- Mechanism: The tangent lines to the non-linear component pull back to genus-1 curves (multisections) that are saliently ramified.
Theorem 1.2 (Non-Thin Rank Jumps):
If the branch sextic $B$ admits a simple singularity defined over $k$ , there exists a field extension $l/k$ of degree at most 18 such that the set of rank jumps $R(\eta, l)$ is not thin in $\mathbb{P}^1_l$ .
- Mechanism: The singularity allows for a genus-1 fibration on $X$ . The fibers of this fibration serve as moving multisections for the genus-2 pencil. The non-thin property is established by showing these fibers are linearly disjoint from potential "thin" sets.

Corollaries and Applications

Corollary 1.4: For a generic K3 surface in the family $\mathcal{L}_d$ (with $d > 3, d \equiv 3 \pmod 4$ ), the rank jump set is non-thin over the base field $k$ itself (no extension needed).
Corollary 1.5: In the coarse moduli space of degree 2 K3 surfaces, there is a dense family of polarized K3 surfaces defined over $\mathbb{Q}$ such that a dense set of curves in their linear system have Jacobians with positive Mordell–Weil rank over $\mathbb{Q}$ .

Explicit Examples (Section 4)

The authors provide three explicit examples of K3 surfaces where their results apply, but previous methods (relying on $k$ -unirationality or the Hilbert property of the total space) do not:

Example 1: A K3 surface with Picard number $\rho=2$ and no genus-1 fibration. They construct a rigid saliently ramified multisection (genus 1) over $\mathbb{Q}$ , proving infinite rank jumps over $\mathbb{Q}$ .
Example 2: A K3 surface with $\rho=5$ admitting a unique genus-1 fibration. They show rank jumps are infinite over $\mathbb{Q}$ and non-thin over $\mathbb{Q}(\sqrt{5})$ .
Example 3: A K3 surface with geometric Picard rank 1. They demonstrate that even with minimal Picard rank, rank jumps occur over $\mathbb{Q}$ and over a quadratic extension.

4. Significance and Relation to Literature

Overcoming Unirationality Constraints: Previous results (e.g., by Hindry–Salgado and Colliot-Thélène) often required the total space of the fibration to be $k$ -unirational or to satisfy the Hilbert property. K3 surfaces generally do not satisfy these properties (especially those with low Picard number). This paper provides a method to prove rank jumps without these hypotheses.
Explicit Bounds: Unlike general existence theorems, this work provides explicit bounds on the degree of the field extension $l/k$ required to observe the rank jumps (degrees 12 and 18).
New Families: The results yield new families of abelian fibrations with the rank-jump property that were previously inaccessible to standard techniques.
Geometric Insight: The paper clarifies the relationship between the singularities and components of the branch curve $B$ and the arithmetic of the Jacobian fibers, showing how specific geometric features (tangents, singularities) force arithmetic complexity (rank jumps).

Conclusion

The paper establishes that for pencils of genus-2 curves on K3 surfaces, the existence of specific geometric features in the branch curve (non-linear components or singularities) guarantees the existence of infinitely many fibers with increased Mordell–Weil rank. By utilizing saliently ramified multisections, the authors bypass the need for global unirationality, providing a robust geometric toolkit for studying rank jumps in higher-genus fibrations over number fields.