Low degree points on singular plane curves

1. Problem Statement

The paper addresses the distribution of low-degree points on algebraic curves defined over a number field $K$. Specifically, it seeks to generalize the celebrated results of Debarre and Klassen (1994), which characterize the existence of points of bounded degree on smooth plane curves.

• Context: Faltings' proof of the Mordell-Lang conjecture implies that for a smooth plane curve $C$ of degree $d \ge 7$, there are only finitely many points of degree $\le d-2$. Furthermore, for $d \ge 8$, all but finitely many points of degree $\le d-1$ arise from the intersection of $C$ with lines defined over $K$.
• The Gap: These results rely heavily on the curve being smooth. The authors investigate whether these finiteness and structural results hold for singular plane curves with a finite number $\delta$ of ordinary nodes and/or cusps.
• Objective: Determine the bounds on the number of singularities $\delta$ (as a function of the degree $d$) such that the curve $C$ still exhibits the same behavior regarding low-degree points as a smooth curve.

2. Methodology

The authors employ a geometric approach centered on the normalization of the singular curve and the properties of the Abel-Jacobi map.

A. Reduction to the Normalization

Let $C \subset \mathbb{P}^2$ be a singular plane curve of degree $d$ with $\delta$ ordinary singularities. Let $\nu: \tilde{C} \to C$ be its normalization (a smooth curve).

• Points of degree $e$ on $C$ correspond to $K$-rational points on the $e$-th symmetric product $\text{Sym}^e(\tilde{C})$.
• The problem is reduced to studying the image of the Abel-Jacobi map:
  $$AJ_e: \text{Sym}^e(\tilde{C}) \to J_{ace}(\tilde{C})$$
  where $W_e(\tilde{C})$ denotes the image of this map inside the Jacobian of $\tilde{C}$.

B. Application of Faltings' Theorem

The core strategy relies on Faltings' Theorem (Theorem 2.2): A subvariety of an abelian variety contains infinitely many rational points if and only if it contains a translate of a positive-dimensional abelian subvariety.

• To prove finiteness of low-degree points, the authors must show that for specific ranges of $e$, the variety $W_e(\tilde{C})$ does not contain any positive-dimensional abelian subvarieties.
• If $W_e(\tilde{C})$ has no such subvarieties, then $\text{Sym}^e(\tilde{C})$ has only finitely many $K$-points (unless the points lie in the fibers of the map, which correspond to linear series).

C. Analysis of Linear Series

The authors analyze the linear series (pencils) on the normalization $\tilde{C}$ that could generate infinite families of points. They utilize results from Coppens and Kato regarding linear series on singular plane curves:

1. Bounds on $\delta$: They derive explicit quadratic bounds on $\delta$ to ensure that $\tilde{C}$ does not admit "unexpected" linear series (e.g., $g^1_e$ for small $e$).
2. Classification of Pencils: They prove that under their bounds, any base-point-free linear series $g^1_n$ on $\tilde{C}$ (for $n \le 2d-2$) must be induced by a pencil of lines ($m=1$) or a pencil of conics ($m=2$) in $\mathbb{P}^2$.
3. Exclusion of Abelian Subvarieties: By analyzing the geometry of these pencils (specifically their base loci), they demonstrate that the image $W_e(\tilde{C})$ cannot contain abelian subvarieties for $e \le d-1$ under the derived conditions.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

The paper establishes that for a plane curve $C$ of degree $d$ with $\delta$ ordinary nodes/cusps, if one of the following conditions holds:

1. Linear Bound: $d \ge 8$ and $\delta \le d - 8$;
2. Quadratic Bound: $d \ge 17$ and $\delta < \frac{d^2}{4} - 2d - \frac{17}{4}$;

Then:

• Finiteness: $C$ contains only finitely many points of degree $\le d-3$.
• Structure: All but finitely many points of degree $\le d-1$ arise as the intersection of $C$ with lines in $\mathbb{P}^2$ defined over $K$.

Technical Refinements

• Density Degree Set: The authors clarify that the minimum of the density degree set $\min \delta(C/K)$ is $\ge d-2$.
• Optimality: In Section 4, they construct an explicit example (based on Coppens-Kato) of a curve with degree $d=2k+4$ and $\delta = (k+1)^2$ nodes. This curve possesses a $g^1_{d-1}$ not induced by lines, proving that the quadratic bound on $\delta$ is asymptotically optimal. The example shows that if $\delta$ exceeds the bound, the structural result (points coming from lines) fails.

Alternative Approach (Section 4.1)

The authors briefly discuss an alternative method using the work of Smith and Vogt on surfaces. By blowing up the singular points to create a surface $S$ where the strict transform $\tilde{C}$ is nef and big, they derive a slightly different bound ($\delta \le \frac{d^2}{4} - \frac{9}{4}d + 2$) using stability conditions on vector bundles, offering a different perspective on the problem.

4. Significance

• Extension of Classical Results: This work successfully extends the Debarre-Klassen theorem from smooth curves to a broad class of singular curves, bridging a significant gap in arithmetic geometry.
• Sharp Bounds: The paper provides precise, explicit bounds on the number of singularities allowed before the arithmetic behavior of the curve changes. The demonstration of asymptotic optimality is a crucial contribution, showing that the quadratic nature of the bound is necessary.
• Methodological Synthesis: The paper effectively synthesizes tools from the theory of linear series on singular curves (Coppens-Kato), the geometry of Jacobians, and Faltings' theorem.
• Implications for Arithmetic Geometry: The results contribute to the broader understanding of the Mordell-Lang conjecture in the context of singular varieties, clarifying how singularities affect the distribution of rational and low-degree points.

In summary, the paper proves that "mild" singularities (bounded by a specific quadratic function of the degree) do not disrupt the fundamental arithmetic structure of plane curves regarding low-degree points, preserving the finiteness and geometric origin (lines) of such points.