On the density of rational lines on diagonal cubic hypersurfaces

Technical Summary: On the Density of Rational Lines on Diagonal Cubic Hypersurfaces

Problem Statement
This paper addresses the quantitative problem of counting rational lines contained within diagonal cubic hypersurfaces. Specifically, it considers the hypersurface defined by the equation $\sum_{i=1}^s c_i z_i^3 = 0$, where $c_i \in \mathbb{Z} \setminus \{0\}$. A rational line on this variety is represented by a pair of integer vectors $\mathbf{x}, \mathbf{y} \in \mathbb{Z}^s$ such that the line $\mathbf{x} + t\mathbf{y}$ lies entirely on the hypersurface. This condition translates to a system of four homogeneous equations:
$$\sum_{i=1}^s c_i x_i^3 = \sum_{i=1}^s c_i x_i^2 y_i = \sum_{i=1}^s c_i x_i y_i^2 = \sum_{i=1}^s c_i y_i^3 = 0.$$
The primary objective is to establish an asymptotic formula for $N_s(X)$, the number of such solution pairs $(\mathbf{x}, \mathbf{y})$ with coordinates bounded by $X$ (i.e., in $[-X, X]^{2s}$).

Methodology
The author employs the Hardy-Littlewood circle method to derive the asymptotic formula. The integral representation of $N_s(X)$ is decomposed into contributions from "major arcs" ($\mathfrak{N}_\delta$) and "minor arcs" ($\mathfrak{n}_\delta$).

1. Major Arcs: The contribution from the major arcs is shown to yield the main term of the asymptotic formula, involving a singular series $\mathfrak{S}$ and a singular integral $\mathfrak{J}$. This part relies on established results from the literature (specifically Zhao [Zha16] and Parsell [Par09]), which confirm that for $s \ge 16$, the major arc integral provides the expected main term $\mathfrak{S}\mathfrak{J}X^{2s-12}$.

2. Minor Arcs: The core technical innovation of the paper lies in bounding the integral over the minor arcs. The author aims to show that for $s \ge 19$, the minor arc contribution is of a lower order than the main term. The proof of this bound (Lemma 2.2) integrates three specific components:

   • Sharp Estimates for Parsell-Vinogradov Systems: The proof utilizes recent sharp estimates for the cubic Parsell-Vinogradov system (specifically [BDG17] and [GZ19]), which provide strong bounds for mean values of exponential sums associated with diagonal systems.
   • Multidimensional Shifting Variables: The paper generalizes the "shifting variables" argument, originally introduced by Wooley [Woo12], to a multidimensional setting. This technique involves defining auxiliary exponential sums $G(\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\theta})$ involving Veronese embeddings of degree 1, 2, and 3. By shifting the integration variables and utilizing orthogonality, the author relates the mean value of the original exponential sum $F(\boldsymbol{\alpha})$ to the mean values of these auxiliary sums.
   • Pruning Argument: A pruning argument is applied to decompose the minor arcs into dyadic intervals. This allows the author to apply the multidimensional shifting variables argument on specific subsets of the minor arcs where the exponential sums can be effectively bounded.

Key Contributions and Results
The paper establishes the following main theorem:

Theorem 1.1: For $s \ge 19$, the number of solutions satisfies the asymptotic formula:
$$N_s(X) = \sigma X^{2s-12} + O(X^{2s-12-\delta}),$$
where $\sigma > 0$ is a constant depending on the coefficients $c_i$, and $\delta > 0$.

This result improves upon the previously best-known bound of $s \ge 21$ established by Zhao [Zha16]. Prior to this, Parsell [Par12] had required $s \ge 29$, and earlier works by Parsell [Par00] required $s \ge 55$ for a lower bound.

The proof relies heavily on Lemma 3.1, which provides a bound for the mean value of the exponential sum $|F(\boldsymbol{\alpha})|^{2s}$ over specific major arc subsets $M_l(H)$. This lemma serves as the engine for the minor arc estimation in Lemma 2.2, demonstrating that the minor arc integral is bounded by $O(X^{2s-12-\eta})$ for $s \ge 19$.

Significance and Claims
The paper claims that the improvement from $s \ge 21$ to $s \ge 19$ is achieved by combining the multidimensional shifting variables argument with a pruning technique and leveraging the sharp estimates for the Parsell-Vinogradov system.

The author notes that the methodology presented is not limited to cubic hypersurfaces. It is suggested that applying these arguments, combined with further generalized shifting variables techniques and sharp estimates for higher-degree Parsell-Vinogradov systems, could improve existing bounds for the number of linear spaces on diagonal hypersurfaces defined by $\sum_{i=1}^s c_i z_i^k = 0$ for degrees $k \ge 3$. However, the paper does not explicitly perform these extensions for $k > 3$, focusing solely on the cubic case ($k=3$).

The work is framed as a refinement of the quantitative theory of rational lines on algebraic varieties, building directly on the foundational inductive approaches initiated by Brauer and Birch and the subsequent quantitative developments by Brandes, Parsell, Wooley, and Zhao.