Manin's conjecture for the chordal cubic fourfold

TL;DR

The paper proves the thin-set version of Manin–Peyre for the chordal cubic fourfold $X$, the determinantal secant variety of the Veronese surface, by identifying $X$ with the symmetric square $\text{Sym}^2(\mathbb{P}^2_\mathbb{Q})$ and reducing the counting problem to Schmidt’s results on quadratic points via the height $H(\mathbf{x}) = \max\{|x_0|,\dots,|x_{12}|\}^3$ and discriminants $\Delta_{ij}=x_{ij}^2-4x_ix_j$. It derives sharp asymptotics for the rational points partitioned into $X_0$, $X_+$, and $X_-$ with $N(X_0,H,B) = c_0 B\log B + O(B)$ and $N(X_\pm,H,B) = c_\pm B\log B + O(B(\log B)^{1/2})$, expressing constants via Schmidt’s data and archimedean volumes. Through desingularization $h:\widetilde{X}\to X$, the study situates the problem within the Manin–Peyre framework for thin sets, identifying $\widetilde{X}$ with $\text{Hilb}^2(\mathbb{P}^2_\mathbb{Q})$, and showing that the sum $c_+ + c_-$ agrees with Peyre’s constant by computing local densities and Tamagawa measures. The results connect the arithmetic of the chordal cubic to the geometry of Hilbert schemes and Severi varieties, providing explicit constants and densities for a nontrivial higher-dimensional example.

Abstract

We prove the thin set version of Manin's conjecture for the chordal (or: determinantal) cubic fourfold, which is the secant variety of the Veronese surface. We reduce this counting problem to a result of Schmidt for quadratic points in the projective plane by showing that the chordal cubic fourfold is isomorphic to the symmetric square of the projective plane over the rational numbers.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3
• Definition 3.1
• Lemma 3.2
• proof