Davenport's geometric-condition conjecture for cubic forms in ten variables

Let $C(\mathbf{x})=\sum_{i,j,k}c_{ijk}x_ix_jx_k$ be a cubic form in $n$ variables with symmetric integer coefficients. Define

$$B_i(\mathbf{x},\mathbf{y})=\sum_{j,k=1}^n c_{ijk}x_jy_k,$$

and let $M(\mathbf{x})$ be the matrix with entries $M(\mathbf{x})_{jk}=\sum_{i=1}^n c_{ijk}x_i$. Write $D(\mathbf{x})=\det M(\mathbf{x})$ and $r(\mathbf{x})=\operatorname{rank}M(\mathbf{x})$. Davenport's Geometric Condition is that, for every integer $r$ with $0\leq r\leq n$,

$$\#\{\mathbf{x}\in\mathbb{Z}^n:\|\mathbf{x}\|_\infty\leq P,\ r(\mathbf{x})=r\}\ll P^{r+\varepsilon}.$$

Davenport's geometric-condition conjecture. If $C$ satisfies Davenport's Geometric Condition, then the asymptotic formula for the number of solutions to $C(\mathbf{x})=0$ holds with positive singular integral and singular series, $\mathfrak{I},\mathfrak{S}>0$, as soon as $n\geq 10$. In particular, $C(\mathbf{x})=0$ has a non-trivial integer solution.

The preceding results establish the corresponding assertion for $n\geq14$, while the paper proves absolute convergence and positivity of the singular series under the geometric condition for $n\geq10$. The conjectural remaining step is the full asymptotic formula in the range $n\geq10$.