The local solubility for homogeneous polynomials with random coefficients over thin sets

Let $d$ and $n$ be natural numbers greater or equal to $2$. Let $⟨\mathit{a},{\nu }_{d,n}(\mathit{x})⟩\in \mathbb{Z}[\mathit{x}]$ be a homogeneous polynomial in $n$ variables of degree $d$ with integer coefficients $\mathit{a}$, where $⟨\cdot ,\cdot ⟩$ denotes the inner product, and ${\nu }_{d,n}:{\mathbb{R}}^n\to {\mathbb{R}}^N$ denotes the Veronese embedding with $N=\left(\genfrac{}{}{0px}{}{n+d-1}{d}\right)$. Consider a variety $V_{\mathit{a}}$ in ${\mathbb{P}}^{n-1}$, defined by $⟨\mathit{a},{\nu }_{d,n}(\mathit{x})⟩=0.$ In this paper, we examine a set of these varieties defined by $${\mathbb{V}}_{d,n}^P(A)=\{V_{\mathit{a}}\subset {\mathbb{P}}^{n-1}\mathrm{\mid }P(\mathit{a})=0,\mathrm{\parallel }\mathit{a}{\mathrm{\parallel }}_{\mathrm{\infty }}\le A\},$$ where $P\in \mathbb{Z}[\mathit{x}]$ is a non-singular form in $N$ variables of degree $k$ with $2\le k\le C(n,d)$ for some constant $C(n,d)$ depending at most on $n$ and $d$. Suppose that $P(\mathit{a})=0$ has a nontrivial integer solution. We confirm that the proportion of varieties $V_{\mathit{a}}$ in ${\mathbb{V}}_{d,n}^P(A)$, which are everywhere locally soluble, converges to a constant $c_P$ as $A\to \mathrm{\infty }\mathrm{.}$ In particular, if there exists $\mathit{b}\in {\mathbb{Z}}^N$ such that $P(\mathit{b})=0$ and the variety $V_{\mathit{b}}$ in ${\mathbb{P}}^{n-1}$ admits a smooth $\mathbb{Q}$-rational point, the constant $c_P$ is positive.