Let $d$ and $n$ be natural numbers. Let $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle\in \mathbb{Z}[\boldsymbol{x}]$ be a homogeneous polynomial in $n$ variables of degree $d$ with integer coefficients $\boldsymbol{a}$, where $\langle\cdot,\cdot\rangle$ denotes the inner product, and $\nu_{d,n}: \mathbb{R}^n\rightarrow \mathbb{R}^N$ denotes the Veronese embedding with $N=\binom{n+d-1}{d}$. Consider a variety $V_{\boldsymbol{a}}$ in $\mathbb{P}^{n-1}$, defined by $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle=0.$ In this paper, we examine a set of these varieties, defined by $$\mathbb{V}^{P}_{d,n}(A)=\{ V_{\boldsymbol{a}}\in \mathbb{P}^{n-1}|\ P(\boldsymbol{a})=0,\ \|\boldsymbol{a}\|_{\infty}\leq A\},$$ where $P\in \mathbb{Z}[\boldsymbol{x}]$ is a non-singular homogeneous polynomial of degree $k\geq 2$. We confirm that when $d\geq 4$, $n$ is sufficiently large in terms of $d$, and $k\leq d,$ the proportion of elements in $\mathbb{V}^{P}_{d,n}(A)$, which do not satisfy the Hasse principle, converges to $0$ as $A\rightarrow \infty$. We make explicit a lower bound on $n$ guaranteeing this conclusion, and in particular, show that when $d\geq 14$ it suffices to take $n\geq 32d+17$. In order to establish it, we mainly use the Hardy-Littlewood circle method., Comment: 92 pages, all comments are welcome!