Let $\mathbb{P}^2_\mathbb{K}$ be the projective space, with $\mathbb{K}=\bar{\mathbb{K}}$. Let $n\geq 7$ be odd and $f_1, \dotsc, f_n$ be $n$ general forms of degree $\frac{n-1}{2}$ in $\mathbb{K}[x_0,x_1,x_2]$ and let $\mathbb{V}$ be the Veronese surface projected in $\mathbb{P}^{n-1}$ obtained as the image of the regular map $[f_1:f_2:\dotso:f_n]:\mathbb{P}^2 \rightarrow \mathbb{P}^{n-1}$.

Define $\mathscr{H}$ to be the union of the irreducible components in the Hilbert scheme containing the general choices of $\mathbb{V}$ (i.e. the general choices of $f_i$).

Can I show that the general point in $\mathscr{H}$ is indeed a projected Veronese surface? In other words, that the rational map $$ H^0\left(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}\left(\frac{n-1}{2}\right)\right)^n \dashrightarrow \mathscr{H} $$ is dominant?