Fix $d\in\mathbb{N}$. Let $S=\mathbb{A}^1_{\mathbb{C}}$ be the affine line over the complex numbers and let $X\subset\mathbb{P}^n_S$ a family of smooth hypersurfaces of degree $d$ over $S$. By this I mean that $X$ is a closed subscheme of $\mathbb{P}^n_S$ such that the fiber $X_s$ of $X\to S$ over any closed point $s$ of $S$ is a smooth hypersurface of degree $d$ in $\mathbb{P}^n_{\mathbb{C}}$.
I say that the family $X$ is essentially constant if for any two closed points $s,s'$ of $S$ there is a linear change of coordinates of $\mathbb{P}^n_{\mathbb{C}}$ that sends $X_s$ to $X_{s'}$.
My question is: Is there a family of smooth hypersurfaces of degree $d$ over $S$ which is not essentially constant?
If such a family would exist, then there would be a non-constant polynomial map $\mathbb{C}\to\mathbb{C}[x_0,\ldots,x_n]_d$ whose image does not intersect the discriminant which is not just given by applying an element of $\textrm{GL}_{n+1}(\mathbb{C}[t])$ to a certain polynomial. I was not able to construct such a map even for $n=1$ and $d=4$ which is probably the first non-trivial case.