Let $k[x_0,x_1,...,x_n]_d$ be a space of all forms (in other words, homogenous polynomials) of degree $d$ of variables $x_0, x_1,...,x_n$ over algebraically closed field $k$. Let's think of $k[x_0,x_1,...,x_n]_d$ as an affine space.
Of course, every such polynomial has a well defined zero-locus on a projective space $\mathbb P^n$, because it is homogenous.
Let $X\subset k[x_0,x_1,...,x_n]_d$ be a subset of polynomials, whose zero-loci are smooth subets of $\mathbb P^n$.
I would like to prove, that $X$ is open and dense subset of $k[x_0,x_1,...,x_n]_d$, but i have no idea how to do that. I guess it has something to do with the fact, that the set of smooth points of an algebraic set is dense and open, but i don't know how to use this fact in this setting.