Let $A_d=\mathbb{C}[x_1,\dots,x_n]_d$ denote the vector space of homogeneous polynomials of degree d. And assume $U\subset A_d$ is a subspace of codimension $d-1$ such that there is no point in $\mathbb{P}^{n-1}$ where every form in $U$ vanishes ($\mathcal{V}(U)=\emptyset$).
After applying a general change of coordinates to $x_1,d\dots,x_n$ we can assume that $\dim (U\cap \mathbb{C}[x_1,x_2]_d)=2$ for dimensional reasons. Let $f_1,f_2$ span this intersection.
I would like to know if the zero set of $f_1,f_2$ in $\mathbb{P}^1$ is empty if we chose generic coordinates (or probably equivalent there exists a change of coordinates such that the zero set of $f_1,f_2$ is empty).
It feels like this should easily be true but since we just intersect in degree d this doesnt seem to be any kind of projection.
Equivalently we can intersect $U$ with $\mathbb{C}[l_1,l_2]_d$ for two generic linear forms instead of applying the coordinate change to $U$.
