Fix an integer $d>0$. Can we find a smooth curve in complex projective space ${\mathbb{CP}}^n$ that does not lie in the zero locus of any polynomial of degree $d$?

In terms of polynomials, this means the following. Can we find $n-1$ polynomials $p_1,\ldots, p_{n-1}$ such that for every polynomial $q$ of degree $d$, the system of equations $p_1=\cdots = p_{n-1}=q=0$ has only finitely many solutions? (I assume these polynomials $p_i$ are generic enough so that their intersection is smooth.)