Let $K$ be an algebraically closed field, and let $X, Y \subset K^{n^2}$ be algebraic sets such that $X$ is an irreducible hypersurface and $Y$ is irreducible of dimension $n$. (Assume $n>1$).
Does there exists some $v \in K^{n^2}$ such that $Y+v=\{ y+v \colon y\in Y\}$ is disjoint from $X$? Is the set of such $v \in K^n$ Zariski open?
Over $\mathbb{C}$ this can be done using topological/analytic tools such as Sard's theorem, but I am not sure if this is true over any $K$ or if we need any condition on $X$, on $Y$, or on $n$?