Bialgebraic varieties of the Gamma function

We characterize the bialgebraic varieties of the $\mathrm{\Gamma }$ function, that is, if $V,W\subseteq {\mathbb{C}}^n$ are irreducible affine algebraic variety which satisfy $\dim V=\dim W$ and $\mathrm{\Gamma }(V)\subseteq W$, then the equations defining $V$ (and hence also $W$) either give an equality between coordinates, or set some coordinates to be constant.