Let $X$ be a complete toric variety and let $D \subset X$ be a toric divisor which is nef and big. Consider the graded algebra $R(D) = \bigoplus_k H^0(X, kD)$. This is a finitely generated graded algebra and we can consider the projective scheme $Y = Proj(R(D))$.
My question is what is Y?
Do I have some kind of contraction birational maps $X \to Y$?
Note that if $D$ is ample (very ample?) then we indeed have an isomorphism $Y \simeq X$.