Let $R$ be a finitely generated positively graded algebra and domain with $X=Proj(R)$. We construct a sequence of $d=\dim(X)$ flat degenerations (over affine line and with reduced and irreducible fibres) that degenerate $X$ to a (not necessarily normal) projective toric variety. As a corollary, we deduce that if $H_R(m)$ is the Hilbert function of $R$, then there is an integer $n>0$ such that the function $m \mapsto H_R(nm)$ is the Hilbert function of a graded finitely generated lattice semigroup. Speaker(s): Devlin Mallory (UM)