Prime ideals and sagbi bases I'm trying to understand the following passage in Combinatorial Commutative Algebra by Miller and Sturmfels:

If $R$ is any subalgebra of a polynomial ring that possesses a finite sagbi basis, then this sagbi basis defines a flat degeneration from $R$ to its initial algebra in($R$). The initial algebra is generated by monomials, so it corresponds to a toric variety. Hence, geometrically, a finite sagbi basis provides a flat family connecting the given variety Spec($R$) to the affine toric variety Spec(in($R$)) - Chapter 14.3, Page 281.

I'm sadly a bit naive about algebraic geometry so my main question is about the last sentence: Does it imply that if $P$ is a prime ideal of $R$, then in$(P)$ is a prime ideal of in$(R)$? 
I haven't been able to prove that statement using just the usual definitions of the terms involved, and, honestly, it seems too good to be true, but hopefully someone out there has some insight.
If it's not true in general, are there any known non-trivial cases in which it holds?

No. What the theorem says is the following: suppose you have any parametrically presented $k$-algebra, that is, it is of the form $k[f_1,\cdots,f_k]$ for some polynomials $f_i$, and the initial terms of these polynomials generate the initial algebra $k[in(f_1),\cdots,in(f_k)]$.
Since the ring $k[in(f_1),\cdots, in(f_k)]$, is generated by monomials, the corresponding ideal (in $k[t_1,\cdots,t_k]$) is a toric prime ideal.
Notice that the statement says nothing about any ideals (which presupposes an embedding somewhere). For the sake of argument, suppose you have an embedding, the most natural one being the one given by the surjection $k[t_1,\cdots,t_k] \to k[f_1,\cdots,f_k]$. Then both ideals $I(R)$ and $I(in(R))$ are prime, since the quotient rings are integral domains.
