Let $R$ be an integral domain. Suppose that $X$⊂ $\Bbb P_R^3$ is givens by a single homogeneous equation $f(x,y,z)＝0$ with coefficients in $R$. Then, the generic fiber of $X$ is the variety defined by the same equation $f(x,y,z)＝0$.
In this case, $X$ and $X$'s generic fiber are the same dimension.
But does this hold in general ?
That is, $R$-Scheme $X$ and $X$'s generic fiber has the same dimension as $S$-scheme?
Thank you in advance.