Consider $A$ a commutative ring, and $I\leq A$ an ideal. Let $X=\mathrm{Spec}(A)$ and $Z=\mathrm{Spec}(A/I)$.
Consider the blowing-up algebra \begin{equation} \mathrm{Bl}_I(A):=\bigoplus_{i=0}^\infty I^i=A\oplus I\oplus I^2\oplus\cdots \end{equation} which is graded such that $I^i$ has degree $i$.
The affine blowing-up algebra is, for $a\in I$ of degree 1 in $\mathrm{Bl}_I(A)$, the homogeneous localisation \begin{equation} A\Big[\frac{I}{a}\Big]:=\Big(\mathrm{Bl}_I(A)\Big)_{(a)}. \end{equation}
The blowing-up is $\mathrm{Bl}_ZX:=\mathrm{Proj}\big(\mathrm{Bl}_I(A)\big)$ and it is covered by \begin{equation} \big(\mathrm{Bl}_ZX\big)_a:=\mathrm{Spec}\Big(A\Big[\frac{I}{a}\Big]\Big). \end{equation} for $a\in I$ of degree 1 in $\mathrm{Bl}_I(A)$.
My questions are:
- Is every $\big(\mathrm{Bl}_ZX\big)_a\to X$ surjective, when $\big(\mathrm{Bl}_ZX\big)_a\ne\emptyset$?
- Is $\mathrm{Bl}_ZX\to X$ surjective when it is not empty?
- If not, what about the images can we say?
Best wishes.