Let $Y$ be a regular surface (therefore a proper, $2$-dimensional scheme over base field $k$) and $y \in Y$ a rational closed point (so $k(y)=k$).
Denote by $I$ the ideal sheaf corresponding to $y$ interpreted as closed subscheme.
Performing the blowup at $y$ we obtain $b: X=Bl_y(Y):= Proj(\oplus_n I^n) \to Y$.
My question is how are the global sections $H^0(Y, O_Y), H^0(X, O_X)$ related to each other?
Indeed higher cohomologies are depending only on derived image sheaf $R^1f_* O_X$ via five term Leray Serre sequence
$$0 \to H^1(Y, f_*O_Y) \to H^1(X, O_X) \to H^0(Y, R^1f_* O_X) \to H^2(Y, f_*O_X) \to H^2(X, O_X) $$
The cruical point is what happens with global sections $H^0(-)$.
My motivatating example was the blowup $X:=Bl_{(0,0)}(\mathbb{A}^2_k)$ of affine surface $\mathbb{A}^2_k$ over basefield $k$ at $(0,0)$ (corresponds to max ideal $(x,y)$). Denote $R:=k[x,y), \mathfrak{a}=(x,y)$.
In order to avoid ambiguity we identify $\oplus_n I^n := R[\mathfrak{a} \cdot T]$ with bookkeeping undeterminant $T$.
Concrete calculations using Cech cohomology provide $H^0(Y, O_Y)=H^0(X, O_X)$.
So my question is under which conditions the blowing up of points preserve the property $H^0(Y, O_Y)=H^0(X, O_X)$?
What are sufficient conditions and why? Is it neccessary that the point $y$ has to be regular or/ and rational? In what dimension are the blowups with this property performed?
Are there recomendable sources which treat this question?
Best Answer
The best general answer I know is that in the case when $Y$ is normal and you're blowing up an ideal sheaf with proper support in every irreducible component, the global sections are the same.
Observation: in a normal scheme, no point is on multiple irreducible components, as all the local rings of a normal scheme are integrally closed domains. So it suffices to treat the case of $Y$ irreducible.
Proof of statement: We'll show that the pushforwards of the structure sheaf is the structure sheaf. Let $\pi:X\to Y$ be the blowup map. This is proper, so $\pi_*\mathcal{O}_X$ is an $\mathcal{O}_Y$ algebra which is finite as a module. But the quotient fields of $X$ and $Y$ are the same at every point because the blowup map is birational (we chose our sheaf of ideals to have proper support, so the blowup map is an isomorphism on the dense open complement of the support of the sheaf of ideals). So at any point, $\mathcal{O}_{Y,y}\subset \pi_*(\mathcal{O}_X)_y$ is an integral extension of an integrally closed ring, and thus an equality. So we have that $\pi_*\mathcal{O}_X = \mathcal{O}_Y$.
To apply this to your stated example at hand, you're blowing up a proper subvariety of a normal variety and you can use exactly this statement.
If you're interested in seeing more arguments like the above, this is the sort of argument that shows up around the various versions of Zariski's main theorem, Zariski's connectedness theorem, and Stein factorization (although different sources will quote from one of something like ten different versions of ZMT - they're all essentially the same, but sometimes getting from one statement to the other can be a little bit of work).
In the case where $Y$ is not normal, interesting things can happen: consider using blowups to resolve the singularities of a curves. Another observation is that if you're working with proper varieties, blowing up won't change the global sections unless you alter the number of connected components. I would be surprised if there were more general things to say in the non-normal case - you'd probably have to do more specific casework and get your hands dirty there.