Serre's conditions under blow-ups, Blowup and normalization Suppose $X = \mathbb{Z}[x, y, z]/(f,g)$ is a 2-dimensional Cohen-Macaulay surface. In particular, $X$ satisfies Serre's condition $S_2$. Suppose it is irreducible, reduced but not normal.
$\bf{Question}$: Are there tools I could use to identify how the $S_2$-locus or Cohen macaulay locus behaves under various kinds of blow-ups. I imagine they are well-behaved under blowups along regular centers. Are there tools that allow up to say more in certain cases? Does Grothendieck duality somehow give a little information?
For example, suppose $p$ is an arbitrary height one prime of $X$. Is the blow up of $X$ at $p$ still an $S_2$ surface.
References will be greatly appreciated!
 A: There is some literature in the commutative algebra world that might be relevant, although I don't know it super-well. 
I don't think anything follows immediately from Grothendieck duality, especially for blowing up height-1 primes.
Say $R$ is a 2-dimensional and Cohen-Macaulay ring (equivalently S2).  Suppose that $p$ is an arbitrary height-1-prime. Since $R$ is 2-dimensional, the analytic spread of $p$ is at most $2$.  (See the book of Swanson-Huneke on integral closure, there are some subtleties with analytic spread in the case of a finite residue field -- be careful).  We have two cases.  
The analytic spread of $p$ is 2  Probably this is the more interesting case.  Suppose there happens to be only one ideal $J$, with two generators such that $J p = p^2$ (this isn't as strong as it might sound, it's a bit stronger than requiring that $J$ and $p$ have the same normalized blowup).  Additionally suppose that $R_p$ is regular (maybe this is too strong).  Then Theorem 3.1 in this paper by Santiago Zarzuela proves that the Rees Algebra of the ideal is Cohen-Macaulay, and hence so is the blowup.  There is also a lot of potentially relevant stuff in this paper of Huckaba and Huneke.  (You can look at the papers which cite it on mathscinet to find even more).

The analytic spread of $p$ is 1  In particular, then the blowup of $p$ is some finite integral extension of $R$ (in particular, the blowup is an affine scheme dominated by the normalization of $R$).  Since $R$ was S2, this implies that $p$ is a prime defining an irreducible component of the non-normal locus.  You want to keep the blowup S2...  I don't know in general if this is possible but I recall that some conditions for such blowups being normalizations appeared towards the end of this paper by Greco and Traverso.  Does your surface happen to be seminormal?
