Assume I have a singular algebraic surface $X$ over an algebraically closed field (characteristic zero if you want) which is singular in a *finite set of points*. I am looking for a condition as to the nature of these singularities which will guarantee that after blowing up $X$ in each of the singular points **once**, I will get a smooth surface.


In my fever dreams, you find a reference for such a statement from a text book with a very laid-out, comprehensible proof. Any reference, however, is welcome. Thanks!



Resolution of "nice" and zero-dimensional singularities on a surface
The condition you are looking for has a name: *absolute isolatedness*.


In fact, a surface singularity is called *absolutely isolated* if it can be resolved by using only quadratic transformations centered at reduced points, that is, no normalizations will be required.


In general, isolated surface singularities are not absolutely isolated. But, for instance, *rational singularities* are so.


Googling "absolutely isolated $2$-dimensional singularities" you can find a lot of references. For example, Tyurina's paper [Absolute isolatedness of rational singularities and triple rational points](https://doi.org/10.1007/BF01075685 "Funkts. Anal. Prilozh. 2, No. 4, 70-81 (1968). zbMATH review at https://zbmath.org/0176.50804")  can be surely useful.


