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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!

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  • $\begingroup$ I guess this will be satisfied for cone singularities (at least those with respect to a very ample divisor inducing a projectively normal embedding). But as discussed below, if you really expect a single blow-up of a closed point to resolve your singularities, this will be rare. $\endgroup$
    – Karl Schwede
    Commented Jul 29, 2011 at 19:30
  • $\begingroup$ Indeed, I want to blow up only once in each point, I edited my post to make that more clear. It is not a problem if it is rare, just as long as the situation can be classified in some way. $\endgroup$
    – Jesko Hüttenhain
    Commented Jul 30, 2011 at 0:35

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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 can be surely useful.

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    $\begingroup$ I think your comment depends on your interpretation of what the OP means by "blowing up in each singular point". Absolutely isolated singularities are singularities which can be resolved by a sequence of blowups of reduced points without blowing up or normalizing any curves (or nonreduced schemes). But my take on the OP's question is he wants to know when a surface singularity germ is resolved by blowng up a single reduced point. $\endgroup$
    – Jason Starr
    Commented Jul 29, 2011 at 18:03
  • $\begingroup$ Well, maybe you are right. But then the condition is much more restrictive. It is not even satisfied by Rational Double Points different from $A_1$. Among hypersurface singularities, I guess it is only verified by ordinary $n$-ple points... $\endgroup$
    – Francesco Polizzi
    Commented Jul 29, 2011 at 18:35
  • $\begingroup$ It is also satisfied by the germ of the origin in the hypersurface $V(xy-z^3)$. That is not an ordinary $n$-uple point. $\endgroup$
    – Jason Starr
    Commented Jul 29, 2011 at 19:11
  • $\begingroup$ True. This is an $A_2$ singularity and in fact my comment above is inaccurate. The blow up af an $A_n$-point introduce a further $A_{n-2}$-point, so both $A_1$ and $A_2$ are risolved by a single blow-up. $\endgroup$
    – Francesco Polizzi
    Commented Jul 29, 2011 at 19:25
  • $\begingroup$ I felt like I should add a comment here as well: Jason Starr's guess is right, I want to blow up no more than once in each singular point. $\endgroup$
    – Jesko Hüttenhain
    Commented Jul 30, 2011 at 0:42

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