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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 the pointreduced 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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The condition you are looking for is that all our singularities are absolutely isolatedhas a name: absolute isolatedness.

In fact, by definition, a normal surface singularity is called absolutely isolated if it can be resolved by using only quadratic transformations centered at the point, that is, no normalization is needednormalizations will be required.

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

Gooling

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.

2 added 307 characters in body

The condition you are looking for is that all our singularities are absolutely isolated.

In fact, by definition, a normal surface singularity is called absolutely isolated if it can be resolved by using only quadratic transformations centered at the point, that is, no normalization is needed.

In general, normal surface singularities of surfaces are not absolutely isolated. But, for instance, rational singularities are so.

Gooling "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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