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We discuss to what extent the local techniques of resolution of singularitiesover fields of characteristic zero can be applied to improve singularities ingeneral. For certain interesting classes of singularities, this leads to anembedded resolution via blowing ups in regular centers. We illustrate this forgeneric determinantal varieties. The article is partially expository and isaddressed to non-experts who aim to construct resolutions for other specialclasses of singularities in positive or mixed characteristic.

Partial local resolution by characteristic zero methods