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Why is it important/useful to resolve singularities generally, and in particular in algebraic geometry? I understand that one gets a nicer geometry, but how does it help one understand the geometry of the variety with singularities that one starts out with? Can this added understanding be gained without 'resolving singularities'? What, if anything, does the process tells me about isomorphism class of the variety, over and above its birational equivalence class?

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    $\begingroup$ Read Theorem 1 of webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/… (which concerns smooth varieties!) and then the top of page 97. Historically, this was the first (rather dramatic) use of resolution as a tool to do things (sort of as a black box) even for the study of smooth varieties. Also look at pp. 120-121 of the Grothendieck-Serre correspondence (which shows that Grothendieck also wondered the same thing). $\endgroup$
    – nfdc23
    Commented Sep 13, 2017 at 19:33

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