Resolution of singularities in characteristic p
Hironaka proved in 1964 that in characteristic zero every singular variety can be replaced by a smooth one, and won a Fields Medal for it. In positive characteristic the method breaks, and the question is settled only up to dimension 3. Many results in arithmetic geometry work around it with alterations, de Jong's weaker substitute.
Posed After Hironaka's 1964 proof in characteristic 0.
Sources
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