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Can you cover a non-singular algebraic variety by pairwise disjoint singular closed subvarieties? Varieties are over an algebraically closed field of characteristic other than $2$ and $3$.

In characteristic $2$ and $3$ this is possible via quasi-elliptic surfaces as pointed out by abx.

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  • $\begingroup$ How are you turning these subvarieties into a partition using axiom of choice? $\endgroup$
    – Wojowu
    Commented Sep 17, 2021 at 16:39
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    $\begingroup$ In characteristic zero such a map does not exist, because the generic fiber is regular and therefore smooth. They do exist in characteristic $>0$: look for quasi-elliptic surfaces. $\endgroup$
    – abx
    Commented Sep 17, 2021 at 16:42
  • $\begingroup$ There is really nothing special about characteristic $2$ or $3$: if you allow fibres of higher arithmetic genus, there should exist smooth surfaces in all positve characteristics mapping to a curve with all (geometric) fibres singular. $\endgroup$
    – naf
    Commented Sep 18, 2021 at 3:55

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