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Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $p>0$. Suppose that the genus of $X$ is >2. Let $Y$ be a non-trivial $\alpha_{p}$-torsor or $\mu_{p}$-torsor of $X$. Write $S$ for the set of singular points of $Y$.

Is the cardinality $\sharp S>2$ ?

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    $\begingroup$ Over the complex numbers, there are branched covers $X$ of elliptic curves $E$ branched over only one point. Some such model can be defined over some number field, and the reduction of that model over all but finitely many primes is a similar model in positive characteristic. The reduction of the elliptic curve $E$ admits purely inseparable isogenous covers by another elliptic curve $E'$. Defining $Y=E'\times_E X$ for the reductions, $Y$ is singular only over the unique branch point of $X\to E$. $\endgroup$
    – Jason Starr
    Commented Jul 11, 2016 at 19:14
  • $\begingroup$ Just to clarify, choose the branching of $X\to E$ to have one branch point in $E$ and one ramification point in $X$. $\endgroup$
    – Jason Starr
    Commented Jul 11, 2016 at 19:41

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