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Let $p$ be a prime number and let $\bar{\mathbb F}_p$ be an algebraic closure of $\mathbb F_p$. Let $C$ be an irreducible singular projective curve over $\bar{\mathbb F}_p$.

Let $P$ be a singularity on $C$, suppose that we get $N(P)$ points after blowing up $P$. If $N(P) = 2$, then do we know which type of singularities $P$ could be?

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    $\begingroup$ By "blowing up" do you mean a single blowup of the maximal ideal of $P$ or do you mean whatever series of blowups it takes to make $P$ smooth? And in the first case, do you assume the blowup of $C$ is smooth? $\endgroup$
    – Will Sawin
    Commented May 23, 2022 at 17:01
  • $\begingroup$ Thanks for your question, I mean whatever series of blowups it takes to make 𝑃 smooth. $\endgroup$
    – Yachen Liu
    Commented May 24, 2022 at 9:52
  • $\begingroup$ $C$ will have two local irreducible components at $P$. For example the vanishing locus of $x^a- y^b$ has this form whenever $\gcd(a,b)=2\neq p$. $\endgroup$
    – Will Sawin
    Commented May 24, 2022 at 20:04

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