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Fix a prime number $p$.

Let $F$ be the geometrically connected Fermat curve of exponent $p$ over a number field $K$, i.e., $F$ is given by the equation $x^p+y^p = z^p$. Consider the same equation over the ring of integers $O_K$. Note that this scheme, denoted by $\mathcal{F}$, is normal. Note that $\mathcal{F}$ is smooth over the complement of the $p$ in $O_K$.

How does one obtain the minimal resolution of singularities of $\mathcal{F}$?

The singular fibre is a genus zero curve $(x+y-z)^p =0$ of multiplicity $p$.

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    See McCallum: The degenerate fibre of the Fermat curve. Progr. Math., {\bf 26}, Birkhäuser, 1982 (when $K=\mathbb Q(\xi_p)$). I don't think you can do it over an arbitrary number field.2012-02-22

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