# Minimal resolution of singularities of Fermat curve

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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Have you referred to this article ?, it has a singularities for fermat hypersurface, and its reduced to your case . Thank you. –  Iyengar Jan 28 '12 at 11:11
@Iyengar. Thnx for the article. It seems to consider more general Fermat equations with singularities. Note that the curve I consider is smooth on the generic fibre. The article seems to be interested in the case where this is not the case. Maybe I'm wrong? –  Ali Jan 28 '12 at 15:08
I have already told that its about fermat hypersurfaces, but it may be useful if you reduce it to your case @Ali –  Iyengar Jan 29 '12 at 4:11
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. –  user18119 Feb 22 '12 at 0:37