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Suppose you have a family of high genus curves $f:X\to C$ over a nonsingular curve $C$, where $X$ is an integral normal 2-dimensional scheme and $f$ is flat and projective.

Suppose I know that $X_c$ is a smooth curve, where $c$ is a closed point in $C$. If I take the minimal resolution of singularities of $X$, is the fibre above $c$ still smooth?

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The minimal resolution of a (normal) surface $X$ consists in blowing-up the (finitely many) singular points of $X$, normalizing the new surface, again blowing-up the singular points in the new surface etc. The theorem is that this process must stop. You see that the minimal resolution doesn't affect the regular locus of $X$.

If $X_c$ is a smooth fiber of $f$, as $C$ is regular, then $X_c$ is contained in the regular locus of $X$, so it is not touched by the resolution of singularities. Therefore $X_c$ is a smooth fiber of the minimal resolution of singularities.

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