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?