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  1. Let $\mathcal{X}$ be a flat family of (proper) algebraic curves. If generic fibers in $\mathcal{X}$ are non-singular of genus $g$, then the geometric genus (i.e. genus of the desingularizations) of special fibers must be $\leq g$. Does this inequality of the geometric genus remain valid also in the case that generic fibers of $\mathcal{X}$ are singular?

  2. More specifically, if generic fibers of $\mathcal{X}$ are singular rational curves, then can the special fiber (assume it is reduced and irreducible) of $\mathcal{X}$ be non-rational?

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If the base of the family is a quasi projective variety, by taking sections and base change you can reduced to the case where $\mathcal X$ is a surface fibered over a smooth curve $B$. Then one can normalize $\mathcal X$ and then solve the remaining singularities. In this way one gets a new suface $\mathcal X'$ fibered over the same base $B$. The general fiber $F$ of $\mathcal X'$ is the normalization the general fiber of $\mathcal X$ and the special fiber of $\mathcal X'$ contains a component $D$ that maps onto the original special fiber $X_0$ (I'm assuming, as in the question, that $X_0$ is reduced and irreducible). Now one can use the adjunction formula on the smooth surface $\mathcal X'$ and Zariski's lemma (Barth-Peters Van de Ven , p.90 of the old edition), to show that $p_a(D)\le p_a(F)=g(F)$.

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