Degeneration of curves inside a family of surfaces

Question

We are interested in understanding how a higher genus curve in a smooth surface of general type can degenerate into the non-normal locus of the limit of a degeneration of surfaces. More precisely:

Let $\mathscr{X} \to B$ be a family of surfaces over a smooth curve $B$ so that the general fiber $X_\eta$ is a smooth surface, the special fiber $X_0$ has semi log canonical singularities, and the canonical sheaf $\omega_{X_b}$ of every fiber is ample. This corresponds to a map from $B$ to the KSBA moduli space of stable surfaces.

Assume that the double locus of $X_0$ (i.e. the non-normal locus of $X_0$) has an irreducible component which is a rational curve $D$.

Question 1. Let $C_\eta \subset X_\eta$ be a smooth curve whose genus is $\geq 2$, and let $\mathscr{C} \to B$ be the family of curves obtains as the closure of $C_\eta$ inside $\mathscr{X}$. Denote by $C_0$ the special fiber $\mathscr{C} \cap X_0$. Can $C_0$ contain $D$ as an irreducible component?

We are additionally interested in how this can generalize to the case of pairs. In this setting, now suppose there is a divisor $\Delta \subset \mathscr{X}$ so that the pair $(X_b, \Delta_b)$ has slc singularities and the log canonical sheaf $\omega_{X_b} + \Delta_b$ is ample for every fiber, where $\Delta_b = X_b \cap \Delta$. Equivalently now there is a map from $B$ into the moduli space of stable pairs of dimension 2.

Question 2. If Question 1 is true, is it possible that $\Delta$ intersects $D$ (the component of $C_0$ inside the double locus of $X_0$) transversally at a smooth point of $C_0$ and that $\Delta \cap C_\eta \neq \emptyset$?

Answer 1

Let $S$ be the "usual" pinch point surface defined by $x^2t=y^2z$ in $\mathbb P^3$ and $T\subseteq \mathbb P^3$ an arbitrary general surface of degree $d-3\geq 2$. Let $X_0=S\cup T$. Note that then $\deg X_0=d$. Let $\ell\subseteq S$ be the double line and $H\subseteq \mathbb P^3$ a general surface of sufficiently high degree such that $\ell\subseteq H$ and let $C_0=H\cap X_0$.

Since $T$ is general, it is smooth and intersects $S$ transversally, so I think $X_0$ is even semi-smooth, but for sure slc. Now choose a smoothing of $X_0$ as a degree $d$ surface in $\mathbb P^3$, i.e., let $f:\mathscr X\to B$ be a family of degree $d$ surfaces in $\mathbb P^3$ parametrized by a smooth curve such that $X_0$ is a fiber of $f$ and the general fiber $X_\eta$ is smooth. Since $d\geq 5$, $\omega_{X_b}$ is an ample line bundle for every $b\in B$.

Now let $\mathscr C$ be the collection of curves $C_b=H\cap X_b$ for $b\in B$. The general member will be smooth and $C_0$ contains $\ell$ as an irreducible component. So this seems to answer your first question.

For the second, take the same example, but add a $\Delta$, i.e., let $\Gamma\subseteq \mathbb P^3$ be a high degree general surface and let $\Delta_b:=\Gamma\cap X_b$ and $\Delta\subseteq \mathscr X$ the preimage of $\Gamma$ in $\mathscr X$. Then $\Gamma$ and hence $\Delta$ intersects $\ell$ transversally at a point which is smooth on $C_0$ and since $\Gamma$ is ample, $\Delta\cap C_\eta\neq\emptyset$.