surface with rational curve in the double locus

Question

I am interested in the existence of a surface $X$ over $\mathbb{C}$ with the following properties (or a reason for why one cannot exist):

• $X$ is slc (and not-normal)
• There is rational curve $C \subset X$ contained in the double locus (i.e. non-normal locus)
• The cotangent $\Omega^1_Y$ is ample, where $Y \to X$ is the normalization

Note that this last condition implies that $Y$ has no rational or elliptic curves. In particular, since the pre-image of the double locus under the normalization is a 2-to-1 cover branched over the pinch points, Riemann-Hurwitz gives that $C$ contains at least 6 pinch points.

The only examples I know of non-normal surfaces with a rational curve in the double locus containing $> 2$ pinch points have non-positive Kodaira dimension.

Answer 1

Here is a possible example building up on what I think was the idea of @Sasha.

Take $Y$ to be a non-isotrivial fibration of genus $g_1 \geq 3$ curves over a genus $g_2 \geq 2$ curve that contains a closed fiber $C'$ which is hyperelliptic. Such surfaces (Kodaira fibrations) have ample cotangent.

Consider the surface $X$ obtained by the pushout of $Y$ given by the quotient by the involution, i.e.

$\require{AMScd}$ \begin{CD} C' @>\sigma>> \mathbb{P}^1\\ @V V V @VV V\\ Y @>>> X \end{CD}

This is an slc surface $X$, and $Y$ has a finite map to $X$ that is birational and finite. Moreover one can show that $Y \to X$ is the normalization. By construction of the pushout the image of the hyperlliptic fiber $C'$ is a $\mathbb{P}^1$ inside the double locus of $X$.

Follow up question: Is this surface contained in the closure of the main component of the moduli space of surfaces of general type (i.e. is $X$ a smoothable surface in the sense of KSBA)?