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.