Relative dualizing sheaf of the family $\varphi: C\rightarrow \mathbb{A}^1$ where $C$ is the vanishing locus of $xy=t^k$ in $\mathbb{A}^3$.

Question

This is an exercise from the book Moduli of Curves.

Let $C$ be the vanishing locus of $xy= t^k$ in $\mathbb{A}^3$. Define a morphism $$\varphi:C\rightarrow \mathbb{A}^1, (x,y,t)\mapsto t.$$ Then $C$ is a family of curves over $\mathbb{A}^1$. Show that the relative cotangent bundle of $\varphi$ on $C^* = C\backslash \{(0,0,0)\}$ extends to $C$, and that the sections on $\varphi^{-1}(0)$ satisfy the residue condition.

I learned how to resolve a node in the case of a curve. But here any neighborhood of the point (0,0,0) is a surface. Could someone give me a hint? Or a detailed solution is much more appreciated :)