I have a (possibly very basic) question about differential forms on nodal curves. After reading Witten's survey "Two-dimensional gravity and intersection theory on moduli space", I am confused by the discussion starting at Equation 2.67 and following.
Let $C$ a nodal genus $0$ curve. Its normalisation $\tilde{C}$ is the disjoint union of $\mathbb{P}^1$s. The dualising sheaf of $C$ is then the space of differential forms on $\tilde{C}$ that can only have poles and these poles can at worst be simple poles at the pre-images of nodes, where the residues of the pre-images of a given node must match.
Now, we consider an $n$-marked smooth curve $(\mathbb{P}^1,x_1,\dots,x_n)$ and a differential form $$dx\left(\frac{1}{x-x_{n-1}}-\frac{1}{x-x_n}\right),$$ i.e. the form with only poles at $x_{n-1}$ and $x_n$; residues $1$ and $-1$; and no zeros.
We then degenerate this curve into a nodal curve $\Sigma$ with two components $\Sigma_1, \Sigma_2$, with let's say $\Sigma_1$ containing $x_{n-1}$ and $x_{n}$. It is then claimed that again there is a unique differential form on the normalisation $\tilde{\Sigma}$ of $\Sigma$, with poles only at the marked points $x_{n-1}$ and $x_n$, and residues $1$ and $-1$.
Now, this is where my confusion stems from. I thought, we should only have poles at nodes, whereas $x_{n-1}$ and $x_n$ are smooth points? How should one think about the differential forms after degenerating?
Also, it is then stated in the survey that on $\Sigma_2$, this differential form vanishes. What is the general behaviour here?
