# Why is this the dualising sheaf of a singular curve?

If $$X$$ is a curve with a nodal singularity at $$x$$, it's referred to here and here that its dualising sheaf is $$\omega_X \ = \ \pi_*(\Omega_{X}(p_1+\cdots+p_n)').$$ Here, $$\pi:X\to X'$$ is the normalisation and $$\Omega_{X}(p_1+\cdots+p_n)'$$ are the forms $$\theta$$ which have at worst simple poles at $$\pi^{-1}(x)=\{p_1,...,p_n\}$$, satisfying $$\sum_i \text{Res}_{p_i}\theta\ = \ 0 .$$

1. What is the proof that this is the dualising sheaf? I can't find a reference, even in the complex-analytic case.
2. What happens when $$X$$ has other types of singularities (and what is the proof)? For instance, working complex-analytically, what if $$X$$ is locally $$(z-\alpha_1w)\cdots(z-\alpha_nw)\ = 0 \ ?$$
• Chapter IV, §3 of Algebraic Groups and Class Fields by Serre is devoted to differentials on singular curves. This is pre-Grothendieck, so the term "dualizing sheaf" does not appear, but Serre proves that his sheaf of differentials is indeed a dualizing sheaf. – abx Oct 10 '18 at 18:19
• I remember there is some discussing of this type on Griffiths and Harris. I do not remember the exact reference. – Bombyx mori Oct 10 '18 at 18:51
• Check the first section of the third chapter in Harris & Morrison's Moduli of Curves. I believe the proof is outlined there in the complex analytic case, as well as some discussion of singularities that cause this sheaf to no longer be a line bundle. – Tabes Bridges Oct 11 '18 at 17:49

2. To compute $$\omega_X$$ even faster, use that if $$X\subseteq \mathbb{P}^n_k$$ is a codimension $$r$$ closed subscheme, then $$\omega_X=\mathcal{E}\text{xt}^r_{\mathbb{P}^n}(\mathcal{O}_X,\Omega^n_{\mathbb{P}^n})$$ is its dualising sheaf (Hartshorne III,7). If $$X$$ is the vanishing set of a degree $$d$$ polynomial $$f$$ for instance, take a resolution of $$\mathcal{O}_X$$ by projective $$\mathcal{O}_{\mathbb{P}^n}$$-modules: $$0 \ \longrightarrow \ \mathcal{O}_{\mathbb{P}^n}(d) \ \stackrel{\cdot f}{\longrightarrow} \ \mathcal{O}_{\mathbb{P}^n} \ \longrightarrow \ \mathcal{O}_X$$ apply $$\mathcal{H}\text{om}_{\mathbb{P}^n}(-,\Omega^n_{\mathbb{P}^n})$$ and take the first cohomology.