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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 $$(z-\alpha_1w)\cdots(z-\alpha_nw)\ = 0 \ ?$$
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    $\begingroup$ 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. $\endgroup$ – abx Oct 10 '18 at 18:19
  • $\begingroup$ I remember there is some discussing of this type on Griffiths and Harris. I do not remember the exact reference. $\endgroup$ – Bombyx mori Oct 10 '18 at 18:51
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    $\begingroup$ 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. $\endgroup$ – Tabes Bridges Oct 11 '18 at 17:49
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There are two ways to compute the dualising sheaf of a curve:

  1. Mimick Serre's proof of Serre duality (p.8-13 of this link) for nonsingular curves, which directly gives the result in the question and explains where the residue map comes from (it comes from the residue pairing which you need to prove Serre duality). Note that it's quite easy to compute.
  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.
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