Let $X$ be a projective nodal curve. Why is the dualizing sheaf of $X$ isomorphic to the log-cotangent bundle of $X$?
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$\begingroup$ Welcome to MO. Your question seems a possible duplicate of mathoverflow.net/questions/312494/… $\endgroup$– Francesco PolizziCommented Oct 9, 2023 at 8:39
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$\begingroup$ @Francesco Polizzi How? $\endgroup$– SKTDCommented Oct 9, 2023 at 8:49
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1$\begingroup$ The answer to your question is, essentially: pass to the normalization $\tilde{X}$ of $X$, compute the dualizing sheaf of $\tilde{X}$ and then deduce the form of the dualizing sheaf of $X$ by using the Residue Theorem. This is explained in the question I linked, and in the answer and in the comments you can also find some standard references. $\endgroup$– Francesco PolizziCommented Oct 9, 2023 at 8:57
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$\begingroup$ @FrancescoPolizzi: Could you maybe make more precise to which technique you are refering to when you say that one can deduce the form of the dualizing sheaf of $X$ by using the Residue Theorem? I faced this problem pondering about an exercise from Jarod Alper's notes (see mathoverflow.net/questions/479996/…) and have troubles to realize the given hint suggesting to reduce it to the normalization. $\endgroup$– user267839Commented Oct 10 at 16:04
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$\begingroup$ (cont) It seems that one should consider an appropriate comparison map between the pairings at $X$ and $\tilde{X}$ and deduce perfectness on level of $X$ from considerations on $\tilde{X}$'s side, but not sure how this actually should be carried out in detail (the main problem seems to be the compatibility of "vertical" maps drawn in linked question) Could you maybe give a sketch how it actually can be carried out to deduce the form of the dualizing sheaf via Residue thm (once having a "promising candidate")? It seems that Jarod Alper refer to that kind of argument there. $\endgroup$– user267839Commented Oct 10 at 16:15
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