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It might be a naive question, but I just realized I had not thought about this before. If $C$ is a smooth curve, for any line bundle $D$ we have the Riemann-Roch formula: $$\chi(D)=\deg D+1-g(C).$$ Does this nicely extend also to singular curves as $\chi(D)=\deg D+1-p_a(C)$ ?

(if yes, is it an easy consequence of the smooth case?)

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    $\begingroup$ One source is in the exercises to Hartshorne IV.1. $\endgroup$
    – Andrew
    Mar 3 '14 at 19:01
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    $\begingroup$ @Andrew: You could turn this into an answer by adding the content of the exercise. $\endgroup$
    – RghtHndSd
    Mar 3 '14 at 22:16
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In the interest of giving an answer for an old question:

Try taking a normalization of your curve, and then studying what happens to Euler characteristics and degrees when you pull back line bundles to the normalization. Each of these two questions have simple answers.

Then upstairs on your new smooth curve you can apply the old Riemann-Roch formula, and end up with the formula you want!

(Hint: "pulling back" commutes nicely with "twisting by divisors," so the only line bundle you ever really need to pull back is the structure sheaf. By definition, it pulls back to the structure sheaf, so degrees don't change. Now what is the difference between the Euler characteristics of the two structure sheaves?)

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  • $\begingroup$ I unerstand how to make sense of this when my curve has nice singularities like double points and simple cusps bu If my curve has arbitrarily complicated singularities is there a way to make this approach applicable? $\endgroup$ Nov 16 '17 at 17:22
  • $\begingroup$ It's been years since I've done any real algebraic geometry, but I think this answer works equally well for any type of singularity. Do you have an example in mind? I believe you can compute one correction term for each singularity, and that the corrections are purely local (i.e., you only need to know about a neighborhood of the singularity, maybe even just a formal neighborhood). These normalizations can be tricky by hand, but it's a tractable problem for computers and there are software packages that can do it no matter how bad your singularities are! Hope this helps. $\endgroup$ Nov 17 '17 at 17:58
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    $\begingroup$ you might look at Serre's Algebraic groups and class fields, chapter 4. $\endgroup$
    – roy smith
    Nov 23 '19 at 4:06

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