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Consider the canonical line bundle $K_C$ for $C$ defined by the compactification of $f=y^3-x^5+1$. $K_C$ is the as a set the set of holomorphic $1$-forms on $C$. How does one go about finding the basis for $K_C$?

I have seen an example for hyperellptic curves that gives the basis as $z^adz/w$, where $w$ is the product of rational functions having divisors precisely at ramification points and infinity. This does not even seem holomorphic to me, and I am thus confused in general.

Jon Doe
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    This is not a hyperelliptic curve. Have you checked if the points at infinity are smooth points? – Ted Shifrin Sep 15 '21 at 23:42
  • I know I was giving the only other example I know of. The point at infinity of the projective closure is not smooth; I'm using its compactification not its closure so the points at infinity are by construction smooth. Edit: theres one point at infinity – Jon Doe Sep 15 '21 at 23:44
  • So if this is not a smooth curve in $\Bbb P^2$, how are you working with the compactification as a concrete curve? How is it smooth? – Ted Shifrin Sep 15 '21 at 23:49
  • I'm not sure what you mean by concrete curve. The closure is not a smooth curve. One can compactify it to a smooth curve in $\mathbb{P}^2$ by adding one point (you can notice that there is a connected region for example), although this smooth curve is not isomorphic to the closure. I don't know if this point can be described explicitly. – Jon Doe Sep 15 '21 at 23:53
  • Are you taking the closure in $\Bbb P^1\times \Bbb P^1$ instead? It seems you’d better talk to your professor and add lots more detail to your post. – Ted Shifrin Sep 15 '21 at 23:57
  • You can take the closure in $\Bbb P^2$ and blow up the singular point a few times to get a smooth curve. This sounds like a very involved exercise if you have only seen one example. – Ted Shifrin Sep 16 '21 at 00:10
  • See section 3.4 of Molin and Neurohr's Computing period matrices and the Abel-Jacobi map of superelliptic curves. Actually, they in turn cite Proposition 2 of Towse's Weierstrass points on cyclic covers of the projective line. – Viktor Vaughn Sep 16 '21 at 00:36
  • @TedShifrin No I'm not taking the closure in $\mathbb{P}^1\times \mathbb{P}^1$ either. This is exercise A-6 of Geometry of Algebraic Curves. It associates to the plane curve a Riemann surface by adding precisely one point. As you pointed out -- this cannot be the closure, as the closure is singular. It is given in the prompt. – Jon Doe Sep 16 '21 at 00:43
  • Also a duplicate of this question, but which has received no answers. – Viktor Vaughn Sep 16 '21 at 00:43
  • @JonDoe - The only way to calculate the global sections of the canonical bundle is to give an explicit construction of the "smooth compactification" $\tilde{C}$ of $C$. Hence you must give information on how to do this: How does the book construct $\tilde{C}$? One you have such a smooth compactification $\tilde{C} \subseteq \mathbb{P}^n$ you may calculate $H^0(\tilde{C}, K_{\tilde{C}})$ using cohomology. As commented: The obvious compactification is not smooth and then $K_{\tilde{C}}$ is a coherent sheaf, still you may calculate the global sections using cohomology. – hm2020 Sep 17 '21 at 15:41

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