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It is a well known fact that a smooth complex projective curve of degree $d$ has topological genus $g = \frac{(d-1)(d-2)}{2}$ and is topologically equivalent to a closed surface with $g$ holes.

Taking for instance $d = 3$ we obtain the famous donut shape of an elliptic curves.

My question is, in the singular case, can we still visualise the topological genus the same way?

For instance, consider the curve $C: xy = 0$ which has a singularity at $(0, 0)$.

Topologically, this curve looks like two copies of a Riemann sphere touching each other at one point.

What is its topological genus and how is it defined? (the definition with the number of holes seems to apply to non-singular closed surfaces only)

What about the other definition of genus (for instance the one which appears in Riemann-Roch theorem), what do they give for this curve?

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  • $\begingroup$ Your $C: xy=0$ is a degenerate conic reduced to the two axes $x=0$ and $y=0$. As all conic its genus is $0$. No hole at all. $\endgroup$
    – Piquito
    Commented Jun 10, 2023 at 12:31
  • $\begingroup$ How do you show it's homeomorphic to the Riemann sphere? $\endgroup$
    – Weier
    Commented Jun 10, 2023 at 15:27
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    $\begingroup$ @Weier it is not topologically homeomorphic to a single Riemann sphere. The genus of a singular curve is usually taken to be the genus of its normalization. $\endgroup$ Commented Jun 12, 2023 at 17:48
  • $\begingroup$ @CraniumClamp what is the normalization? (For instance of the curve xy) $\endgroup$
    – Weier
    Commented Jun 12, 2023 at 18:59
  • $\begingroup$ @Weier a pair of disjoint Riemann spheres. It’s like de-attaching them at the node. $\endgroup$ Commented Jun 12, 2023 at 19:31

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There isn't really a sensible notion of a genus of a singular curve except via normalisation. It is worth noting that $\frac{(d-1)(d-2)}2$ still holds as a lower bound for the genus, for all smooth surfaces with the same $d$ (defined via homology), by the Kronheimer-Mrowka theorem; see e.g., https://people.math.harvard.edu/~kronheim/thomconj.pdf

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