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There are two notion of genus in algebraic geometry, namely arithmetic genus $p_a=(-1)^{\dim X}(\chi(\mathcal{O}_X)-1)$ and geometric genus $p_g=\dim H^0(X,\Omega^{\dim X})$. I keep forgetting definition of these, or being confused which is which. Are there any good ways to remember them?

More precisely I would like to associate these definition with these names "arithmetic" and "geometric".

M. K.
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    You might look at this answer – Georges Elencwajg Jan 04 '13 at 21:19
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    As a sort of mnemonic, I tell myself that geometric genus counts something geometric, like holes or independant loops or independant cycles and is thus always positive or zero, whereas arithmetic genus has some pleasant formal properties (which is why Hirzebruch introduced them) but has no geometric interpretation and in particular may even be negative. – Georges Elencwajg Jan 04 '13 at 21:30
  • Thank you for the comment. – M. K. Jan 05 '13 at 08:56

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If your curves becomes singular (say, nodal), then the arithmetic genus stays the same, while geometric genus drops.

From another point of view, your curve is arithmetically the same, (say, still a degree d plane curve), while geometrically (and topologically!) is different.

I believe that this is not only a natural way to remember, but, possibly, also how these terms appeared historically. (Severi studying nodal curves)

MLStudent
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  • When you say “becomes singular, then the geometric genus drops”, do you mean something in the sense of a semicontinuity in a flat family? – Lucas Henrique Jan 12 '25 at 01:55