Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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".

share|improve this question
    
You might look at this answer –  Georges Elencwajg Jan 4 '13 at 21:19
8  
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 4 '13 at 21:30
    
Thank you for the comment. –  M. K. Jan 5 '13 at 8:56

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.