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I was reading Bloch's conjecture for projective algebraic curvers(algebraic curves that are homogenous, for example $x^2+y^2-z^2=0$), and stumbled across this notation, $p_g(X)$. What does this denotion mean? I was first thinking it would be the vanishing of the Albanese kernel, but it is sadly conjectural only. Thanks for all your answers.

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Its the geometric genus. – Manos Oct 17 '12 at 17:31
So it is still given by the Riemann-Roch theorem just like a nonsingular curve. – Jaivir Baweja Oct 17 '12 at 17:34
up vote 3 down vote accepted

Let $X$ be a curve (e.g. an integral scheme o dimension 1, proper over the base field, all of whose local rings are regular; such a curve is projective). Then $p_g(X)$ denotes its geometric genus, which is also equal to the arithmetic genus. Thus we talk about the genus $g=p_g(X)$ of the curve, which is the quantity appearing in the Riemann-Roch Theorem.

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