Arithmetic genus

Remarks

The sum is finite because $X$ is projective, so each $H^i(X, \mathcal{O}_X)$ is finite-dimensional and vanishes for $i > n$. Equivalently, $p_a(X) = (-1)^n (P_X(0) - 1)$, where $P_X$ is the Hilbert polynomial of $X$ for a chosen embedding. For a curve $C$ with $H^0(C, \mathcal{O}_C) = k$, for example a connected reduced projective curve over an algebraically closed field, $p_a(C) = \dim_k H^1(C, \mathcal{O}_C)$. Example: $H^1(\mathbb{P}^1, \mathcal{O}) = 0$, so $p_a(\mathbb{P}^1) = 0$. Example: a plane cubic curve, smooth or nodal, has $p_a = 1$; this follows from the exact sequence $0 \to \mathcal{O}_{\mathbb{P}^2}(-3) \to \mathcal{O}_{\mathbb{P}^2} \to \mathcal{O}_C \to 0$. Non-example: the affine line $\mathbb{A}^1$ is not projective, and $H^0(\mathbb{A}^1, \mathcal{O}) = k[t]$ is infinite-dimensional, so $\chi$ is not defined.