Definition 53.11.1. Let $k$ be a field. Let $X$ be a geometrically irreducible curve over $k$. The geometric genus of $X$ is the genus of a smooth projective model of $X$ possibly defined over an extension field of $k$ as in Lemma 53.2.9.
53.11 Geometric genus
If $X$ is a proper and smooth curve over $k$ with $H^0(X, \mathcal{O}_ X) = k$, then
\[ p_ g(X) = \dim _ k H^0(X, \Omega _{X/k}) \]
is called the geometric genus of $X$. By Lemma 53.8.4 the geometric genus of $X$ agrees with the (arithmetic) genus. However, in higher dimensions there is a difference between the geometric genus and the arithmetic genus, see Remark 53.11.2.
For singular curves, we will define the geometric genus as follows.
If $k$ is perfect, then the nonsingular projective model $Y$ of $X$ is smooth (Lemma 53.2.8) and the geometric genus of $X$ is just the genus of $Y$. But if $k$ is not perfect, this may not be true. In this case we choose an extension $K/k$ such that the nonsingular projective model $Y_ K$ of $(X_ K)_{red}$ is a smooth projective curve and we define the geometric genus of $X$ to be the genus of $Y_ K$. This is well defined by Lemmas 53.2.9 and 53.8.2.
Remark 53.11.2. Suppose that $X$ is a $d$-dimensional proper smooth variety over an algebraically closed field $k$. Then the arithmetic genus is often defined as $p_ a(X) = (-1)^ d(\chi (X, \mathcal{O}_ X) - 1)$ and the geometric genus as $p_ g(X) = \dim _ k H^0(X, \Omega ^ d_{X/k})$. In this situation the arithmetic genus and the geometric genus no longer agree even though it is still true that $\omega _ X \cong \Omega _{X/k}^ d$. For example, if $d = 2$, then we have where $h^ i(X, \mathcal{F}) = \dim _ k H^ i(X, \mathcal{F})$ and where the last equality follows from duality. Hence for a surface the difference $p_ g(X) - p_ a(X)$ is always nonnegative; it is sometimes called the irregularity of the surface. If $X = C_1 \times C_2$ is a product of smooth projective curves of genus $g_1$ and $g_2$, then the irregularity is $g_1 + g_2$.
\begin{align*} p_ a(X) - p_ g(X) & = h^0(X, \mathcal{O}_ X) - h^1(X, \mathcal{O}_ X) + h^2(X, \mathcal{O}_ X) - 1 - h^0(X, \Omega ^2_{X/k}) \\ & = - h^1(X, \mathcal{O}_ X) + h^2(X, \mathcal{O}_ X) - h^0(X, \omega _ X) \\ & = - h^1(X, \mathcal{O}_ X) \end{align*}
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