Definition 53.8.1. Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_ X) = k$. Then the genus of $X$ is $g = \dim _ k H^1(X, \mathcal{O}_ X)$.
53.8 The genus of a curve
If $X$ is a smooth projective geometrically irreducible curve over a field $k$, then we've previously defined the genus of $X$ as the dimension of $H^1(X, \mathcal{O}_ X)$, see Picard Schemes of Curves, Definition 44.6.3. Observe that $H^0(X, \mathcal{O}_ X) = k$ in this case, see Varieties, Lemma 33.26.2. Let us generalize this as follows.
This is sometimes called the arithmetic genus of $X$. In the literature the arithmetic genus of a proper curve $X$ over $k$ is sometimes defined as
\[ p_ a(X) = 1 - \chi (X, \mathcal{O}_ X) = 1 - \dim _ k H^0(X, \mathcal{O}_ X) + \dim _ k H^1(X, \mathcal{O}_ X) \]
This agrees with our definition when it applies because we assume $H^0(X, \mathcal{O}_ X) = k$. But note that
$p_ a(X)$ can be negative, and
$p_ a(X)$ depends on the base field $k$ and should be written $p_ a(X/k)$.
For example if $k = \mathbf{Q}$ and $X = \mathbf{P}^1_{\mathbf{Q}(i)}$ then $p_ a(X/\mathbf{Q}) = -1$ and $p_ a(X/\mathbf{Q}(i)) = 0$.
The assumption that $H^0(X, \mathcal{O}_ X) = k$ in our definition has two consequences. On the one hand, it means there is no confusion about the base field. On the other hand, it implies the scheme $X$ is Cohen-Macaulay and equidimensional of dimension $1$ (Lemma 53.6.1). If $\omega _ X$ denotes the dualizing module as in Lemmas 53.4.1 and 53.4.2 we see that
53.8.1.1
\begin{equation} \label{curves-equation-genus} g = \dim _ k H^1(X, \mathcal{O}_ X) = \dim _ k H^0(X, \omega _ X) \end{equation}
by duality, see Remark 53.4.3.
If $X$ is proper over $k$ of dimension $\leq 1$ and $H^0(X, \mathcal{O}_ X)$ is not equal to the ground field $k$, instead of using the arithmetic genus $p_ a(X)$ given by the displayed formula above we shall use the invariant $\chi (X, \mathcal{O}_ X)$. In fact, it is advocated in [page 276, FAC] and [Introduction, Hirzebruch] that we should call $\chi (X, \mathcal{O}_ X)$ the arithmetic genus.
Lemma 53.8.2. Let $k'/k$ be a field extension. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_ X) = k$. Then $X_{k'}$ is a proper scheme over $k'$ having dimension $1$ and $H^0(X_{k'}, \mathcal{O}_{X_{k'}}) = k'$. Moreover the genus of $X_{k'}$ is equal to the genus of $X$.
Proof. The dimension of $X_{k'}$ is $1$ for example by Morphisms, Lemma 29.28.3. The morphism $X_{k'} \to \mathop{\mathrm{Spec}}(k')$ is proper by Morphisms, Lemma 29.41.5. The equality $H^0(X_{k'}, \mathcal{O}_{X_{k'}}) = k'$ follows from Cohomology of Schemes, Lemma 30.5.2. The equality of the genus follows from the same lemma. $\square$
Lemma 53.8.3. Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_ X) = k$. If $X$ is Gorenstein, then
\[ \deg (\omega _ X) = 2g - 2 \]
where $g$ is the genus of $X$ and $\omega _ X$ is as in Lemma 53.4.1.
Proof. Immediate from Lemma 53.5.2. $\square$
Lemma 53.8.4. Let $X$ be a smooth proper curve over a field $k$ with $H^0(X, \mathcal{O}_ X) = k$. Then
\[ \dim _ k H^0(X, \Omega _{X/k}) = g \quad \text{and}\quad \deg (\Omega _{X/k}) = 2g - 2 \]
where $g$ is the genus of $X$.
Proof. By Lemma 53.4.1 we have $\Omega _{X/k} = \omega _ X$. Hence the formulas hold by (53.8.1.1) and Lemma 53.8.3. $\square$
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