Proposition 53.17.3. Let $k$ be an algebraically closed field. Let $X$ be a proper scheme over $k$ which is reduced, connected, and has dimension $1$. Let $g$ be the genus of $X$ and let $g_{geom}$ be the sum of the geometric genera of the irreducible components of $X$. For any prime $\ell$ different from the characteristic of $k$ we have

$\dim _{\mathbf{F}_\ell } \mathop{\mathrm{Pic}}\nolimits (X)[\ell ] \leq g + g_{geom}$

and equality holds if and only if all the singularities of $X$ are multicross.

Proof. Let $\nu : X^\nu \to X$ be the normalization (Varieties, Lemma 33.41.2). Choose a factorization

$X^\nu = X_ n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X$

as in Lemma 53.15.4. Let us denote $h^0_ i = \dim _ k H^0(X_ i, \mathcal{O}_{X_ i})$ and $h^1_ i = \dim _ k H^1(X_ i, \mathcal{O}_{X_ i})$. By Lemmas 53.15.5 and 53.15.6 for each $n > i \geq 0$ we have one of the following there possibilities

1. $X_ i$ is obtained by glueing $a, b \in X_{i + 1}$ which are on different connected components: in this case $\mathop{\mathrm{Pic}}\nolimits (X_ i) = \mathop{\mathrm{Pic}}\nolimits (X_{i + 1})$, $h^0_{i + 1} = h^0_ i + 1$, $h^1_{i + 1} = h^1_ i$,

2. $X_ i$ is obtained by glueing $a, b \in X_{i + 1}$ which are on the same connected component: in this case there is a short exact sequence

$0 \to k^* \to \mathop{\mathrm{Pic}}\nolimits (X_ i) \to \mathop{\mathrm{Pic}}\nolimits (X_{i + 1}) \to 0,$

and $h^0_{i + 1} = h^0_ i$, $h^1_{i + 1} = h^1_ i - 1$,

3. $X_ i$ is obtained by squishing a tangent vector in $X_{i + 1}$: in this case there is a short exact sequence

$0 \to (k, +) \to \mathop{\mathrm{Pic}}\nolimits (X_ i) \to \mathop{\mathrm{Pic}}\nolimits (X_{i + 1}) \to 0,$

and $h^0_{i + 1} = h^0_ i$, $h^1_{i + 1} = h^1_ i - 1$.

To prove the statements on dimensions of cohomology groups of the structure sheaf, use the exact sequences in Examples 53.15.2 and 53.15.3. Since $k$ is algebraically closed of characteristic prime to $\ell$ we see that $(k, +)$ and $k^*$ are $\ell$-divisible and with $\ell$-torsion $(k, +)[\ell ] = 0$ and $k^*[\ell ] \cong \mathbf{F}_\ell$. Hence

$\dim _{\mathbf{F}_\ell } \mathop{\mathrm{Pic}}\nolimits (X_{i + 1})[\ell ] - \dim _{\mathbf{F}_\ell }\mathop{\mathrm{Pic}}\nolimits (X_ i)[\ell ]$

is zero, except in case (2) where it is equal to $-1$. At the end of this process we get the normalization $X^\nu = X_ n$ which is a disjoint union of smooth projective curves over $k$. Hence we have

1. $h^1_ n = g_{geom}$ and

2. $\dim _{\mathbf{F}_\ell } \mathop{\mathrm{Pic}}\nolimits (X_ n)[\ell ] = 2g_{geom}$.

The last equality by Lemma 53.17.1. Since $g = h^1_0$ we see that the number of steps of type (2) and (3) is at most $h^1_0 - h^1_ n = g - g_{geom}$. By our comptation of the differences in ranks we conclude that

$\dim _{\mathbf{F}_\ell } \mathop{\mathrm{Pic}}\nolimits (X)[\ell ] \leq g - g_{geom} + 2g_{geom} = g + g_{geom}$

and equality holds if and only if no steps of type (3) occur. This indeed means that all singularities of $X$ are multicross by Lemma 53.16.3. Conversely, if all the singularities are multicross, then Lemma 53.16.3 guarantees that we can find a sequence $X^\nu = X_ n \to \ldots \to X_0 = X$ as above such that no steps of type (3) occur in the sequence and we find equality holds in the lemma (just glue the local sequences for each point to find one that works for all singular points of $x$; some details omitted). $\square$

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