In this section we bound the torsion in the Picard group of a $1$-dimensional proper scheme over a field. We will use this in our study of semistable reduction for curves.
There does not seem to be an elementary way to obtain the result of Lemma 53.17.1. Analyzing the proof there are two key ingredients: (1) there is an abelian variety classifying degree zero invertible sheaves on a smooth projective curve and (2) the structure of torsion points on an abelian variety can be determined.
Lemma 53.17.1. Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve of genus $g$ over $k$.
If $n \geq 1$ is invertible in $k$, then $\mathop{\mathrm{Pic}}\nolimits (X)[n] \cong (\mathbf{Z}/n\mathbf{Z})^{\oplus 2g}$.
If the characteristic of $k$ is $p > 0$, then there exists an integer $0 \leq f \leq g$ such that $\mathop{\mathrm{Pic}}\nolimits (X)[p^ m] \cong (\mathbf{Z}/p^ m\mathbf{Z})^{\oplus f}$ for all $m \geq 1$.
Proof. Let $\mathop{\mathrm{Pic}}\nolimits ^0(X) \subset \mathop{\mathrm{Pic}}\nolimits (X)$ denote the subgroup of invertible sheaves of degree $0$. In other words, there is a short exact sequence
The group $\mathop{\mathrm{Pic}}\nolimits ^0(X)$ is the $k$-points of the group scheme $\underline{\mathrm{Pic}}^0_{X/k}$, see Picard Schemes of Curves, Lemma 44.6.7. The same lemma tells us that $\underline{\mathrm{Pic}}^0_{X/k}$ is a $g$-dimensional abelian variety over $k$ as defined in Groupoids, Definition 39.9.1. Thus we conclude by the results of Groupoids, Proposition 39.9.11. $\square$
Lemma 53.17.2. Let $k$ be a field. Let $n$ be prime to the characteristic of $k$. Let $X$ be a smooth proper curve over $k$ with $H^0(X, \mathcal{O}_ X) = k$ and of genus $g$.
If $g = 1$ then there exists a finite separable extension $k'/k$ such that $X_{k'}$ has a $k'$-rational point and $\mathop{\mathrm{Pic}}\nolimits (X_{k'})[n] \cong (\mathbf{Z}/n\mathbf{Z})^{\oplus 2}$.
If $g \geq 2$ then there exists a finite separable extension $k'/k$ with $[k' : k] \leq (2g - 2)(n^{2g})!$ such that $X_{k'}$ has a $k'$-rational point and $\mathop{\mathrm{Pic}}\nolimits (X_{k'})[n] \cong (\mathbf{Z}/n\mathbf{Z})^{\oplus 2g}$.
Proof. Assume $g \geq 2$. First we may choose a finite separable extension of degree at most $2g - 2$ such that $X$ acquires a rational point, see Lemma 53.13.9. Thus we may assume $X$ has a $k$-rational point $x \in X(k)$ but now we have to prove the lemma with $[k' : k] \leq (n^{2g})!$. Let $k \subset k^{sep} \subset \overline{k}$ be a separable algebraic closure inside an algebraic closure. By Lemma 53.17.1 we have
By Picard Schemes of Curves, Lemma 44.7.2 we conclude that
By Picard Schemes of Curves, Lemma 44.7.2 there is a continuous action
and the lemma is true for the fixed field $k'$ of the kernel of this map. The kernel is open because the action is continuous which implies that $k'/k$ is finite. By Galois theory $\text{Gal}(k'/k)$ is the image of the displayed arrow. Since the permutation group of a set of cardinality $n^{2g}$ has cardinality $(n^{2g})!$ we conclude by Galois theory that $[k' : k] \leq (n^{2g})!$. (Of course this proves the lemma with the bound $|\text{GL}_{2g}(\mathbf{Z}/n\mathbf{Z})|$, but all we want here is that there is some bound.)
If the genus is $1$, then there is no upper bound on the degree of a finite separable field extension over which $X$ acquires a rational point (details omitted). Still, there is such an extension for example by Varieties, Lemma 33.25.6. The rest of the proof is the same as in the case of $g \geq 2$. $\square$
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 and equality holds if and only if all the singularities of $X$ are multicross.
\[ \dim _{\mathbf{F}_\ell } \mathop{\mathrm{Pic}}\nolimits (X)[\ell ] \leq g + g_{geom} \]
Proof. Let $\nu : X^\nu \to X$ be the normalization (Varieties, Lemma 33.41.2). Choose a factorization
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
$X_ i$ is obtained by glueing $a, b \in X_{i + 1}$ which are on different connected components: in this case $\mathop{\mathrm{Pic}}\nolimits (X_ i) = \mathop{\mathrm{Pic}}\nolimits (X_{i + 1})$, $h^0_{i + 1} = h^0_ i + 1$, $h^1_{i + 1} = h^1_ i$,
$X_ i$ 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
and $h^0_{i + 1} = h^0_ i$, $h^1_{i + 1} = h^1_ i - 1$,
$X_ i$ is obtained by squishing a tangent vector in $X_{i + 1}$: in this case there is a short exact sequence
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
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
$h^1_ n = g_{geom}$ and
$\dim _{\mathbf{F}_\ell } \mathop{\mathrm{Pic}}\nolimits (X_ n)[\ell ] = 2g_{geom}$.
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
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). $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)