Let $C$ be a hyperelliptic curve of genus $g$ and let $S = C^{(2)}$ denote the symmetric square of $C$. Let $\nabla$ be the divisor on $C^2$ defined by $\{(P, \overline{P}) \mid P \in C\}$ where $\overline{P}$ denotes the image of $P$ under the hyperelliptic involution. Finally let $\nabla_S$ be the push-forward of $\nabla$ under the quotient map $C^2 \to S$.

If $g = 2$, then $S$ is the blowup of the Jacobian $J_C$ of $C$ at the origin with exceptional divisor $\nabla_S$. So, by results from Hartshorne Section V.3 on the relationships between blowups and the intersection pairing, we immediately conclude that $\nabla_S$ is a canonical divisor of $S$ and it has self-intersection $-1$. I would like to know how to do these calculations without the crutch of using blowups, and hence, with any luck, obtain the analogous results for when $g>2$. My question is therefore

How do I calculate a canonical divisor, and its self-intersection number, of the symmetric square of a hyperelliptic curve of genus $g>2$?


Let $\pi: C^2\to S$ be the canonical quotient morphism. It is étale outside of the diagonal $\Delta$ of $C^2$. Let $K$ be the base field. The canonical map of differentials $$ \pi^*\Omega_{S/K}^1\to \Omega^1_{C^2/K}$$ is then an isomorphism outside of $\Delta$, and induces an injective homomorphism $\pi^*\omega_{S/K}\to \omega_{C^2/K}$ of canonical sheaves. So $\pi^*\omega_{S/K}=\omega_{C^2/K}(-D)$ for some effective divisor $D$ on $C^2$, with support in $\Delta$, hence $D=r\Delta$ for some integer $r\ge 0$. The multiplicity $r$ can be computed locally for the Zariski topology, and even for étale topology on $C$. So we can work with Spec ($K[x]$) and find $r=1$: $$ \pi^*\omega_{S/K}=\omega_{C^2/K}(-\Delta).$$ This should be enough to describe a canonical divisor on $S$ interms of the pushforward of a canonical divisor of $C^2$ and of $\pi(\Delta)$. The self-intersection should be easily computed with the projection formula. Otherwise ask for more details.

Edit Computation of the multiplicity $r$. Let $\xi$ be the generic point of $\Delta$ and $\eta=\pi(\xi)$. Then $$ \omega_{S/K, \xi}\otimes O_{C^2,\xi}=(\pi^*\omega_{S/K})_{\xi}=\omega_{C^2/K}(-r\Delta)_{\xi}=\omega_{C^2/K,\xi}(-r\Delta).$$ This explains why $r$ can be computed Zariski locally. Let $U$ be a dense open subset of $C$. Then one can compute $r$ on $U^2\to U^{(2)}$. If we can write $U$ as an étale cover $U\to V\subseteq \mathbb A^1_K$, then the map $$ \pi^*\Omega_{U^{(2)}/K}^1\to \Omega^1_{U^2/K}$$ is just the pull-back of the map $$ \pi^*\Omega_{V^{(2)}/K}^1\to \Omega^1_{V^2/K}.$$ If you take a local basis $dx, dy$ for $\Omega^1_{V^2/K}$, then $d(x+y), d(xy)$ is a local basis for $\Omega^1_{V^{(2)}/K}$, and their pull-backs to $U^2$ (resp. $U^{(2)}$) are local bases, and $r$ can be computed with these local bases. Now $\omega_{V^2/K}$ is generated by $dx\wedge dy$, and $\omega_{V^{(2)}/K}$ is generated by $d(x+y)\wedge d(xy)$ who image in $\omega_{V^2/K}$ is $(x-y)(dx\wedge dy)$. As $x-y$ generates locally the ideal of $\Delta$, we see that $r=1$.

  • $\begingroup$ Could you elaborate on what you mean by computing the multiplicity $r$ locally for the Zariski/étale topology? I don't really understand what that means. Thanks! $\endgroup$ – Hamish May 27 '12 at 20:52

The self intersection of the diagonal of a curve $C$ in $C \times C$ is just $2 - 2g$. (This is a form of the Poincare--Hopf index theorem.) The self-intersection of the graph of the hyperelliptic involution is also $2 - 2g$ (since it the image of the diagonal under the automorphism $(P,Q) \mapsto (P,\bar{Q})$ of $C^2$.

If we quotient out by the involution of $C^2$ given by transposition, to get $Sym^2 C$, then the self-intersection is halved, and so we get that the image of the graph of the hyperelliptic involution has self-intersection $1 - g$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.