I am interested in this claim:
The $n$th symmetric power $C^{(n)}$ of a genus $g$ curve $C$ is isomorphic to the projectivization $\mathbb{P}(E_n)$ of the sheaf $E_n := \pi_\ast(P_n)$ over the Jacobian $J(C)$, where $P_n$ is a degree $n$ Poincare bundle over $C \times J(C)$ and $\pi$ is the projection $C \times J(C) \to J(C)$.
Moreover, under this isomorphism, the standard line bundle $\mathcal{O}(1)$ over $\mathbb{P}(E_n)$ corresponds to the line bundle $\mathcal{O}(D)$, where $D$ is the divisor corresponding to the image of the map $C^{(n-1)} \hookrightarrow C^{(n)}$ given by $p_1 + \cdots + p_{n-1} \mapsto p_1 + \cdots + p_{n-1} + p$, where $p$ is some fixed point.
(Also, the isomorphism $\phi : C^{(n)} \to \mathbb{P}(E_n)$ is compatible with the Abel-Jacobi map $u: C^{(n)} \to J(C)$, that is, $u = p \circ \phi$, where $p : \mathbb{P}(E_n) \to J(C)$.)
My questions:
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This is claimed on page 309 of the book "Geometry of Algebraic Curves" by Arbarello-Cornalba-Grifiths-Harris, for $n \geq 2g-1$ (so that $E_n$ is a vector bundle, by Riemann-Roch; for smaller $n$ it isn't necessarily a vector bundle and they don't address this case). Their proof is pretty sketchy. It basically just says that, since the fibers of $\mathbb{P}(E_n) \to J(C)$ correspond to effective degree $n$ divisors, it follows that $C^{(n)} \cong \mathbb{P}(E_n)$. But this seems to me to only prove a set theoretic bijection between the two. So, how do I prove that I actually have an isomorphism of varieties? Or, is there a(nother) reference?
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I believe the claim should still be true for $n < 2g-1$. Again, how do I prove this? Is there a reference? The sheaf $E_n$ will no longer be locally free, so $\mathbb{P}(E_n)$ will no longer be a bundle of projective spaces, but one should still be able to take the projectivization of a sheaf…
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On page 7 of the paper http://arxiv.org/abs/0805.3621 by Moonen and Polishchuk, they talk about the families version of this statement. To be precise, they consider a family $\pi: C \to S$ of curves, and everything is done relative to the base $S$ (take relative symmetric product, take relative Jacobian, and so on). In this case, we must have a section $s: S \to C$ of $\pi$, corresponding to picking a point in each fiber, in order for the map $C^{(n-1)} \hookrightarrow C^{(n)}$ and the divisor $D$ to make sense. Anyway, Moonen and Polishchuk claim that in this families situation, $\mathbb{P}(E_n)$ is still isomorphic to the symmetric product $C^{(n)}$, and that under this isomorphism the line bundle $\mathcal{O}(1)$ corresponds to the line bundle $\mathcal{O}(D + n\psi)$, where $\psi$ is given by $\psi = \pi^\ast s^\ast K$, where $K$ is the relative canonical class of $\pi$. But how do I prove these statements?
Best Answer
This is worked out in excruciating detail in the article Jacobians and Symmetric products by Schwarzenberger. I think the arguments there are perfectly good in the families setting as well.