Suppose we consider the second symmetric product S of a smooth projective curve C over complex numbers. How do we compute the Picard group of S ?

For product $C\times C$ It is shown that Picard group is a product $ Pic(C)\times Pic(C)\times End(Jac(C))$. What will be Invariants of this under $S_2$ action ?

  • $\begingroup$ Welcome new contributor. Definitely there are other references, but one reference is Proposition 2.3, p. 10 of the PhD. thesis of Yusuf Mustopa (if Mustopa is online, he can certainly give you more details): math.stonybrook.edu/alumni/2008-Yusuf-Mustopa.pdf $\endgroup$ – Jason Starr Aug 22 '18 at 11:59
  • $\begingroup$ My guess is that the transposition $\sigma$ is given by $(a,b,\phi) \mapsto (b,a,\phi^\intercal)$, where $(-)^\intercal$ is the Rosati involution. This should allow you to compute the invariants. I also vaguely recall that there could be a subtle difference between invariant $\operatorname{Pic}$ and equivariant $\operatorname{Pic}$; the latter consisting of line bundles $\mathscr L$ together with an isomorphism $\mathscr L \cong \sigma^* \mathscr L$ satisfying a cocycle condition. $\endgroup$ – R. van Dobben de Bruyn Aug 22 '18 at 12:04

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