Let $M_g$ denote the moduli space of smooth genus $g$ curves over $\mathbb{C}$, where $g \geq 2$. Let $Pic^{d,g}$ denote the universal picard variety over $M_g$, parameterizing pairs $(C,L)$ where $C$ is of genus $g$ and $L$ is a degree $d$ line bundle on $C$. Is it true that $Pic^{d,g}$ is irreducible? If so, is there a reference?
2
-
2$\begingroup$ This is a straightforward deformation theory computation, but if you need a reference, it is probably in Caporaso’s article on compactifications of $\text{Pic}^{d,g}.$ $\endgroup$– Jason StarrCommented Oct 31 at 17:34
-
$\begingroup$ @JasonStarr Thank you Professor Starr. I am a beginner to all of this so can I ask: in Caporaso's article, they consider the universal Picard variety over $M^0_g$ (smooth genus $g$ curves which are automorphism free), and claim that this is irreducible. Here I am considering $M_g$, which includes curves with automorphisms. Does irreducibility still hold? $\endgroup$– maxoCommented Nov 13 at 23:54
Add a comment
|