Given a smooth curve $C$, denote by $\text{Sym}^d(C)$ its $d$-th symmetric power. Let $\Delta$ be the diagonal subvariety which is defined as the codimension $1$ subvariety that at least two of the points coincide. Let $\text{Sym}^{d-1}(C)$ be the closed variety that its embedding is given by adding some extra fixed point. Let $Z$ be either $\Delta$ or $\text{Sym}^{d-1}(C)$.



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> **Question 1.** Is $Z$ an ample divisor?
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> **Question 2.** Is there any interesting interpretation of coherent modules/vector bundles on the formal completion of $Z$ in
> $\text{Sym}^d(C)$?
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Edit: Is it true that for $Z=\text{Sym}^{d-1}(C)$, the formal completion is going to be a bundle on $Z$? Is it like completing a zero section of a vector bundle?



Symmetric powers of curves and completion along the diagonal
$\operatorname{Sym}^{d-1}(C) $ is ample, and $\Delta $ is not unless $C\cong \mathbb{P}^1$. To see this, consider the finite map $\pi :C^d\rightarrow \operatorname{Sym}^{d}(C) $, and the projections $\pi\_i:C^d\rightarrow C$. Your divisor $Z$ is ample if and only if $\pi ^\*Z$ is ample. Now $\pi ^\*\operatorname{Sym}^{d-1}(C) =\sum p\_i^\*[p]$, where $p$ is your fixed point. Since $[p]$ is an ample divisor on $C$, $\operatorname{Sym}^{d-1}(C) $ is ample.


On the other hand, take $d=2$; then $\pi ^\*\Delta =2 \Delta '$ where $\Delta '$ is the diagonal in $C^2$, and $\Delta '^2=2-2g$ is $\leq 0$ if $g(C)\geq 1$.


