A remark on some punctual Quot schemes on smooth projective curves
Atsushi Ito
TL;DR
The paper analyzes punctual Quot schemes $F_n$ on a smooth projective curve, showing they are normal $\mathbb{Q}$-factorial Fano varieties of dimension $n(r-1)$ with Picard number $1$ and Fano index $r$. It constructs a natural embedding $F_n \hookrightarrow \mathrm{Gr}(nr,n)$ so that the restricted Plücker line $\mathcal{O}(1)|_{F_n}$ is the ample generator of $\mathrm{Pic}(F_n)$, and proves that the singular locus has codimension $2$ for $n\ge 2$, while $\mathrm{Cl}(F_n)$ is generated by a single divisor $H$ with $nH\sim \mathcal{O}_{F_n}(1)$. The argument uses an iterated $\mathbb{P}^{r-1}$-bundle resolution and a divisorial contraction to advance by induction on $n$, and explicitly describes the divisor class group in terms of $H$ and the Plücker generator. In the special case $r=2$, the paper provides a precise description of the exceptional divisor as a $\mathbb{P}^1$-bundle over $F_{n-1}$, clarifying the contraction structure. These results unify the geometry of punctual Quot schemes with Grassmannian embeddings and birational contractions, highlighting their Fano nature and simple divisor theory.
Abstract
For a locally free sheaf $\mathcal{E}$ on a smooth projective curve, we can define the punctual Quot scheme which parametrizes torsion quotients of $\mathcal{E}$ of length $n$ supported at a fixed point. It is known that the punctual Quot scheme is a normal projective variety with canonical Gorenstein singularities. In this note, we show that the punctual Quot scheme is a $\mathbb{Q}$-factorial Fano variety of Picard number one.