\newaliascnt

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A remark on some punctual Quot schemes on smooth projective curves

Atsushi Ito Department of Mathematics, Institute of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan [email protected]

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.

Key words and phrases:

Punctual Quot scheme, Quot-to-Chow morphism

2020 Mathematics Subject Classification:

14H60, 14E05

1. Introduction

Throughout this paper, we work over an algebraically closed field $k$ of any characteristic. For a locally free sheaf $\mathcal{E}$ of rank $r$ on a smooth projective curve $C$ and $n\geqslant 0$ , let $\operatorname{Quot}_{C}^{n}(\mathcal{E})$ be the Quot scheme which parametrizes torsion quotients of $\mathcal{E}$ of length $n$ . It is known that $\operatorname{Quot}_{C}^{n}(\mathcal{E})$ is a smooth projective variety of dimension $nr$ (see [BFP20, Lemma 2.2, Corollary 4.7] for instance). We can define the Quot-to-Chow morphism

(1.1)

\displaystyle\pi:\operatorname{Quot}_{C}^{n}(\mathcal{E})\to\operatorname{Sym}^{n}C

sending the quotient $[\mathcal{E}\twoheadrightarrow\mathcal{Q}]$ to the effective divisor on $C$ determined by the torsion sheaf $\mathcal{Q}$ . For $q\in C$ , the punctual Quot scheme $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ is defined to be the scheme-theoretic fiber of $\pi$ over $nq\in\operatorname{Sym}^{n}C$ . Recently, the fibers of $\pi$ are studied by many authors. For example, the following are known:

•

([Ric20, §2.1]) The isomorphism class of $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ depends only on $r$ and $n$ . In particular, it is independent of $C$ and $q$ .
•

([GS20, Lemma 6.5], [BJS24, Theorem 1.2]) The fiber of $\pi$ over $\sum_{i=1}^{l}m_{i}q_{i}\in\operatorname{Sym}^{n}C$ with $q_{i}\neq q_{j}\ (i\neq j)$ is isomorphic to the product $\prod_{i=1}^{l}\operatorname{Quot}_{C}^{m_{i}}(\mathcal{E})_{q_{i}}$ .
•

([GS20, Corollary 6.6], [BJS24, Theorem 1.2]) $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ is a normal projective variety of dimension $n(r-1)$ with Cartier canonical divisor.
•

([GS20, §4,5]) $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ is birational to $\mathbb{P}^{n(r-1)}$ .
•

([BGS24, Lemma 6.2]) $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ has a crepant resolution. In particular, $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ has canonical Gorenstein singularities.

If $r=1$ , 1.1 is an isomorphism and hence $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ is a point. If $n=1$ , 1.1 coincides with the $\mathbb{P}^{1}$ -bundle $\mathbb{P}_{C}(\mathcal{E})\to C$ and hence $\operatorname{Quot}_{C}^{1}(\mathcal{E})_{q}=\mathbb{P}(\mathcal{E}\otimes k(q))\simeq\mathbb{P}^{r-1}$ .

In [BJS24], the authors investigate the geometry of $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ for $r=2$ in detail. In particular, they prove that $\operatorname{Quot}_{C}^{n}(\mathcal{O}_{C}^{\oplus 2})_{q}$ is

•

$\mathbb{P}^{1}$ if $n=1$ ,
•

a singular quadric in $\mathbb{P}^{3}$ if $n=2$ ,
•

a normal $\mathbb{Q}$ -factorial Fano $3$ -fold of Picard number one with canonical singularities along a copy of $\mathbb{P}^{1}$ if $n=3$

in characteristic zero [BJS24, Theorems 1.3, 1.4, 1.5]. The purpose of this note is to show a similar statement for any $n,r$ as follows.

Theorem 1.1.

Let $\mathcal{E}$ be a locally free sheaf on a smooth projective curve $C$ of rank $r\geqslant 2$ and $q\in C$ . For $n\geqslant 1$ , the following hold.

(1)

$\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ is a normal $\mathbb{Q}$ -factorial Fano $n(r-1)$ -fold of Picard number one.
(2)

There exists an embedding $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}\hookrightarrow\mathrm{Gr}(nr,n)$ to a Grassmannian such that $\mathcal{O}(1)\coloneqq\mathcal{O}_{\mathrm{Gr}(nr,n)}(1)|_{\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}}$ is the ample generator of the Picard group $\operatorname{Pic}(\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q})\simeq\mathbb{Z}$ .
(3)

The Fano index of $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ is $r$ , that is, $K_{\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}}=\mathcal{O}(-r)$ .
(4)

If $n\geqslant 2$ , the singular locus of $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ is irreducible of codimension two in $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ .
(5)

There exists a prime divisor $H\subset\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ such that the divisor class group $\operatorname{Cl}(\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q})$ is generated by the class $[H]$ and $nH\sim\mathcal{O}(1)$ .

The idea of the proof essentially follows from [GS20], where the authors construct a resolution of $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ as an iterated $\mathbb{P}^{r-1}$ -bundle. Following their construction, we define a $\mathbb{P}^{r-1}$ -bundle $f_{n}:\mathbb{P}_{\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}}(\mathcal{F})\to\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ and a divisorial contraction $\mu_{n+1}:\mathbb{P}_{\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}}(\mathcal{F})\to\operatorname{Quot}_{C}^{n+1}(\mathcal{E})_{q}$ . Then we can prove the theorem by the induction on $n$ .

This paper is organized as follows. In §2, we recall some notation and give an embedding of $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ to a Grassmannian. In §3 and §4, we investigate the Picard group and the divisor class group of $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ respectively. In §5, we give a description of the exceptional divisor of the divisorial contraction $\mu_{n+1}:\mathbb{P}_{\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}}(\mathcal{F})\to\operatorname{Quot}_{C}^{n+1}(\mathcal{E})_{q}$ for $r=2$ .

Acknowledgments

The author was supported by JSPS KAKENHI Grant Numbers 17K14162, 21K03201.

2. Embedding to a Grassmannian

For a $k$ -vector space $E$ , $\mathrm{Gr}(E,s)$ (resp. $\mathrm{Gr}(s,E)$ ) denotes the Grassmannian of $s$ -dimensional quotients (resp. subspaces) of $E$ . More generally, for a locally free sheaf $\mathcal{E}$ on a variety $X$ , $\mathrm{Gr}_{X}(\mathcal{E},s)$ (resp. $\mathrm{Gr}_{X}(s,\mathcal{E})$ ) denotes the Grassmannian bundle which parametrizes quotient bundles (resp. subbundles) of $\varphi^{*}\mathcal{E}$ of rank $s$ for each $\varphi:T\to X$ . We use the notation $\mathbb{P}(E)\coloneqq\mathrm{Gr}(E,1)=\operatorname{Proj}\operatorname{Sym}E$ and $\mathbb{P}_{X}(\mathcal{E})\coloneqq\mathrm{Gr}_{X}(\mathcal{E},1)=\operatorname{Proj}_{X}\operatorname{Sym}\mathcal{E}$ .

For a coherent sheaf of $\mathcal{F}$ on $X$ , $\operatorname{Quot}_{X}^{n}(\mathcal{F})$ denotes the Quot scheme which parametrizes quotients of $\mathcal{F}$ with zero-dimensional supports of degree $n$ . The point in $\operatorname{Quot}_{X}^{n}(\mathcal{F})$ corresponding to an exact sequence $0\to\mathcal{A}\to\mathcal{F}\to\mathcal{B}\to 0$ on $C$ is denoted by $[\mathcal{A}\hookrightarrow\mathcal{F}]$ or $[\mathcal{F}\twoheadrightarrow\mathcal{B}]$ . If the context is clear, we write it as $[\mathcal{A}]$ or $[\mathcal{B}]$ simply .

Let $\mathcal{E}$ be a locally free sheaf of rank $r$ on a smooth projective curve $C$ and $n\geqslant 0$ . Throughout this paper, $p_{C}:C\times T\to C$ and $p_{T}:C\times T\to T$ are the natural projections for a locally noetherian scheme $T$ over $k$ . Then a morphism $T\to\operatorname{Quot}_{C}^{n}(\mathcal{E})$ corresponds to an exact sequence

(2.1)

\displaystyle 0\to\mathscr{A}\to p_{C}^{*}\mathcal{E}\to\mathscr{B}\to 0

on $C\times T$ such that $\mathscr{B}$ is locally free of rank $n$ as an $\mathcal{O}_{T}$ -module. Since $C$ is a smooth curve, $\mathscr{A}$ is locally free of rank $r$ .

Recall the definition of the Quot-to-Chow morphism $\pi:\operatorname{Quot}_{C}^{n}(\mathcal{E})\to\operatorname{Sym}^{n}C$ (see [GS20, Section 2] for the details). Let $Q=\operatorname{Quot}_{C}^{n}(\mathcal{E})$ and let $0\to\mathscr{A}_{Q}\to p_{C}^{*}\mathcal{E}\to\mathcal{B}_{Q}\to 0$ be the universal exact sequence on $C\times Q$ . Since $C$ is a smooth curve, $\mathscr{A}_{Q}$ is locally free of rank $r=\operatorname{rank}\mathcal{E}$ and hence we obtain an exact sequence

\displaystyle 0\to\det\mathscr{A}_{Q}\to\det p_{C}^{*}\mathcal{E}\to\mathscr{C}\to 0.

We can check that $\mathscr{C}$ is flat over $Q$ and hence

0\to\det\mathscr{A}_{Q}\otimes(\det p_{C}^{*}\mathcal{E})^{-1}\to\mathcal{O}_{C\times Q}\to\mathscr{C}\otimes(\det p_{C}^{*}\mathcal{E})^{-1}\to 0

induces the Quot-to-Chow morphism $\pi:Q=\operatorname{Quot}_{C}^{n}(\mathcal{E})\to\operatorname{Sym}^{n}C$ .

For $q\in C$ , let $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ be the scheme theoretic fiber of $\pi:\operatorname{Quot}_{C}^{n}(\mathcal{E})\to\operatorname{Sym}^{n}C$ over $nq$ . By the definition of $\pi$ , the morphism $T\to\operatorname{Quot}_{C}^{n}(\mathcal{E})$ corresponding to 2.1 factors through $\pi^{-1}(nq)=\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}\subset\operatorname{Quot}_{C}^{n}(\mathcal{E})$ if and only if $\det\mathscr{A}=p_{C}^{*}\mathfrak{m}^{n}\det\mathcal{E}$ , where $\mathfrak{m}=\mathcal{O}_{C}(-q)$ is the maximal ideal sheaf corresponding to $q\in C$ .

The following proposition is essentially explained in [BJS24, §6.4], at least set-theoretically.

Proposition \theprop.

Under the above setting, $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ coincides with $\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})_{\mathrm{red}}\subset\operatorname{Quot}_{C}^{n}(\mathcal{E})$ , where we embed¹¹1See [FGI⁺05, §5.5.3] for the embedding between Quot schemes induced by a surjection of coherent sheaves. $\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})$ to $\operatorname{Quot}_{C}^{n}(\mathcal{E})$ by the natural surjection $\mathcal{E}\to\mathcal{E}/\mathfrak{m}^{n}\mathcal{E}$ and $\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})_{\mathrm{red}}$ means the reduced scheme structure on $\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})$ .

In particular, there exists an embedding $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}=\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})_{\mathrm{red}}\hookrightarrow\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)=\mathrm{Gr}(nr,n)$ .

Proof.

Consider a morphism $T\to\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ , which corresponds to an exact sequence $0\to\mathscr{A}\to p_{C}^{*}\mathcal{E}\to\mathscr{B}\to 0$ on $C\times T$ . Then $\mathscr{A}$ is locally free of rank $r$ with $\det\mathscr{A}=p_{C}^{*}\mathfrak{m}^{n}\det\mathcal{E}$ . By Cramer’s rule, $\mathscr{A}$ contains $p_{C}^{*}\mathfrak{m}^{n}\mathcal{E}$ and hence the morphism $T\to\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}\subset\operatorname{Quot}_{C}^{n}(\mathcal{E})$ factors through the subscheme $\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})\subset\operatorname{Quot}_{C}^{n}(\mathcal{E})$ . This means that $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ is a subscheme of $\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})$ .

A closed point of $\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})$ is a quotient $[\mathcal{E}/\mathfrak{m}^{n}\mathcal{E}\to\mathcal{B}]$ on $C$ whose length is $n$ . As a point in $\operatorname{Quot}_{C}^{n}(\mathcal{E})$ , this is the point $[\mathcal{E}\to\mathcal{E}/\mathfrak{m}^{n}\mathcal{E}\to\mathcal{B}]$ , which is contained in $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ since $\operatorname{Supp}(\mathcal{B})=\{q\}$ . Since $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ is reduced by [GS20] or [BJS24], $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}=\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})_{\mathrm{red}}$ holds.

Since $\mathcal{E}/\mathfrak{m}^{n}\mathcal{E}$ is a $k$ -vector space of dimension $nr$ , the Quot scheme $\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})$ is naturally embedded to the Grassmannian $\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)=\mathrm{Gr}(nr,n)$ . In fact, a morphism $T\to\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})$ corresponds to a quotient $(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})\otimes\mathcal{O}_{T}\to\mathscr{B}$ of $\mathcal{O}_{T}\otimes\mathcal{O}_{C,q}$ -modules such that $\mathscr{B}$ is locally free of rank $n$ as $\mathcal{O}_{T}$ -module. On the other hand, a morphism $T\to\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)$ corresponds to a quotient $(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})\otimes\mathcal{O}_{T}\to\mathscr{B}$ of $\mathcal{O}_{T}$ -modules such that $\mathscr{B}$ is locally free of rank $n$ . Hence there exists a natural injection $\operatorname{Hom}_{k\text{-}sch}(T,\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E}))\to\operatorname{Hom}_{k\text{-}sch}(T,\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n))$ . Thus $\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})$ is embedded into $\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)=\mathrm{Gr}(nr,n)$ . ∎

Remark \therem.

Let $0\to\mathscr{A}_{n}\to p_{C}^{*}\mathcal{E}\to\mathscr{B}_{n}\to 0$ be the universal exact sequence on $C\times\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ . The embedding $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}\to\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)$ is induced by

\displaystyle 0\to\overline{\mathscr{A}_{n}/p_{C}^{*}\mathfrak{m}^{n}\mathcal{E}}\to\overline{p_{C}^{*}\mathcal{E}/p_{C}^{*}\mathfrak{m}^{n}\mathcal{E}}=(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})\otimes\mathcal{O}_{\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}}\to\overline{\mathscr{B}_{n}}\to 0,

where $\overline{\mathcal{F}}$ is the pushforward of $\mathcal{F}$ by $p_{\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}}:C\times\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}\to\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ . In particular, $\mathcal{O}_{\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)}(1)|_{\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}}=\det\overline{\mathscr{B}_{n}}$ holds, where $\mathcal{O}_{\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)}(1)$ is the Plücker line bundle of the Grassmannian $\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)$ .

Remark \therem.

The Picard group of the Quot scheme $Q=\operatorname{Quot}_{C}^{n}(\mathcal{E})$ is computed by [GS21] as follows. Let $0\to\mathscr{A}_{Q}\to p_{C}^{*}\mathcal{E}\to\mathscr{B}_{Q}\to 0$ be the universal exact sequence on $C\times Q$ and let $\mathcal{O}_{Q}(1)=\det({p_{Q}}_{*}(\mathscr{B}_{Q}))$ . Then $\pi^{*}:\operatorname{Pic}^{0}(\operatorname{Sym}^{n}C)\to\operatorname{Pic}(Q)$ induced by the Quot-to-Chow morphism $\pi:Q\to\operatorname{Sym}^{n}C$ is injective and $\operatorname{Pic}(Q)=\pi^{*}\operatorname{Pic}^{0}(\operatorname{Sym}^{n}C)\oplus\mathbb{Z}[\mathcal{O}_{Q}(1)]$ by [GS21, Theorem 3.7].

Then $\mathcal{O}_{\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)}(1)|_{\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}}$ coincides with $\mathcal{O}_{Q}(1)|_{\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}}$ since

\mathcal{O}_{\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)}(1)|_{\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}}=\det\overline{\mathscr{B}_{n}}=\det({p_{Q}}_{*}(\mathscr{B}_{Q})|_{\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}})=\mathcal{O}_{Q}(1)|_{\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}}

for $\overline{\mathscr{B}_{n}}$ in § 2.

Remark \therem.

In general, $\operatorname{Quot}_{C}^{n}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E})$ is non-reduced. For example, let $\mathcal{E}=\mathcal{O}_{C}^{\oplus 2},n=2$ . Then $\mathcal{E}/\mathfrak{m}^{2}\mathcal{E}=(k[t]/(t^{2}))^{\oplus 2}$ , where $t$ is a local coordinate of $C$ at $q$ . For $T=\operatorname{Spec}R=\operatorname{Spec}k[\varepsilon]/(\varepsilon^{2})$ , a quotient

\displaystyle p_{C}^{*}(\mathcal{E}/\mathfrak{m}^{2}\mathcal{E})=(R[t]/(t^{2}))^{\oplus 2}\to R^{\oplus 2}:(f(t),g(t))\mapsto(f(\varepsilon),g(\varepsilon))

on $C\times T$ gives a morphism $T\to\operatorname{Quot}_{C}^{2}(\mathcal{E}/\mathfrak{m}^{2}\mathcal{E})$ . This does not factor through $\operatorname{Quot}_{C}^{2}(\mathcal{E})_{q}$ if $\mathrm{char}\ k\neq 2$ since the kernel of $p_{C}^{*}\mathcal{E}=\mathcal{O}_{C\times T}^{\oplus 2}\to(R[t]/(t^{2}))^{\oplus 2}\to R^{\oplus 2}$ is $(t-\varepsilon)\mathcal{O}_{C\times T}^{\oplus 2}$ , whose determinant is $(t^{2}-2\varepsilon t)\mathcal{O}_{C\times T}\neq t^{2}\mathcal{O}_{C\times T}$ .

3. Picard groups

Throughout this section, we fix a locally free sheaf $\mathcal{E}$ of rank $r$ on a smooth projective curve $C$ and $q\in C$ . Since the punctual Quot scheme $\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ is a point if $r=1$ , we assume $r\geqslant 2$ in the rest of this section. For simplicity, we set $F_{n}=\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ and $\mathfrak{m}=\mathcal{O}_{C}(-q)\subset\mathcal{O}_{C}$ . As in § 2, $\overline{\mathcal{F}}$ denotes the pushforward of a coherent sheaf $\mathcal{F}$ on $C\times F_{n}$ by the projection $p_{F_{n}}:C\times F_{n}\to F_{n}$ .

Let

\displaystyle 0\to\mathscr{A}_{n}\to p_{C}^{*}\mathcal{E}\to\mathscr{B}_{n}\to 0

be the universal exact sequence on $C\times F_{n}$ . Recall that $\mathscr{A}_{n}$ is locally free of rank $r$ with $\det\mathscr{A}_{n}=p_{C}^{*}\mathfrak{m}^{n}\det\mathcal{E}$ . Then $\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}}=\mathscr{A}_{n}|_{\{q\}\times F_{n}}$ is locally free of rank $r$ on $F_{n}$ and hence we can define a $\mathbb{P}^{r-1}$ bundle $f_{n}:\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})\to F_{n}$ . Let $f_{n}^{*}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})\to\mathcal{O}_{f_{n}}(1)$ be the tautological line bundle on $\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})$ .

Lemma \thelem.

For $n\geqslant 0$ , $\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})$ is isomorphic to

\displaystyle F_{n,n+1}

\displaystyle\coloneqq\{([\mathcal{A}_{n}],[\mathcal{A}_{n+1}])\in F_{n}\times F_{n+1}\mid\mathcal{A}_{n+1}\subset\mathcal{A}_{n}\subset\mathcal{E}\}

with the reduced structure over $F_{n}$ .

Proof.

Let $\operatorname{pr}_{n}:F_{n,n+1}\to F_{n}$ and $\operatorname{pr}_{n+1}:F_{n,n+1}\to F_{n+1}$ be the natural projections. We first explain this lemma set-theoretically. Fix $[\mathcal{A}_{n}]\in F_{n}$ . Then a point in $\operatorname{pr}_{n}^{-1}([\mathcal{A}_{n}])$ corresponds to a quotient $\mathcal{O}_{C}$ -module $\mathcal{A}_{n}\to\mathcal{V}$ of length one with $\operatorname{Supp}\mathcal{V}=\{q\}$ . Since such $\mathcal{V}$ is isomorphic to $\mathcal{O}_{C}/\mathfrak{m}$ , such quotient $\mathcal{A}_{n}\to\mathcal{V}$ corresponds to a quotient $k$ -vector space $\mathcal{A}_{n}/\mathfrak{m}\mathcal{A}_{n}\to V$ with $\dim_{k}V=1$ , which is nothing but a point in $\mathbb{P}(\mathcal{A}_{n}/\mathfrak{m}\mathcal{A}_{n})=\mathbb{P}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}}\otimes k([\mathcal{A}_{n}]))=f_{n}^{-1}([\mathcal{A}_{n}])$ . Hence there exists a canonical bijection between $F_{n,n+1}$ and $\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})$ .

We can construct this bijection as an isomorphism as follows. For simplicity, $\mathbb{P}_{F_{n}}$ denotes $\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})$ . Let

\iota:\mathbb{P}_{F_{n}}=\{q\}\times\mathbb{P}_{F_{n}}\hookrightarrow C\times\mathbb{P}_{F_{n}}

be the natural immersion. Since $\left((\operatorname{id}_{C}\times f_{n})^{*}\mathscr{A}_{n}\right)|_{\{q\}\times\mathbb{P}_{F_{n}}}=(\operatorname{id}_{C}\times f_{n})^{*}(\mathscr{A}_{n}|_{\{q\}\times F_{n}})=\iota_{*}f_{n}^{*}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})$ , we can consider the composite map

(3.1)

\displaystyle(\operatorname{id}_{C}\times f_{n})^{*}\mathscr{A}_{n}\to\iota_{*}f_{n}^{*}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})\to\iota_{*}\mathcal{O}_{f_{n}}(1)

on $C\times\mathbb{P}_{F_{n}}$ . Let $\mathscr{A}^{\prime}$ be the kernel of 3.1. Then we have a diagram

on $C\times\mathbb{P}_{F_{n}}$ . By the snake lemma, we have an exact sequence

(3.2)

\displaystyle 0\to\iota_{*}\mathcal{O}_{f_{n}}(1)\to p_{C}^{*}\mathcal{E}/\mathscr{A}^{\prime}\to(\operatorname{id}_{C}\times f_{n})^{*}\mathscr{B}_{n}\to 0.

Since $\iota_{*}\mathcal{O}_{f_{n}}(1)$ and $(\operatorname{id}_{C}\times f_{n})^{*}\mathscr{B}_{n}$ are flat over $\mathbb{P}_{F_{n}}$ , so is $p_{C}^{*}\mathcal{E}/\mathscr{A}^{\prime}$ . Since $\mathscr{A}^{\prime}$ is the kernel of 3.2, we see that $\mathscr{A}^{\prime}$ is a locally free of rank $r$ with $\det\mathscr{A}^{\prime}=\mathfrak{m}(\operatorname{id}_{C}\times f_{n})^{*}\det\mathscr{A}_{n}=p_{C}^{*}\mathfrak{m}^{n+1}\det\mathcal{E}$ . Hence $0\to\mathscr{A}^{\prime}\to p_{C}^{*}\mathcal{E}\to p_{C}^{*}\mathcal{E}/\mathscr{A}^{\prime}\to 0$ induces a morphism $\mu_{n+1}:\mathbb{P}_{F_{n}}\to F_{n+1}=\operatorname{Quot}_{C}^{n+1}(\mathcal{E})_{q}$ . Since $\mathscr{A}^{\prime}\subset(\operatorname{id}_{C}\times f_{n})^{*}\mathscr{A}_{n}$ , the image of $f_{n}\times\mu_{n+1}:\mathbb{P}_{F_{n}}\to F_{n}\times F_{n+1}$ is contained in $F_{n,n+1}$ .

The inverse of $f_{n}\times\mu_{n+1}$ is constructed as follows. The composite morphism $C\times F_{n,n+1}\xrightarrow{\operatorname{id}_{C}\times\operatorname{pr}_{i}}C\times F_{i}\xrightarrow{\operatorname{pr}_{i}}F_{i}$ is denoted by $\tilde{\operatorname{pr}}_{i}$ . Then we have a diagram

on $C\times F_{n,n+1}$ . Hence

(3.3)

\displaystyle\overline{\tilde{\operatorname{pr}}_{n}^{*}\mathscr{A}_{n}/\tilde{\operatorname{pr}}_{n+1}^{*}\mathscr{A}_{n+1}}\simeq\overline{\ker\varpi}

is locally free of rank one on $F_{n,n+1}$ . Since $\det\tilde{\operatorname{pr}}_{n+1}^{*}\mathscr{A}_{n+1}=p_{C}^{*}\mathfrak{m}^{n+1}\det\mathcal{E}=\mathfrak{m}\det\tilde{\operatorname{pr}}_{n}^{*}\mathscr{A}_{n}$ , we have $\mathfrak{m}\tilde{\operatorname{pr}}_{n}^{*}\mathscr{A}_{n}\subset\tilde{\operatorname{pr}}_{n+1}^{*}\mathscr{A}_{n+1}$ by Cramer’s rule and hence 3.3 is a quotient of

\displaystyle\overline{\tilde{\operatorname{pr}}_{n}^{*}\mathscr{A}_{n}/\mathfrak{m}\tilde{\operatorname{pr}}_{n}^{*}\mathscr{A}_{n}}=\operatorname{pr}_{n}^{*}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}}).

Hence $\operatorname{pr}_{n}^{*}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})\to\overline{\tilde{\operatorname{pr}}_{n}^{*}\mathscr{A}_{n}/\tilde{\operatorname{pr}}_{n+1}^{*}\mathscr{A}_{n+1}}$ induces a morphism $F_{n,n+1}\to\mathbb{P}_{F_{n}}$ . By construction, this is the inverse of $f_{n}\times\mu_{n+1}:\mathbb{P}_{F_{n}}\to F_{n,n+1}$ . ∎

By § 3, $\operatorname{pr}_{n}:F_{n,n+1}\to F_{n}$ is a $\mathbb{P}^{r-1}$ -bundle. On the other hand, $\operatorname{pr}_{n+1}:F_{n,n+1}\to F_{n+1}$ is birational as follows.

Lemma \thelem.

Set $U\coloneqq\{[\mathcal{E}\twoheadrightarrow\mathcal{B}_{n+1}]\in F_{n+1}\mid\mathcal{B}_{n+1}\simeq\mathcal{O}_{C}/\mathfrak{m}^{n+1}\}$ . Then

(1)

$\operatorname{pr}_{n,n+1}:F_{n,n+1}\to F_{n+1}$ is an isomorphism over $U$ .
(2)

The dimension of the fiber of $\operatorname{pr}_{n+1}$ over a point in $F_{n+1}\setminus U$ is positive.
(3)

The codimension of $\operatorname{pr}_{n+1}^{-1}(F_{n+1}\setminus U)$ in $F_{n,n+1}$ is one.

Proof.

Let $[\mathcal{A}_{n+1}]\in F_{n+1}$ be a point corresponding to $0\to\mathcal{A}_{n+1}\to\mathcal{E}\to\mathcal{B}_{n+1}\to 0$ . Let $\mathcal{B}^{\prime}_{n+1}=\{b\in\mathcal{B}_{n+1}\mid\mathfrak{m}b=0\}$ . Then the fiber $\operatorname{pr}_{n+1}^{-1}([\mathcal{A}_{n+1}])$ is canonically identified with $\mathrm{Gr}(1,\mathcal{B}^{\prime}_{n+1})$ as follows: A point in the fiber $\operatorname{pr}_{n+1}^{-1}([\mathcal{A}_{n+1}])$ corresponds to a quotient $\mathcal{B}_{n+1}\to\mathcal{B}_{n}$ of length $n$ . Such quotient corresponds to a submodule $\mathcal{C}\subset\mathcal{B}_{n+1}$ of length one. Since such a submodule $\mathcal{C}$ is isomorphic to $\mathcal{O}_{C}/\mathfrak{m}$ and hence $\mathfrak{m}\mathcal{C}=0$ , a submodule $\mathcal{C}\subset\mathcal{B}_{n+1}$ of length one is nothing but a one dimensional subspace of $\mathcal{B}^{\prime}_{n+1}$ . Hence there exists a bijection between the fiber $\operatorname{pr}_{n+1}^{-1}([\mathcal{A}_{n+1}])$ and $\mathrm{Gr}(1,\mathcal{B}^{\prime}_{n+1})$ .
(1) If $\mathcal{B}_{n+1}\simeq\mathcal{O}_{C}/\mathfrak{m}^{n+1}$ , it holds that $\mathcal{B}^{\prime}_{n+1}=\mathfrak{m}^{n}\mathcal{B}_{n+1}$ and $\mathcal{B}_{n+1}/\mathcal{B}^{\prime}_{n+1}\simeq\mathcal{O}_{C}/\mathfrak{m}^{n}$ . Hence for the universal exact sequence $0\to\mathscr{A}_{n+1}\to p_{C}^{*}\mathcal{E}\to\mathscr{B}_{n+1}\to 0$ on $C\times F_{n+1}$ , the quotient $\mathscr{B}_{n+1}/\mathfrak{m}^{n}\mathscr{B}_{n+1}$ is flat of length $n$ over $U$ . Hence $p_{C}^{*}\mathcal{E}|_{C\times U}\to(\mathscr{B}_{n+1}/\mathfrak{m}^{n}\mathscr{B}_{n+1})|_{C\times U}$ gives a morphism $g:U\to F_{n}$ . By construction, $(g,\operatorname{id}_{U}):U\to F_{n,n+1}$ is the inverse of $\operatorname{pr}_{n,n+1}$ over $U$ .
(2) If $\mathcal{B}_{n+1}\not\simeq\mathcal{O}_{C}/\mathfrak{m}^{n+1}$ , it holds that $\mathcal{B}_{n+1}\simeq\bigoplus_{i=1}^{l}\mathcal{O}_{C}/\mathfrak{m}^{n_{i}}$ with $\sum_{i=1}^{l}n_{i}=n+1$ for some $l\geqslant 2$ and $n_{i}\geqslant 1$ by the classification of modules over PID. Then $\mathcal{B}^{\prime}_{n+1}\simeq\bigoplus_{i=1}^{l}\mathfrak{m}^{n_{i}-1}/\mathfrak{m}^{n_{i}}\simeq k^{l}$ and hence the dimension of $\mathrm{Gr}(1,\mathcal{B}^{\prime}_{n+1})=\mathbb{P}^{l-1}$ is positive.
(3) Since $\operatorname{codim}_{F_{n+1}}(F_{n+1}\setminus U)=2$ by [BJS24, §5], the codimension of the exceptional locus $\operatorname{pr}_{n+1}^{-1}(F_{n+1}\setminus U)$ is one by (1), (2) and the irreducibility of $F_{n,n+1}$ . ∎

Proposition \theprop.

For each $n\geqslant 1$ , the following hold.

(1)

$F_{n}$ is $\mathbb{Q}$ -factorial and $\operatorname{Pic}F_{n}=\mathbb{Z}\mathcal{O}_{F_{n}}(1)$ , where $\mathcal{O}_{F_{n}}(1)=\mathcal{O}_{\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)}(1)|_{F_{n}}$ is the restriction of $\mathcal{O}_{\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)}(1)$ to $F_{n}\subset\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)$ .
(2)

$K_{F_{n}}=\mathcal{O}_{F_{n}}(-r)$ and $K_{F_{n,n+1}}=\operatorname{pr}_{n+1}^{*}K_{F_{n}}$ hold.
(3)

$\operatorname{pr}_{n+1}:F_{n,n+1}\to F_{n+1}$ is a divisorial contraction.
(4)

For $n\geqslant 2$ , the singular locus of $F_{n}$ is $\{[\mathcal{E}\twoheadrightarrow\mathcal{B}_{n}]\in F_{n}\mid\mathcal{B}_{n}\not\simeq\mathcal{O}_{C}/\mathfrak{m}^{n}\}$ , which is irreducible of codimension two in $F_{n}$ .

Proof.

We show (1) and (2) by the induction of $n$ . Since $F_{1}=\mathbb{P}(\mathcal{E}/\mathfrak{m}\mathcal{E})\simeq\mathbb{P}^{r-1}$ , (1), (2) hold for $n=1$ . We assume (1), (2) for $n$ and show (1), (2) for $n+1$ .

By induction hypothesis, $\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})=F_{n,n+1}$ is $\mathbb{Q}$ -factorial with $\operatorname{Pic}\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})=\mathbb{Z}\mathcal{O}_{f_{n}}(1)\oplus\mathbb{Z}f_{n}^{*}\mathcal{O}_{F_{n}}(1)$ , where $\mathcal{O}_{f_{n}}(1)$ is the tautological line bundle of $f_{n}:\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})\to F_{n}$ . Since $\operatorname{pr}_{n+1}:F_{n,n+1}\to F_{n+1}$ is birational and contracts a divisor by § 3, $F_{n+1}$ is $\mathbb{Q}$ -factorial with Picard number one.

Recall that the embedding $F_{i}\hookrightarrow\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{i}\mathcal{E},i)$ is induced by the quotient

\displaystyle(\mathcal{E}/\mathfrak{m}^{i}\mathcal{E})\otimes\mathcal{O}_{F_{i}}=\overline{p_{C}^{*}\mathcal{E}/p_{C}^{*}\mathfrak{m}_{q}^{i}\mathcal{E}}\to\overline{\mathscr{B}_{i}}

on $F_{i}$ and hence $\mathcal{O}_{F_{i}}(1)=\det\overline{\mathscr{B}_{i}}$ by § 2 for $i=n,n+1$ . On the other hand, $\operatorname{pr}_{n+1}:\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})=F_{n,n+1}\to F_{n+1}$ is induced by the quotient $p_{C}^{*}\mathcal{E}\to p_{C}^{*}\mathcal{E}/\mathscr{A}^{\prime}$ on $C\times\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})$ , where $\mathscr{A}^{\prime}$ in the kernel of 3.1. Hence $\operatorname{pr}_{n+1}:\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})=F_{n,n+1}\to F_{n+1}\subset\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n+1}\mathcal{E},n+1)$ is induced by the quotient

\displaystyle(\mathcal{E}/\mathfrak{m}^{n+1}\mathcal{E})\otimes\mathcal{O}_{\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})}={p}_{*}(p_{C}^{*}\mathcal{E}/p_{C}^{*}\mathfrak{m}_{q}^{n+1}\mathcal{E})\to p_{*}(p_{C}^{*}\mathcal{E}/\mathscr{A}^{\prime})

on $\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})$ , where $p:C\times\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})\to\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})$ is the second projection. Taking $p_{*}$ of 3.2, we obtain

\displaystyle 0\to\mathcal{O}_{f_{n}}(1)\to p_{*}(p_{C}^{*}\mathcal{E}/\mathscr{A}^{\prime})\to f_{n}^{*}\overline{\mathscr{B}_{n}}\to 0.

Thus it holds that

\displaystyle\operatorname{pr}_{n+1}^{*}\mathcal{O}_{F_{n+1}}(1)=\det p_{*}(p_{C}^{*}\mathcal{E}/\mathscr{A}^{\prime})=\mathcal{O}_{f_{n}}(1)\otimes\det f_{n}^{*}\overline{\mathscr{B}_{n}}=\mathcal{O}_{f_{n}}(1)\otimes f_{n}^{*}\mathcal{O}_{F_{n}}(1),

which is primitive in $\operatorname{Pic}\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})=\mathbb{Z}\mathcal{O}_{f_{n}}(1)\oplus\mathbb{Z}f_{n}^{*}\mathcal{O}_{F_{n}}(1)$ . Hence $\operatorname{Pic}F_{n+1}$ is generated by $\mathcal{O}_{F_{n+1}}(1)$ , which proves (1) for $n+1$ .

To show (2), we determine $\det(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})\in\operatorname{Pic}F_{n}$ first. For a generator $t\in\mathfrak{m}\mathcal{O}_{C,q}$ , the kernel of $p_{C}^{*}\mathcal{E}/p_{C}^{*}\mathfrak{m}^{n+1}\mathcal{E}\xrightarrow{t\times}p_{C}^{*}\mathcal{E}/p_{C}^{*}\mathfrak{m}^{n+1}\mathcal{E}$ on $C\times F_{n}$ is $p_{C}^{*}\mathfrak{m}^{n}\mathcal{E}/p_{C}^{*}\mathfrak{m}^{n+1}\mathcal{E}$ , which is contained in $\mathscr{A}_{n}/p_{C}^{*}\mathfrak{m}^{n+1}\mathcal{E}$ . Hence we have an exact sequence

\displaystyle 0\to p_{C}^{*}\mathfrak{m}^{n}\mathcal{E}/p_{C}^{*}\mathfrak{m}^{n+1}\mathcal{E}\to\mathscr{A}_{n}/p_{C}^{*}\mathfrak{m}^{n+1}\mathcal{E}\xrightarrow{t\times}\mathscr{A}_{n}/p_{C}^{*}\mathfrak{m}^{n+1}\mathcal{E}\to\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}\to 0.

Taking pushforwards, we have an exact sequence

\displaystyle 0\to\overline{p_{C}^{*}\mathfrak{m}^{n}\mathcal{E}/p_{C}^{*}\mathfrak{m}^{n+1}\mathcal{E}}\to\overline{\mathscr{A}_{n}/p_{C}^{*}\mathfrak{m}^{n+1}\mathcal{E}}\to\overline{\mathscr{A}_{n}/p_{C}^{*}\mathfrak{m}^{n+1}\mathcal{E}}\to\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}}\to 0

of locally free sheaves on $F_{n}$ . Since $\overline{p_{C}^{*}\mathfrak{m}^{n}\mathcal{E}/p_{C}^{*}\mathfrak{m}^{n+1}\mathcal{E}}=(\mathfrak{m}^{n}\mathcal{E}/\mathfrak{m}^{n+1}\mathcal{E})\otimes\mathcal{O}_{F_{n}}$ is a trivial bundle of rank $r$ , it holds that $\det(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})=\mathcal{O}_{F_{n}}$ .

Since $K_{F_{n}}=\mathcal{O}_{F_{n}}(-r)$ by induction hypothesis, we have

K_{\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})}=\mathcal{O}_{f_{n}}(-r)\otimes f_{n}^{*}(K_{F_{n}}\otimes\det(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}}))=\mathcal{O}_{f_{n}}(-r)\otimes f_{n}^{*}\mathcal{O}_{F_{n}}(-r)=\operatorname{pr}_{n+1}^{*}\mathcal{O}_{F_{n+1}}(-r).

Thus $K_{F_{n+1}}={\operatorname{pr}_{n+1}}_{*}K_{\mathbb{P}_{F_{n}}(\overline{\mathscr{A}_{n}/\mathfrak{m}\mathscr{A}_{n}})}=\mathcal{O}_{F_{n+1}}(-r)$ , which proves (2) for $n+1$ .

Hence (1) and (2) are proved for any $n\geqslant 1$ . Since $F_{n,n+1}$ and $F_{n+1}$ are $\mathbb{Q}$ -factorial with Picard number two and one respectively, (3) holds.
(4) Assume $n\geqslant 2$ . By § 3, $Z=\{[\mathcal{E}\twoheadrightarrow\mathcal{B}_{n}]\in F_{n}\mid\mathcal{B}_{n}\not\simeq\mathcal{O}_{C}/\mathfrak{m}^{n}\}$ is the image of the exceptional divisor of $\operatorname{pr}_{n}:F_{n-1,n}\to F_{n}$ . Since the discrepancy of the exceptional divisor of $\operatorname{pr}_{n}:F_{n-1,n}\to F_{n}$ is zero by (2) of this proposition, $\{[\mathcal{E}\twoheadrightarrow\mathcal{B}_{n}]\in F_{n}\mid\mathcal{B}_{n}\not\simeq\mathcal{O}_{C}/\mathfrak{m}^{n}\}$ is contained in the singular locus of $F_{n}$ . On the other hand, $F_{n}$ is smooth at $[\mathcal{E}\twoheadrightarrow\mathcal{B}_{n}]$ if $\mathcal{B}_{n}\simeq\mathcal{O}_{C}/\mathfrak{m}^{n}$ by [GS20, Lemma 3.3]. Thus $Z$ is the singular locus of $F_{n}$ . Since $Z$ is the image of the exceptional divisor of the divisorial contraction $\operatorname{pr}_{n+1}$ , $Z$ is irreducible. By [BJS24, §5], $\operatorname{codim}_{F_{n}}Z=2$ . ∎

4. Divisor class groups

In this section, let $F_{n}=\operatorname{Quot}_{C}^{n}(\mathcal{E})_{q}$ be the punctual Quot scheme with $r=\operatorname{rank}\mathcal{E}\geqslant 2$ as in the previous section.

Proposition \theprop.

There exists a prime divisor $H\subset F_{n}$ such that the divisor class group $\operatorname{Cl}(F_{n})$ is generated by the class $[H]$ and $nH\sim\mathcal{O}_{F_{n}(1)}=\mathcal{O}_{\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n)}(1)|_{F_{n}}$ .

Proof.

If $n=1$ , $F_{1}\simeq\mathbb{P}^{r-1}$ and hence we can take $H\sim\mathcal{O}_{\mathbb{P}^{r-1}}(1)$ .

Let $n\geqslant 2$ . We may assume that $\mathcal{E}=V\otimes\mathcal{O}_{C}$ for $V=k^{r}$ . Let $e_{1},\dots,e_{r}$ be the standard basis of $V$ .

The smooth locus of $F_{n}$ is $\{[V\otimes\mathcal{O}_{C}\twoheadrightarrow\mathcal{B}_{n}]\in F_{n}\mid\mathcal{B}_{n}\simeq\mathcal{O}_{C}/\mathfrak{m}^{n}\}$ by § 3 (4). Hence the smooth locus is covered by open subsets $U_{1},\dots,U_{r}$ defined by

U_{i}\coloneqq\{[V\otimes\mathcal{O}_{C}\stackrel{{\scriptstyle\beta}}{{\twoheadrightarrow}}\mathcal{O}_{C}/\mathfrak{m}^{n}]\in F_{n}\mid\text{the image $\beta(e_{i})$ is invertible in $\mathcal{O}_{C}/\mathfrak{m}^{n}$}\}

as explained in [BJS24, §5]. Furthermore, each $U_{i}$ is isomorphic to $\mathbb{A}^{n(r-1)}$ . For example, we have an isomorphism $\mathbb{A}^{n(r-1)}\to U_{1}$ defined by

	$\displaystyle\beta(e_{1})$	$\displaystyle=1,$
	$\displaystyle\beta(e_{2})$	$\displaystyle=a_{2,0}+a_{2,1}t+\cdots+a_{2,n-1}t^{n-1},$
		$\displaystyle\vdots$
	$\displaystyle\beta(e_{r})$	$\displaystyle=a_{r,0}+a_{r,1}t+\cdots+a_{r,n-1}t^{n-1},$

where $t$ is a generator of the ideal $\mathfrak{m}/\mathfrak{m}^{n}\subset\mathcal{O}_{C}/\mathfrak{m}^{n}$ and $a_{i,j}$ ’s are the coordinates of $\mathbb{A}^{n(r-1)}$ . Then $U_{1}\setminus U_{i}=(a_{i,0}=0)\subset U_{1}=\mathbb{A}^{n(r-1)}$ and hence $U_{1}\setminus U_{i}\simeq\mathbb{A}^{n(r-1)-1}$ . Set

\displaystyle H\coloneqq\overline{U_{1}\setminus U_{2}}\subset F_{n},

which is a prime divisor of $F_{n}$ .

Consider the composite morphism $\mathbb{A}^{n(r-1)}\simeq U_{1}\subset F_{n}\hookrightarrow\mathrm{Gr}(V\otimes\mathcal{O}_{C}/\mathfrak{m}^{n},n)$ . Since $V\otimes\mathcal{O}_{C}/\mathfrak{m}^{n}$ has a basis $\{e_{i}\otimes t^{j}\mid 1\leqslant i\leqslant r,0\leqslant j\leqslant n-1\}$ and $\beta(e_{i}\otimes t^{j})=t^{j}\beta(e_{i})$ for $\beta:V\otimes\mathcal{O}_{C}\twoheadrightarrow\mathcal{O}_{C}/\mathfrak{m}^{n}$ , the morphism $\mathbb{A}^{n(r-1)}\hookrightarrow\mathrm{Gr}(V\otimes\mathcal{O}_{C}/\mathfrak{m}^{n},n)$ is described by the matrix of size $n\times nr$

\displaystyle\begin{pmatrix}A_{1}&A_{2}&\cdots&A_{r}\end{pmatrix},

where $A_{1}=E_{n}$ is the identity matrix of size $n$ and

\displaystyle A_{i}=\begin{pmatrix}a_{i,0}&0&\cdots&\cdots&0\\ a_{i,1}&a_{i,0}&\ddots&&\vdots\\ a_{i,2}&a_{i,1}&a_{i,0}&\ddots&\vdots\\ \vdots&\vdots&&\ddots&0\\ a_{i,r-1}&a_{i,r-2}&\cdots&\cdots&a_{i,0}\end{pmatrix}

for $2\leqslant i\leqslant r$ . For the Plücker coordinates $p_{1,\dots,n}$ and $p_{n+1,\dots,2n}$ on $\mathrm{Gr}(V\otimes\mathcal{O}_{C}/\mathfrak{m}^{n},n)$ , we have

\displaystyle p_{1,\dots,n}|_{U_{1}}=\det A_{1}=1,\quad p_{n+1,\dots,2n}|_{U_{1}}=\det A_{2}=a_{2,0}^{n}.

Hence it holds that

\displaystyle\operatorname{div}(p_{1,\dots,n})|_{U_{1}}=0,\quad\operatorname{div}(p_{n+1,\dots,2n})|_{U_{1}}=nH|_{U_{1}}.

By symmetry, we have $\operatorname{div}(p_{n+1,\dots,2n})|_{U_{2}}=0$ . Since $U_{2}\cap H=U_{2}\cap\overline{U_{1}\setminus U_{2}}=\emptyset$ , it holds that $\operatorname{div}(p_{n+1,\dots,2n})|_{U_{1}\cup U_{2}}=nH|_{U_{1}\cup U_{2}}$ .

Recall that the singular locus $F_{n}\setminus(U_{1}\cup\cdots\cup U_{r})$ has codimension two in $F_{n}$ . For $i\geqslant 3$ , $U_{i}\setminus(U_{1}\cup U_{2})$ is isomorphic to $\mathbb{A}^{n(r-1)-2}$ and hence $(U_{1}\cup\cdots\cup U_{r})\setminus(U_{1}\cup U_{2})$ has codimension two in $U_{1}\cup\cdots\cup U_{r}$ . Thus $F_{n}\setminus(U_{1}\cup U_{2})$ has codimension two in $F_{n}$ and hence $\operatorname{Cl}(F_{n})=\operatorname{Cl}(U_{1}\cup U_{2})$ . Since $U_{1}\cap H=U_{1}\setminus U_{2}$ and $U_{2}\cap H=\emptyset$ , we have $(U_{1}\cup U_{2})\setminus H=U_{2}$ . Then there exists an exact sequence

\displaystyle\mathbb{Z}[H|_{U_{1}\cup U_{2}}]\to\operatorname{Cl}(U_{1}\cup U_{2})\to\operatorname{Cl}(U_{2})\to 0.

Since $\operatorname{Cl}(U_{2})\simeq\operatorname{Cl}(\mathbb{A}^{n(r-1)})=0$ , it holds that $\operatorname{Cl}(F_{n})=\operatorname{Cl}(U_{1}\cup U_{2})=\mathbb{Z}[H]$ . Since $\operatorname{div}(p_{n+1,\dots,2n})|_{U_{1}\cup U_{2}}=nH|_{U_{1}\cup U_{2}}$ and $\operatorname{codim}_{F_{n}}(U_{1}\cup U_{2})=2$ , it holds that $nH=\operatorname{div}(p_{n+1,\dots,2n})\sim\mathcal{O}_{F_{n}}(1)$ . ∎

Proof of Theorem 1.1.

(1)-(4) follow from Propositions 2 and 3. (5) is nothing but § 4. ∎

5. The case $r=2$

We use the notation in §3. The purpose of this section is to give a description of the exceptional divisor of $\operatorname{pr}_{n+1}:F_{n,n+1}\to F_{n+1}$ for $r=2$ . Throughout this section, we assume $r=\operatorname{rank}\mathcal{E}=2$ and hence $\dim F_{n}=n(r-1)=n$ .

Lemma \thelem.

For $n\geqslant 1$ , there exists a natural embedding

(5.1)

\displaystyle F_{n-1}\hookrightarrow F_{n+1}\ :\ [\mathcal{A}_{n-1}]\mapsto[\mathfrak{m}\mathcal{A}_{n-1}].

Proof.

Let $0\to\mathscr{A}_{n-1}\to p_{C}^{*}\mathcal{E}\to\mathscr{B}_{n-1}\to 0$ be the universal exact sequence on $C\times F_{n-1}$ . Then we have an exact sequence

0\to\mathscr{A}_{n-1}/\mathfrak{m}\mathscr{A}_{n-1}\to p_{C}^{*}\mathcal{E}/\mathfrak{m}\mathscr{A}_{n-1}\to\mathscr{B}_{n-1}\to 0.

Since $\mathscr{A}_{n-1}/\mathfrak{m}\mathscr{A}_{n-1}$ and $\mathscr{B}_{n-1}$ are flat over $F_{n-1}$ of length $2$ and $n-1$ respectively, $p_{C}^{*}\mathcal{E}/\mathfrak{m}\mathscr{A}_{n-1}$ is flat over $F_{n-1}$ of length $n+1$ . Since $\det\mathfrak{m}\mathscr{A}_{n-1}=\mathfrak{m}^{2}\det\mathscr{A}_{n}=p_{C}^{*}\mathfrak{m}^{n+1}\det\mathcal{E}$ , the exact sequence $0\to\mathfrak{m}\mathscr{A}_{n-1}\to p_{C}^{*}\mathcal{E}\to p_{C}^{*}\mathcal{E}/\mathfrak{m}\mathscr{A}_{n-1}\to 0$ induces the morphism 5.1.

Furthermore, 5.1 is an embedding since it is the restriction to $F_{n-1}\subset\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n-1}\mathcal{E},n-1)$ of the embedding

	$\displaystyle\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n-1}\mathcal{E},n-1)$	$\displaystyle\hookrightarrow\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n}\mathcal{E},n-1)$
		$\displaystyle\simeq\mathrm{Gr}(\mathfrak{m}\mathcal{E}/\mathfrak{m}^{n+1}\mathcal{E},n-1)\hookrightarrow\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n+1}\mathcal{E},n+1),$

induced by the surjection $\mathcal{E}/\mathfrak{m}^{n}\mathcal{E}\to\mathcal{E}/\mathfrak{m}^{n-1}\mathcal{E}$ and an isomorphism $\mathcal{E}/\mathfrak{m}^{n}\mathcal{E}\simeq\mathfrak{m}\mathcal{E}/\mathfrak{m}^{n+1}\mathcal{E}\subset\mathcal{E}/\mathfrak{m}^{n+1}\mathcal{E}$ . The last embedding is obtained as $\mathrm{Gr}(\mathfrak{m}\mathcal{E}/\mathfrak{m}^{n+1}\mathcal{E},n-1)=\mathrm{Gr}(n+1,\mathfrak{m}\mathcal{E}/\mathfrak{m}^{n+1}\mathcal{E})\subset\mathrm{Gr}(n+1,\mathcal{E}/\mathfrak{m}^{n+1}\mathcal{E})=\mathrm{Gr}(\mathcal{E}/\mathfrak{m}^{n+1}\mathcal{E},n+1)$ . ∎

Remark \therem.

We can check that the embedding 5.1 is the same as the one constructed in [BJS24, §6.4, Proposition 9.1].

Lemma \thelem.

For $n\geqslant 1$ , the embedding 5.1 induces an embedding

(5.2)

\displaystyle F_{n-1,n}\hookrightarrow F_{n,n+1}\ :\ ([\mathcal{A}_{n-1}],[\mathcal{A}_{n}])\mapsto([\mathcal{A}_{n}],[\mathfrak{m}\mathcal{A}_{n-1}]).

Proof.

The embedding 5.1 induces an embedding $F_{n-1}\times F_{n}\to F_{n}\times F_{n+1}:([\mathcal{A}_{n-1}],[\mathcal{A}_{n}])\mapsto([\mathcal{A}_{n}],[\mathfrak{m}\mathcal{A}_{n-1}])$ . If $([\mathcal{A}_{n-1}],[\mathcal{A}_{n}])\in F_{n-1,n}$ , it holds that $\mathcal{A}_{n-1}/\mathcal{A}_{n}\simeq\mathcal{O}_{C}/\mathfrak{m}$ and hence $\mathfrak{m}\mathcal{A}_{n-1}\subset\mathcal{A}_{n}$ . Thus $([\mathcal{A}_{n}],[\mathfrak{m}\mathcal{A}_{n-1}])$ is contained in $F_{n,n+1}$ . ∎

The following proposition shows that the exceptional divisor of $\operatorname{pr}_{n+1}:F_{n,n+1}\to F_{n+1}$ is a $\mathbb{P}^{1}$ -bundle over $F_{n-1}\subset F_{n+1}$ .

Proposition \theprop.

If $n\geqslant 1$ , $F_{n-1,n}$ embedded in $F_{n,n+1}$ by 5.2 is the exceptional divisor of $\operatorname{pr}_{n+1}:F_{n,n+1}\to F_{n+1}$ . The restriction $\operatorname{pr}_{n+1}|_{F_{n-1,n}}:F_{n-1,n}\to F_{n+1}$ coincides with the $\mathbb{P}^{1}$ -bundle $\operatorname{pr}_{n-1}:F_{n-1,n}\to F_{n-1}\subset F_{n+1}$ .

Proof.

By 5.2, $\operatorname{pr}_{n+1}$ maps $([\mathcal{A}_{n-1}],[\mathcal{A}_{n}])\in F_{n-1,n}$ to $[\mathfrak{m}\mathcal{A}_{n-1}]\in F_{n+1}$ , which is regarded as $[\mathcal{A}_{n-1}]\in F_{n-1}\subset F_{n+1}$ under the embedding 5.1. Hence the restriction $\operatorname{pr}_{n+1}|_{F_{n-1,n}}$ coincides with $\operatorname{pr}_{n-1}:F_{n-1,n}\to F_{n-1}\subset F_{n+1}$ . Since $\dim F_{n-1,n}=n=\dim F_{n,n+1}-1$ , $F_{n-1,n}$ is the exceptional divisor of $\operatorname{pr}_{n,n+1}$ . ∎

References

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A remark on some punctual Quot schemes on smooth projective curves

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

Theorem 1.1.

Acknowledgments

2. Embedding to a Grassmannian

Proposition \theprop.

Proof.

Remark \therem.

Remark \therem.

Remark \therem.

3. Picard groups

Lemma \thelem.

Proof.

Lemma \thelem.

Proof.

Proposition \theprop.

Proof.

4. Divisor class groups

Proposition \theprop.

Proof.

Proof of Theorem 1.1.

5. The case r=2r=2

Lemma \thelem.

Proof.

Remark \therem.

Lemma \thelem.

Proof.

Proposition \theprop.

Proof.

References

5. The case $r=2$