Proof. See [Reference Huybrechts and LehnHL97, Chapter 8], we use the assumption that ${\mathrm {rk}}(\mathbf {v})>0$ .

Proof. By the functoriality of the determinant line bundle construction, we have

$$\begin{align*}\beta(\lambda(u))= \deg(\lambda_{F}(u)),\end{align*}$$

where $\lambda _{F}(u)$ is the determinant line bundle associated to the family F on $S\times C$ and a class $u \in K_{0}(S)$ . Using the Grothendieck–Riemann–Roch theorem and the projection formula, we obtain

$$ \begin{align*} \deg(\lambda_F(u))=\deg(p_{C*}(p^{*}_{S}u\cdot [F]))&=\int_{C} {\mathrm{ch}}(p_{C*}(p^{*}_{S}u\cdot [F]))\\ &=\int_{S\times C}{\mathrm{ch}}(p^*_{S}u\cdot [F]) \cdot p^*_{S}\mathrm{td}_S\\ &=\int_S {\mathrm{ch}}(u) \cdot p_{S*}{\mathrm{ch}}(F) \cdot \mathrm{td}_S\\ &=\int_{S} {\mathrm{ch}}(u) \cdot {\mathrm{ch}}(F)_{\mathrm{d}} \cdot \mathrm{td}_S. \end{align*} $$

Proof. Since $\chi (\mathbf {v} \cdot u_i )=0$ , we get that

$$\begin{align*}{\mathrm{rk}}(p_{C*}(p^{*}_{S}u_i\cdot [F]))=0,\end{align*}$$

and hence,

$$ \begin{align*}{\mathcal L}_{i}\cdot_{f} C =\deg(p_{C*}(p^{*}_{S}u_i\cdot [F]))=\chi(p_{C*}(p^{*}_{S}u_i\cdot [F])) &=\chi(p^{*}_{S}u_i\cdot [F]) \\ &=\chi(u_i\cdot p_{S*}[F]). \end{align*} $$

The claim then follows from the following computation for a class $u\in K_0(S)$ :

$$ \begin{align*} \chi( u_{1}\cdot u) &=-{\mathrm{rk}}(\mathbf{v})\chi(u \cdot h)+\chi(\mathbf{v}\cdot h)\chi([{\mathcal O}_{\mathrm{pt}}]\cdot u) \nonumber \\ &=-{\mathrm{rk}}(\mathbf{v})\left(\deg(u)-\frac{{\mathrm{rk}}(u)}{2}H^{2}-\frac{{\mathrm{rk}}(u)}{2}H\cdot \mathrm{c}_{1}(S) \right) \nonumber\\ &\quad+\left(\deg(\mathbf{v})-\frac{{\mathrm{rk}}(\mathbf{v})}{2}H^{2}-\frac{{\mathrm{rk}}(\mathbf{v})}{2}H\cdot \mathrm{c}_{1}(S)\right){\mathrm{rk}}(u)\nonumber \\ & ={\mathrm{rk}}(u)\deg(\mathbf{v})-{\mathrm{rk}}(\mathbf{v})\deg(u), \nonumber \\ \chi( u_{0}\cdot u) &= -{\mathrm{rk}}(\mathbf{v})\chi(u)+\chi(\mathbf{v})\chi([{\mathcal O}_{\mathrm{pt}}]\cdot u) \nonumber \\ &= \chi(\mathbf{v}){\mathrm{rk}}(u)-{\mathrm{rk}}(\mathbf{v})\chi(u). \end{align*} $$

This finishes the proof.

Proof. Let F be the family of sheaves on $S\times C$ associated to f. Assume for simplicity that f has one base point $b \in C$ . By Langton’s semistable reduction [Reference LangtonLan75], the sheaf F can be modified at a point b to a sheaf which is stable over b and is isomorphic to F away from $S\times \{b\} \subset S\times C$ . The modification is given by a finite sequence of short exact sequences,

$$ \begin{align*} 0 \rightarrow F^{1}\rightarrow &F^0 \rightarrow Q^{1} \rightarrow 0, \\ &\vdots \\ 0\rightarrow F^{k} \rightarrow &F^{k-1} \rightarrow Q^{k} \rightarrow 0, \end{align*} $$

where $F^0=F$ , the sheaf $F^{k}$ is stable over $b \in C$ , and $Q^{i}$ is the maximal destabilising quotient sheaf of $F^{i-1}_{b}$ ; that is, the quotient of $F^{i-1}_{b}$ by the maximal destabilising subsheaf $E^i$ ,

$$\begin{align*}0\rightarrow E^i \rightarrow F^{i-1}_{b} \rightarrow Q^i \rightarrow 0.\end{align*}$$

More precisely, if $F^{i-1}_b$ does not have torsion, we refer to [Reference Huybrechts and LehnHL97, Section 1.3] for the construction of the maximal destabilising subsheaf $E^i$ . If $F^{i-1}_{b}$ has torsion $T^i$ , then $E^i$ is defined to be the preimage of the maximal destabilizing subsheaf of the torsion-free sheaf $F^{i-1}_{b}/T^i$ . If $ F^{i-1}_{b}/T^i$ is stable. Then $Q^i=F^{i-1}_{b}/T^i$ . Note that by the maximality, $Q^i$ is torsion-free.

Since $Q^{i}$ is a destabilizing quotient of a sheaf in the class $\mathbf {v}$ , we have

(3.2) $$ \begin{align} \mathrm{deg}(\mathbf{v})\mathrm{rk}(Q^{i})-\mathrm{rk}(\mathbf{v})\mathrm{deg}(Q^{i})\geq 0. \end{align} $$

Applying the derived pushforward $p_{S*}$ to each sequence, we get distinguished triangles

$$\begin{align*}p_{S*}(F^{i}) \rightarrow p_{S*}(F^{i-1}) \rightarrow Q^{i} \xrightarrow{}. \end{align*}$$

By Lemma 3.6, we obtain that

(3.3) $$ \begin{align} {\mathcal L}_{1} \cdot_{f^{i-1}}C={\mathcal L}_{1} \cdot_{f^i} C+\mathrm{deg}(\mathbf{v})\mathrm{rk}(Q^{i})-\mathrm{rk}(\mathbf{v})\mathrm{deg}(Q^{i}), \end{align} $$

where $f^i$ is the quasimap associated to $F^i$ . The line bundle ${\mathcal L}_{1}$ is nef on $M(\mathbf {v})$ by Theorem 2.1; therefore,

(3.4) $$ \begin{align} {\mathcal L}_{1} \cdot_{f^k}C\geq 0 \end{align} $$

because $f^k$ does not have base points. The claim for a quasimap with one base point now follows from (3.2), (3.3) and (3.4). By applying the semistable reduction to all base points at the same time, we extend the argument to a quasimap with arbitrarily many base points.

Proof. Let us denote ${\mathcal L}_1 \cdot _f C$ by $\beta ({\mathcal L}_1)$ . By Lemma 3.8, (3.2) and (3.3), we have

$$\begin{align*}\beta({\mathcal L}_{1}) \geq \mathrm{deg}(\mathbf{v})\mathrm{rk}(Q^{i})-\mathrm{rk}(\mathbf{v})\mathrm{deg}(Q^{i})\geq0. \end{align*}$$

This bounds the degrees of $Q^i$ as follows:

(3.5) $$ \begin{align} \frac{\deg(\mathbf{v}){\mathrm{rk}}(Q^i)-\beta({\mathcal L}_{1}) }{{\mathrm{rk}}(\mathbf{v})}\leq\deg(Q^i)\leq \frac{\deg(\mathbf{v}){\mathrm{rk}}(Q^i)}{{\mathrm{rk}}(\mathbf{v})}. \end{align} $$

We therefore get a uniform bound on $\deg (Q^i)$ depending on the sign of $\deg (\mathbf {v})$ ,

(3.6) $$ \begin{align} \begin{aligned} -\frac{\beta({\mathcal L}_{1})}{{\mathrm{rk}}(\mathbf{v})}<\deg(Q^i)\leq \deg(\mathbf{v}),\quad &\text{if } \deg(\mathbf{v})\geq 0, \\ \deg(\mathbf{v})-\frac{\beta({\mathcal L}_{1})}{{\mathrm{rk}}(\mathbf{v})}\leq \deg(Q^i) < 0, \quad& \text{if } \deg(\mathbf{v})< 0. \end{aligned} \end{align} $$

Combing these two bounds together, we get the first inequality in the statement of the corollary.

We now deal with $\mu _{\mathrm {max}}(Q^i)$ . Let $E^i$ be the maximal destabilizing subsheaf of $F^{i-1}_{b}$ from the proof of Lemma 3.8. If $E^i$ is torsion, then

$$\begin{align*}\mu_{\mathrm{max}}(Q^i)=\mu(Q^i)\leq \mu(\mathbf{v})\end{align*}$$

since $F^{i-1}_{b}/ E^i=Q^i$ and $Q^i$ is stable. If $E^i$ is non-torsion, then by the maximality, we must have

$$\begin{align*}\mu_{\mathrm{max}}(Q^i) \leq \mu(E^i).\end{align*}$$

Indeed, if it was not the case, then by taking the preimage of the maximal destabilizing subsheaf of $Q^i$ , we would obtain a subsheaf of $F^{i-1}_{b}$ , whose slope is greater than $\mu (E^i)$ , contradicting the maximality of $E^i$ . Using the fact that $\deg (\mathbf {v})=\deg (Q^i)+\deg (E^i)$ , and (3.5), we obtain a uniform upper bound on $\mu (E^i)$ , which therefore bounds $\mu _{\mathrm {max}}(Q^i)$ ,

(3.7) $$ \begin{align} \begin{aligned} \mu_{\mathrm{max}}(Q^i) \leq \mu(E^i)< \deg(\mathbf{v})+\frac{\beta({\mathcal L}_{1})}{{\mathrm{rk}}(\mathbf{v})},\quad &\text{if } \deg(\mathbf{v})\geq 0, \\ \mu_{\mathrm{max}}(Q^i) \leq \mu(E^i)\leq \frac{\beta({\mathcal L}_1)}{{\mathrm{rk}}(\mathbf v)}, \quad& \text{if } \deg(\mathbf{v})< 0. \end{aligned} \end{align} $$

Combining these two bounds together with the bound for torsion $E^i$ , we get the second inequality in the statement of the corollary.

Proof of Proposition 3.7.

We now deal with the claim of the proposition. Assume f is constant. Then ${\mathcal L}_{0}\otimes {\mathcal L}_{1}^{m} \cdot _{f} C=0$ for all m.

Conversely, assume f is non-constant, and let F be the associated sheaf. We apply the semistable reduction to F at all points at once. Then by Lemma 3.6, we obtain

(3.8) $$ \begin{align} {\mathcal L}_{0}\otimes{\mathcal L}_{1}^{m} \cdot_{f} C={\mathcal L}_{0}&\otimes{\mathcal L}_{1}^{m} \cdot_{f^k} C\notag \\ &+m\sum_{i} \mathrm{deg}(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\mathrm{deg}(Q^i)+\sum_{i}\chi(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\chi(Q^i), \end{align} $$

where $f^k$ is a stable map. As in Lemma 3.8, $Q^i$ is destabilizing; hence,

(3.9) $$ \begin{align} \mathrm{deg}(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\mathrm{deg}(Q^i)\geq 0. \end{align} $$

We therefore have to analyse the terms

(3.10) $$ \begin{align} \chi(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\chi(Q^i). \end{align} $$

We will split our analysis, depending on whether (3.9) is positive or zero. If

$$\begin{align*}\mathrm{deg}(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\mathrm{deg}(Q^i)=0,\end{align*}$$

then since $Q^i$ is a destabilizing quotient, it must be destabilizing in the second coefficient of the Hilbert polynomial (see Section A.3 for the definition of Hilbert polynomials); that is,

(3.11) $$ \begin{align} \chi(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\chi(Q^i)>0. \end{align} $$

Consider now the case of $Q^i$ , such that

$$\begin{align*}\mathrm{deg}(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\mathrm{deg}(Q^i)>0.\end{align*}$$

By [Reference Huybrechts and LehnHL97, Corollary 3.3.3] and Serre’s duality, dimensions of $H^0( Q^i)$ and $H^2(Q^i)$ can be bounded as follows:

(3.12) $$ \begin{align} \begin{aligned} h^0(Q^i)&\leq \frac{\deg(S){\mathrm{rk}}(\mathbf{v})}{2}\left[\frac{\mu_{\mathrm{max}}(Q^i)}{\deg(S)}+\frac{{\mathrm{rk}}(\mathbf{v})-1}{2}+2\right]^2_+ ,\\ h^2(Q^i)&\leq \frac{\deg(S){\mathrm{rk}}(\mathbf{v})}{2} \left[\frac{\mu_{\mathrm{max}}((Q^i)^\vee\otimes\omega_S)}{\deg(S)}+\frac{{\mathrm{rk}}(\mathbf{v})-1}{2}+2\right]^2_+, \end{aligned} \end{align} $$

where $\deg (S)=\mathrm {c}_1({\mathcal O}_S(1))^2$ , and $[x]_+=\mathrm {max}\{0,x\}$ .

By Corollary 3.9, $\mu _{\mathrm {max}}(Q^i)$ is bounded from above. On the other hand, since $Q^i$ is torsion-free, we can also bound $\mu _{\mathrm {max}}((Q^i)^\vee \otimes \omega _S)$ from above in terms of $\deg (Q^i)$ and $\mu _{\mathrm {max}}(Q^i)$ . First,

$$\begin{align*}\mu_{\mathrm{max}}((Q^i)^\vee\otimes\omega_S)= \mu_{\mathrm{max}}((Q^i)^\vee)+\deg(\omega_S).\end{align*}$$

Now let $E \hookrightarrow (Q^i)^\vee $ be a saturated subsheaf.Footnote ⁸ Then by dualizing, we get a map $(Q^i)^{\vee \vee } \rightarrow E^\vee $ , which is surjective away from a codimension 2 locus. We also have an embedding $Q^i \hookrightarrow (Q^i)^{\vee \vee }$ which is an isomorphism away from a codimension 2 locus. For the purpose of calculating the slope, we therefore may assume that we have a surjection $Q^i \twoheadrightarrow E^\vee $ . Let K be its kernel, the slope of $E^\vee $ can be bounded as follows:

$$ \begin{align*} \mu(E^\vee)=\frac{\deg(Q^i)}{{\mathrm{rk}}(E^\vee)}-\frac{{\mathrm{rk}} (K)}{{\mathrm{rk}}(E^\vee)}\mu(K) &\geq \frac{\deg(Q^i)}{{\mathrm{rk}}(E^\vee)}-\frac{{\mathrm{rk}} (K)}{{\mathrm{rk}}(E^\vee)}\mu_{\mathrm{max}}(Q^i). \end{align*} $$

This bounds the slope of E from above,

$$\begin{align*}\mu(E)=-\mu(E^\vee)\leq \frac{{\mathrm{rk}} (K)}{{\mathrm{rk}}(E^\vee)}\mu_{\mathrm{max}}(Q^i)-\frac{\deg(Q^i)}{{\mathrm{rk}}(E^\vee)}.\end{align*}$$

Using Corollary 3.9, we obtain a uniform bound,

$$\begin{align*}\mu(E) <\left({\mathrm{rk}}(\mathbf{v})+1\right)\cdot \left(|\deg(\mathbf{v})|+\frac{\beta({\mathcal L}_{1})}{{\mathrm{rk}}(\mathbf{v})}\right).\end{align*}$$

We conclude that

(3.13) $$ \begin{align} \mu_{\mathrm{max}}((Q^i)^\vee\otimes\omega_S)<\left({\mathrm{rk}}(\mathbf{v})+1\right)\cdot \left(|\deg(\mathbf{v})|+\frac{\beta({\mathcal L}_{1})}{{\mathrm{rk}}(\mathbf{v})}\right)+\deg(\omega_S). \end{align} $$

Overall, using Corollary 3.9, (3.12) and (3.13), we obtain that $\chi (Q^i)$ can be bounded by an explicit constant $\chi _0$ that depends only on $\beta $ , $\mathbf {v}$ and ${\mathcal O}_S(1)$ , assuming S is fixed,

$$\begin{align*}\chi(Q^i) \leq h^0(Q^i)+h^2(Q^i)<\chi_0.\end{align*}$$

This allows us to uniformly bound the terms (3.10),

(3.14) $$ \begin{align} \chi(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\chi(Q^i)>-\mathrm{rk}(\mathbf{v}) \cdot (|\chi(\mathbf{v})|+\chi_0 ). \end{align} $$

Now let $m_{0} \in {\mathbb {N}}$ be such that ${\mathcal L}_{0}\otimes {\mathcal L}_{1}^{m_{0}}$ is ample on $M(\mathbf {v})$ , which is possible by Theorem 2.1, and

$$ \begin{align*} \begin{aligned} m_{0}\cdot(\mathrm{deg}(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\mathrm{deg}(Q^i))-\mathrm{rk}(\mathbf{v}) \cdot (|\chi(\mathbf{v})|+\chi_0 )>0,\quad \end{aligned} \end{align*} $$

for all $Q^{i}$ satisfying

$$\begin{align*}\mathrm{deg}(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\mathrm{deg}(Q^i)>0.\end{align*}$$

Such $m_0$ exists because the quantity

$$\begin{align*}\mathrm{deg}(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\mathrm{deg}(Q^i)\end{align*}$$

is an integer, and, in particular, it is at least 1, if it is positive. More explicitly, we can choose $m_0$ such that ${\mathcal L}_{0}\otimes {\mathcal L}_{1}^{m_{0}}$ is ample on $M(\mathbf {v})$ , and

$$\begin{align*}m_0 \geq \mathrm{rk}(\mathbf{v}) \cdot (|\chi(\mathbf{v})|+\chi_0 )+1.\end{align*}$$

By the definition of $m_0$ and the fact that $\mathrm{deg}(\mathbf {v})\mathrm {rk}(Q^i)-\mathrm {rk}(\mathbf {v})\mathrm{deg}(Q^i)$ is an integer, the choice of $m_{0}$ depends only on $\mathbf {v}$ , ${\mathcal O}_S(1)$ (via the ampleness requirement) and the constant $\chi _0$ . The latter in turn depends on $\beta $ , $\mathbf {v}$ and ${\mathcal O}_S(1)$ by Corollary 3.9 and bounds presented above, assuming S is fixed. We conclude that $m_0$ depends only on $\beta $ , $\mathbf {v}$ and ${\mathcal O}_S(1)$ .

By (3.11), the choice of $m_0$ and (3.14), for all $m\geq m_0$ and for all i, we have

$$\begin{align*}m\cdot (\mathrm{deg}(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\mathrm{deg}(Q^i))+\chi(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\chi(Q^i)>0.\end{align*}$$

Hence, by (3.8) and Theorem 2.1, for all $m\geq m_0$ , we have

$$\begin{align*}{\mathcal L}_{0}\otimes {\mathcal L}_{1}^{m} \cdot_{f} C>0\end{align*}$$

for all non-constant prestable quasimaps of degree $\beta $ . By Lemma 3.8 and Lemma 3.9, the same bounds hold for the restriction of f to a subcurve $C'\subseteq C$ , since ${\mathcal L}_{1}\cdot _{f_{|C'}} C'\leq {\mathcal L}_{1}\cdot _{f} C$ . Hence, for all $m\geq m_0$ ,

$$\begin{align*}{\mathcal L}_{0}\otimes {\mathcal L}_{1}^{m}\cdot_{f_{|C'}} C'>0\end{align*}$$

if and only if $f_{|C'}$ is not constant.

Proof. By (2.1), we have an isomorphism

$$\begin{align*}(\tau,\lambda(\mathbf{u}))\colon \mathfrak{Coh}(S,\mathbf{v})\xrightarrow{\sim} \mathfrak{Coh}_{r}(S,\mathbf{v})\times B{\mathbb{C}}^*,\end{align*}$$

which identifies maps from a base scheme B to $\mathfrak {Coh}_{r}(S,\mathbf {v})$ , that is, the B-valued points, with maps from B to $\mathfrak {Coh}(S,\mathbf {v})$ together with a trivialisation $\phi \colon \lambda (\mathbf {u})_{|B}\xrightarrow {\sim } {\mathcal O}_{B}$ . By construction of $\mathfrak {Coh}(S,\mathbf {v})$ and $\lambda (\mathbf {u})$ , maps from B to $\mathfrak {Coh}(S,\mathbf {v})$ are given by sheaves F on $S\times B$ and $\lambda (\mathbf {u})_{|B} =\det (p_{B*}(p_{S}^{*}\mathbf {u}\otimes F))$ . Applying this to families of curves ${\mathcal C}$ over a base scheme $B'$ , and using Lemma 3.17 to match flatness of sheaves over ${\mathcal C}$ with flatness over $B'$ , we obtain that p and q must be inverses of each other.

Proof. By the local criteria of flatness, a sheaf F is flat over a closed point $p \in {\mathcal C}$ if and only if

$$\begin{align*}L\iota^*_pF= \iota_p^*F,\end{align*}$$

where $\iota _p \colon S\times \{p\} \hookrightarrow S\times {\mathcal C}$ is the natural inclusion. The inclusion $\iota _p$ can be factored as follows:

where ${\mathcal C}_b$ is the fiber in which p is contained; hence,

$$\begin{align*}L\iota^*_pF= L\iota^{\prime *}_p (L\iota^*_bF).\end{align*}$$

Since ${\mathcal C}$ and F are flat over B, we conclude that $F_b$ is flat over ${\mathcal C}_b$ for all closed points $b\in B$ if and only if F is flat over ${\mathcal C}$ .

Proof. The restriction of a stable quasimap to an unstable component (a rational bridge or a rational tail) must be non-constant by stability, and it must pair positively with ${\mathcal L}_{\beta }$ by Proposition 3.7. Therefore, the number of unstable components of the domain curve of a stable quasimap is bounded in terms of $\beta $ . Hence, the projection $Q^{\epsilon }_{g,N}(M(\mathbf {v}),\beta ) \rightarrow {\mathfrak {M}}_{g,N}$ factors through a substack of finite type.

Proof. Let $F^0$ be an $\epsilon $ -stable sheaf with the associated quasimap $ f \colon C \rightarrow \mathfrak {Coh}(S,\mathbf {v})$ . The semistable reduction applied to all base points at once gives a sequence of short exact sequences,

$$ \begin{align*} 0 \rightarrow F^{1}\rightarrow &F^0 \rightarrow Q^{1} \rightarrow 0, \\ &\vdots \\ 0\rightarrow F^{k} \rightarrow &F^{k-1} \rightarrow Q^{k} \rightarrow 0, \end{align*} $$

such that $F^k$ defines a map $f^k\colon C \rightarrow M(\mathbf {v})$ . By the associated long exact sequence of cohomologies, $F^0$ is m-regular if and only if $F^k$ and $Q^i$ are m-regular. Using the fact that a family of sheaves is bounded if and only if its Mumford–Castelnuovo regularity is bounded and the set of Hilbert polynomials is finite [Reference Huybrechts and LehnHL97, Lemma 1.7.6], we have to show

1. (i) the number of steps k is bounded,

2. (ii) the family of possible $Q^i$ is bounded,

3. (iii) the family of possible $F^k$ is bounded.

(i) By the proof of Proposition 3.7, at each step of Langton’s semistable reduction, the difference between ${\mathcal L}_\beta \cdot _{f^i}C$ and ${\mathcal L}_\beta \cdot _{f^{i-1}}C$ is a strictly positive integer, and ${\mathcal L}_\beta \cdot _{f^k}C\geq 0$ . Hence, the number of steps k must be bounded by the intersection number of C with the line bundle ${\mathcal L}_\beta $ ,

$$\begin{align*}k\leq {\mathcal L}_\beta \cdot_f C=\beta({\mathcal L}_\beta).\end{align*}$$

(ii) We will show that the family of $Q^i$ is bounded by bounding the Hilbert polynomial and using the bound on $\mu _{\mathrm {max}}(Q^i)$ . By the Grothendieck–Riemann–Roch theorem, to bound the Hilbert polynomial, we need to bound the rank, the degree and the Euler characteristics of $Q^i$ .

The rank of $Q^i$ is bounded by ${\mathrm {rk}}(\mathbf {v})$ . The degree of $Q^{i}$ is bounded by Lemma 3.9. The Euler characteristics are bounded from above by the bounds from the proof of Proposition 3.7 and by Lemma 3.9. The bound from below for the Euler characteristics also follows from Proposition 3.7 and its proof as we will now demonstrate. First, by $\mathbf {(i)}$ , the number of steps k is bounded. Moreover, the family of possible $f^k \colon C \rightarrow M(\mathbf {v})$ is also bounded since its degree with respect to an ample line bundle ${\mathcal L}_\beta $ is bounded, and the moduli space of maps from a bounded family of curves of bounded degree is boundedFootnote ⁹ (boundedness of $F^k$ requires an extra argument, but for this part, boundedness of $f^k$ suffices because intersection numbers with ${\mathcal L}_0$ and ${\mathcal L}_1$ are invariant with respect to tensoring families of sheaves with line bundles; see the proof of the part (iii)). This implies that ${\mathcal L}_0 \cdot _{f^k} C$ is also bounded. By Lemma 3.6, we have the following expression for the difference between intersection numbers with the line bundle ${\mathcal L}_0$ ,

(4.1) $$ \begin{align} {\mathcal L}_0 \cdot_{f^i} C-{\mathcal L}_0 \cdot_{f^{i-1}} C=\chi(\mathbf{v})\mathrm{rk}(Q^i)-\mathrm{rk}(\mathbf{v})\chi(Q^i). \end{align} $$

It must be bounded from below because $\chi (Q^i)$ is bounded from above. We conclude that the difference (4.1) must be also bounded from above because $\beta ({\mathcal L}_0)$ is fixed, there are bounded number of steps, and ${\mathcal L}_0 \cdot _{f^k} C$ is bounded. This bounds the Euler characteristics of $Q^i$ from below. We thereby bounded the Hilbert polynomial of $Q^i$ . Finally, $\mu _{\mathrm {max}}(Q^i)$ is bounded by Lemma 3.9. Using [Reference Huybrechts and LehnHL97, Theorem 3.3.7], we conclude that the family of possible $Q^i$ is bounded.

(iii) For this part, we need the boundedness of corresponding maps $f^k\colon C \rightarrow M(\mathbf {v})$ and the boundedness of line bundles L such that

$$\begin{align*}\det(p_{C*}(p_{S}^{*}\mathbf{u}\otimes F^k))\cong L.\end{align*}$$

The latter is needed because the semistable reduction does not preserve the condition $\det (p_{C*}(p_{S}^{*}\mathbf {u}\otimes F))\cong {\mathcal O}_{C}$ . On the other hand, the sheaf $F^k\otimes p_C^*L^{-1}$ satisfies $\det (p_{C*}(p_{S}^{*}\mathbf {u}\otimes F^k\otimes p^*_CL^{-1}))\cong {\mathcal O}_{C}$ ; hence, the associated quasimap is a lift of a quasimap from $M(\mathbf {v})$ by the section $s_{\mathbf {u}}$ . We refer to Remark 3.2 for more on how different lifts are related.

The boundedness of $f^k\colon C \rightarrow M(\mathbf {v})$ was established in the proof of the part $(\mathbf {ii})$ above. It remains to deal with line bundles L. By the exact sequences above and the boundedness of $Q^i$ from the part $(\mathbf {ii})$ , the boundedness of possible L follows from the boundedness of the number of steps k, which was proved in $(\mathbf {i})$ .

Here, we crucially rely on the fact that our target is the rigidified stack $\mathfrak {Coh}_{r}(S,\mathbf {v})$ , which means that we start with a sheaf that satisfies the condition $\det (p_{C*}(p_{S}^{*}\mathbf {u}\otimes F^0))\cong {\mathcal O}_C$ . Otherwise, the line bundle $\det (p_{C*}(p_{S}^{*}\mathbf {u}\otimes F^0))$ can be arbitrary; the family of possible L is therefore unbounded.

Overall, we conclude that the number of steps k is bounded, and the families of possible $Q^i$ and $F^k$ are also bounded. This implies that the Mumford–Castelnuovo regularity of $\epsilon $ -stable sheaves with the given Chern character is bounded by some number A,

$$\begin{align*}\mathrm{reg}(F)\leq A, \end{align*}$$

and the set of Hilbert polynomials of such sheaves is finite.

Proof. The condition of prestability of quasimaps can be easily seen to be open. The condition $\mathbf {(i)}$ of $\epsilon $ -stability is also open since ampleness of line bundles is open. We therefore have to deal with the condition $\mathbf {(ii)}$ . Let

$$\begin{align*}{\mathcal U} \subset \mathrm{Map}_{{\mathfrak{M}}_{g,N}}(\mathfrak{C}_{g,N}, \mathfrak{Coh}_r(S,\mathbf{v})\times {\mathfrak{M}}_{g,N}) \end{align*}$$

be an open quasi-compact substack. We may assume that ${\mathcal U}$ contains only prestable quasimaps. Let ${\mathcal C}_{{\mathcal U}}$ be the universal curve over ${\mathcal U}$ and let

be the universal map. We want to stratify the space ${\mathcal U}$ in the way that can be stabilized at base points to be able to evaluate their length. First, we stratify ${\mathcal U}$ according to the number of base points. Namely, let ${\mathcal U}_k$ be moduli space of maps from prestable nodal curves to $\mathfrak {Coh}_{r}(S,\mathbf {v})$ with $N+k$ markings, such that

• ∘ there are exactly k base points,

• ∘ all k base points are marked by the last k markings,

• ∘ the underlying map with N markings is contained in ${\mathcal U}$ .

These are locally closed substacks of mapping stacks with $N+k$ markings. Since ${\mathcal U}$ is of finite type, the number of base points is bounded from below; hence, a finite disjoint union $\bigcup _{k} {\mathcal U}_k$ covers ${\mathcal U}$ (i.e., there exists a surjection,

$$\begin{align*}p\colon \bigcup_{k} {\mathcal U}_k \rightarrow {\mathcal U},\end{align*}$$

given by forgetting the last k makings). We now further stratify ${\mathcal U}_k$ . Consider the universal maps,

since all base points are contained in the regular locus of the universal curve, the maps can be stabilized at base points over the generic points of irreducible components of ${\mathcal U}_k$ (we can reduce the components, if necessary, to talk about generic points, as this does not affect the topology). Note that stabilisation just removes indeterminacies of the associated rational map

In particular, there exists a dense open subset of ${\mathcal U}_k$ , such that the quasimap can be stabilised at $i^{\mathrm {th}}$ base point. By passing to the complement of that open subset, repeating the argument and using the fact that ${\mathcal U}_k$ is quasicompact, we obtain a finite stratification of ${\mathcal U}_k$ into locally closed subsets,

$$\begin{align*}\bigcup_{j,i} {\mathcal U}^j_{k,i}={\mathcal U}_k, \end{align*}$$

such that the quasimap map can be stabilized at $i^{\mathrm {th}}$ base point over each stratum ${\mathcal U}^j_{k,i}$ . We now can evaluate the length of base points at each stratum ${\mathcal U}^j_{k,i}$ . The condition $\mathbf {(ii)}$ of $\epsilon $ -stability involves bounding a degree of line bundles from above; hence, this condition just picks some connected components (or none at all) of the strata ${\mathcal U}^j_{k,i}$ , which we denote by ${\mathcal U}^{j'}_{k,i}$ . Since ${\mathcal U}^j_{k,i}$ are locally closed subsets, ${\mathcal U}^{j'}_{k,i}$ are also locally closed and, in particular, constructible. The locus of $\epsilon $ -stable quasimaps ${\mathcal U}^{\epsilon }_k$ in ${\mathcal U}_k$ is given by intersections of $\cup _{j'}{\mathcal U}^{j'}_{k,i}$ for all i,

$$\begin{align*}{\mathcal U}^{\epsilon}_k :=\bigcap_i \left( \bigcup_{j'}{\mathcal U}^{j'}_{k,i}\right), \end{align*}$$

which is constructible, since it involves finite intersections and finite unions. The locus of $\epsilon $ -stable quasimaps in ${\mathcal U}$ is the image of the union $\cup _k{\mathcal U}^{\epsilon }_k$ ; that is, on the level of geometric points, we have

$$\begin{align*}|Q^{\epsilon}_{g,N}(M(\mathbf{v}),\beta)\cap {\mathcal U}|=\lvert p\left(\bigcup_k{\mathcal U}^{\epsilon}_k\right)\rvert.\end{align*}$$

Since p is a map between stacks of finite type, the image of a constructible set is constructible. We conclude that $Q^{\epsilon }_{g,N}(M(\mathbf {v}),\beta )$ is locally constructible.

Proof. As we said before, the only nontrivial part of the claim is the condition $\mathbf {(ii)}$ of $\epsilon $ -stability. By Lemma 4.4, we can use the valuative criterion of openess for (locally) constructible sets. We therefore have to show that if a family of quasimaps $f \colon {\mathcal C} \rightarrow \mathfrak {Coh}_{r}(S,\mathbf {v})$ over the spectrum of a discrete valuation ring $\Delta $ does not satisfy the condition $\mathbf {(ii)}$ at the generic fiber, it also does not satisfy it at the special fiber.

If the generic fiber $f^\circ \colon {\mathcal C}^\circ \rightarrow \mathfrak {Coh}_{r}(S,\mathbf {v})$ does not satisfy the condition $\mathbf {(ii)}$ , then there exists a base point $b^\circ \in {\mathcal C}^\circ $ which violates it. We can stabilize it to obtain another family $f^{\prime \circ } \colon {\mathcal C}^\circ \rightarrow \mathfrak {Coh}_{r}(S,\mathbf {v})$ over $\Delta ^\circ $ , which does not have a base point at $b^\circ $ . It extends to the family $f' \colon {\mathcal C} \rightarrow \mathfrak {Coh}_{r}(S,\mathbf {v})$ over $\Delta $ by Lemma 4.9. By definition, the length of the base point $b^\circ $ is given as follows:

$$\begin{align*}\ell(b^\circ)={\mathcal L}_{\beta} \cdot_{f^\circ} {\mathcal C}^\circ-{\mathcal L}_{\beta} \cdot_{f^{\prime\circ}} {\mathcal C}^\circ.\end{align*}$$

Let $b_0 \in {\mathcal C}_0$ be the limit of $b^\circ $ in the central fiber ${\mathcal C}_0$ . Then

$$\begin{align*}\ell(b_0)\geq {\mathcal L}_{\beta} \cdot_{f_0} {\mathcal C}_0-{\mathcal L}_{\beta} \cdot_{f^{\prime}_0} {\mathcal C}_0={\mathcal L}_{\beta} \cdot_{f^\circ} {\mathcal C}^\circ-{\mathcal L}_{\beta} \cdot_{f^{\prime\circ}} {\mathcal C}^\circ.\end{align*}$$

We use the sign of inequality because $b_0$ might still be a base point of $f^{\prime }_0$ whose length with respect to $f^{\prime }_0$ is smaller than the length with respect to $f_0$ . We therefore conclude that $f_0$ also does not satisfy the condition $\mathbf {(ii)}$ . This implies that the complement of $Q^{\epsilon }_{g,N}(M(\mathbf {v}), \beta )$ is closed; hence, $Q^{\epsilon }_{g,N}(M(\mathbf {v}), \beta )$ is open.

Proof. First, $Q^{\epsilon }_{g,N}(M(\mathbf {v}), \beta )$ is algebraic, quasi-separated and locally of finite presentation by Proposition 4.5, since mapping stacks have these properties. Second, by Lemma 4.3, Lemma 4.2 and [Reference Huybrechts and LehnHL97, Lemma 1.7.6], the stack $M_{\mathbf {v},\check {\beta }}^{\epsilon }(S\times C_{g,N})$ is quasi-compact. Hence by Proposition 3.7, the stack $Q^{\epsilon }_{g,N}(M(\mathbf {v}), \beta )$ is also quasi-compact, and therefore of finite presentation.

Proof. Assume F is flat, such that $F_{p}$ is torsion-free for a general $p \in C$ . Let $T(F) \subset F$ be the maximal torsion subsheaf. First, $T(F)\neq F$ because ${\mathrm {rk}}(F)> 0$ (a general fiber of F is torsion-free and therefore of nonzero rank). It also cannot be supported on fibers of $S\times C \rightarrow C$ due to the flatness of F over C; therefore, Supp $(T(F))$ intersects a general fiber. Since $F/T(F)$ is generically flat, restricting $T(F) \subset F$ to a general fiber $p\in C$ , we get a torsion subsheaf of $F_{p}$ for a general $p \in C$ , which is zero; therefore, $T(F)$ is zero. The converse follows from the fact that torsion freeness implies flatness over a discrete valuation ring, and torsion in a general fiber $F_p$ would produce torsion in F.

Proof. It is enough to show that ${\mathcal H} om_{p_{\mathcal U}} (F_1,F_2)=p_{{\mathcal U}*}{\mathcal H} om(F_1,F_2)$ is reflexive because being reflexive is equivalent to being locally free on regular surfaces.

Since $F_1$ is flat over ${\mathcal U}$ , by the construction from the proof of [Reference Huybrechts and LehnHL97, Proposition 2.1.10], there is an exact sequence

$$\begin{align*}H_1 \rightarrow H_0 \rightarrow F_1 \rightarrow 0,\end{align*}$$

such that $H_i=p_{{\mathcal U}}^*H_i' \otimes {\mathcal O}_S(-k_i)$ for a locally free sheaf $H_i'$ on ${\mathcal U}$ and for a sufficiently big integer $k_i$ . We apply ${\mathcal H} om_{p_{{\mathcal U}}}(-,F_2)$ to obtain an exact sequence

$$\begin{align*}0\rightarrow {\mathcal H} om_{p_{{\mathcal U}}}(F_1,F_2) \rightarrow {\mathcal H} om_{p_{{\mathcal U}}}(H_0,F_2) \rightarrow {\mathcal H} om_{p_{{\mathcal U}}}(H_1,F_2).\end{align*}$$

By using the identification

$$\begin{align*}{\mathcal H} om(H_i,F_2)=H_i^*\otimes F_2=p_{{\mathcal U}}^*H_i^{\prime *} \otimes F_2 \otimes {\mathcal O}_S(k_i) \end{align*}$$

and choosing big enough $k_i$ , the flatness of $F_2$ and [Reference Huybrechts and LehnHL97, Proposition 2.1.2] imply that

$$\begin{align*}{\mathcal H} om_{p_{{\mathcal U}}}(H_i,F_2)=H_i^{\prime *} \otimes p_{{\mathcal U}*} (F_2 \otimes {\mathcal O}_S(k_i)) \end{align*}$$

is a locally free sheaf. Hence, by [Sta24, Lemma 0EB8], we obtain that $ {\mathcal H} om_{p_{{\mathcal U}}}(F_1,F_2)$ is reflexive.

Proof. Let $\tilde {F}$ be the family on $S \times \tilde {{\mathcal C}}$ corresponding to the lift of $\tilde {f}$ by $s_{\mathbf {u}}$ . We then extend $\tilde {F}$ to a coherent sheaf F on $S\times {\mathcal C}$ , quotienting our torsion, if necessary. The sheaf F is therefore flat over $\Delta $ . Moreover, by Lemma 3.17, the sheaf F is flat over ${\mathcal C}$ if and only if the central fiber $F_{0}$ is flat over ${\mathcal C}_0$ . By Lemma 4.7, the central fiber $F_{0}$ is flat over ${\mathcal C}_0$ if and only if it is torsion-free, because ${\mathcal C}_{0}$ is regular at $p_i$ . If $F_{0}$ is not torsion-free, we can remove the torsion inductively as follows. Let $F^{0}=F$ and $F^{i}$ be defined by short exact sequences,

$$\begin{align*}0 \rightarrow F^{i} \rightarrow F^{i-1} \rightarrow Q^{i} \rightarrow 0,\end{align*}$$

such that $Q^{i}$ is the quotient of $F^{i-1}_{0}$ by the maximal torsion subsheaf. At each step, the torsion of $F_{0}^{i}$ is supported at slices $S\times \{ p_{i}\}$ ; therefore, all $F^i$ are isomorphic to $F^0$ over $S\times \tilde {{\mathcal C}}$ . By the argument from [Reference Huybrechts and LehnHL97, Theorem 2.B.1], this process terminates (i.e., $F^i=F^{i+1}$ and $F^i_{0}$ is torsion-free for $i\gg 0$ ). Let us redefine the sheaf F. We set $F=F^i$ for some $i\gg 0$ . Then the sheaf F induces a quasimap to $\mathfrak {Coh}(S,\mathbf {v})$ , and composing it with the projection to $\mathfrak {Coh}_{r}(S,\mathbf {v})$ , we thereby obtain an extension $f\colon {\mathcal C} \rightarrow \mathfrak {Coh}_{r}(S,\mathbf {v})$ of $\tilde {f}$ .

Consider now another extension $f'\colon {\mathcal C} \rightarrow \mathfrak {Coh}_{r}(S,\mathbf {v})$ . We lift both f and $f'$ to $\mathfrak {Coh}(S,\mathbf {v})$ with $s_{\mathbf {u}}$ . Let F and $F'$ be the corresponding families on $S\times {\mathcal C}$ , which satisfy

$$\begin{align*}F_{|\tilde{{\mathcal C}}}=F^{\prime}_{|\tilde{{\mathcal C}}}.\end{align*}$$

We will show that F and $F'$ are in fact isomorphic, using the argument from [Reference KuhnKuh23, Lemma 3.25]. Let ${\mathcal U}\subseteq {\mathcal C}$ be a regular open neighbourhood of $\{p_1, \dots , p_m\}\subset {\mathcal C}$ . By Lemma 4.8, ${\mathcal H} om_{p_{{\mathcal U}}}(F,F')$ is locally free. Moreover, since F and $F'$ are equal away from $\{p_1 ,\dots , p_m\}$ and a general fiber is simple, ${\mathcal H} om_{p_{{\mathcal U}}}(F,F')$ must be a line bundle. It has a nonvanishing section given by the identity morphism, defined away from $\{p_1 ,\dots , p_m\}$ . Since $\{p_1 ,\dots , p_m\}$ is of codimension 2 and ${\mathcal H} om_{p_{{\mathcal U}}}(F,F')$ is locally free, the section extends to the whole ${\mathcal U}$ , providing a trivialisation ${\mathcal H} om_{p_{{\mathcal U}}}(F,F')\cong {\mathcal O}_{\mathcal U}$ . Using the tautological morphismFootnote ¹¹

$$\begin{align*}F_{|{\mathcal U}} \otimes p_{{\mathcal U}}^*({\mathcal H} om_{p_{{\mathcal U}}}(F,F')) \rightarrow F^{\prime}_{|{\mathcal U}}, \end{align*}$$

and the trivialisaiton above, we therefore obtain a morphism

$$\begin{align*}F_{|{\mathcal U}} \rightarrow F^{\prime}_{|{\mathcal U}},\end{align*}$$

which is an identity away $\{p_1 ,\dots , p_m\}$ . Gluing it with the identity morphism over $\tilde {{\mathcal C}}$ , we obtain a morphism defined over the entire ${\mathcal C}$ ,

(4.2) $$ \begin{align} F \rightarrow F'. \end{align} $$

It is injective, and its cokernel Q is supported over $\{p_1 ,\dots , p_m\}$ . By pulling back to the closed fiber ${\mathcal C}_0$ of ${\mathcal C}$ , we obtain an exact sequence

$$\begin{align*}0 \rightarrow F_{|{\mathcal C}_0} \rightarrow F^{\prime}_{|{\mathcal C}_0} \rightarrow Q_{|{\mathcal C}_0}\rightarrow0.\end{align*}$$

Note that it is exact from the left because $F_{|{\mathcal C}_0}$ is torsion-free by Lemma 4.7. Since $F_{|{\mathcal C}_0}$ and $F^{\prime }_{|{\mathcal C}_0}$ have the same Chern character (equal to the one of the generic fiber), we obtain that Q must be zero; hence, the morphism (4.2) is an isomorphism. Finally, for a fixed isomorphism of F and $F'$ over $\tilde {{\mathcal C}}$ , the isomorphism over ${\mathcal C}$ is unique, as the isomorphism over $\tilde {{\mathcal C}}$ determines a section of ${\mathcal H} om_{p_{{\mathcal U}}}(F,F')$ away from $\{p_1 ,\dots , p_m\}$ , which uniquely extends to the whole ${\mathcal U}$ .

Proof. We have to show that

$$\begin{align*}\operatorname{Hom}(F,F)={\mathbb{C}}.\end{align*}$$

By passing to the normalisation of C, we may assume that C is smooth. The sheaf ${\mathcal H} om_{p_{C}}(F,F)=p_{C*}{\mathcal H} om(F,F)$ is torsion-free on C, and a general fiber of F over C is simple; hence, ${\mathcal H} om_{p_{C}}(F,F)$ is a line bundle. Moreover, it has a nonvanishing section given by the identity morphism ${\mathcal O}_C \rightarrow {\mathcal H} om_{p_{C}}(F,F)$ . We conclude that ${\mathcal H} om_{p_{C}}(F,F)$ is a trivial line bundle, and therefore,

$$\begin{align*}\operatorname{Hom}(F,F)=H^0(S\times C, {\mathcal H} om(F,F))=H^0(C, {\mathcal H} om_{p_{C}}(F,F))={\mathbb{C}}, \end{align*}$$

which finishes the proof.

Proof. We use identification from Theorem 3.16. By Corollary 4.6 and Lemma 4.11, the forgetful map

$$\begin{align*}M_{\mathbf{v},\check{\beta}}^{\epsilon}(S\times C_{g,N}) \rightarrow {\mathfrak{M}}_{g,N}\end{align*}$$

is representable by algebraic spaces. By $\epsilon $ -stability, $\epsilon $ -stable sheaves are fixed by at most a finite discrete subgroup of the automorphism group of a curve. We therefore conclude that $M_{\mathbf {v},\check {\beta }}^{\epsilon }(S\times C_{g,N})$ is a quasi-separated Deligne–Mumford stack of finite type. Using the valuative criteria of properness for quasi-separated Deligne–Mumford stacks of finite type [Sta24, Section 0CLY] and Lemma 4.9, the proof of properness then proceeds as in the GIT case [Reference Ciocan-Fontanine, Kim and MaulikCKM14, Proposition 4.3.1].

Proof. In fact, we will show that p is a finite-group torsor. Let

$$\begin{align*}\Gamma:=\mathrm{Pic}_0(C)[{\mathrm{rk}}(\mathbf{v})]\end{align*}$$

be the ${\mathrm {rk}}(\mathbf {v})$ -torsion points of $ \mathrm { Pic} _0(C)$ . The group $\Gamma $ acts on the moduli space $\widetilde M_{\mathbf {v},\check {\beta }}(S \times C)$ by tensoring with line bundles because $\det (F\otimes A)=\det (F)\otimes A^{\otimes {\mathrm {rk}}(\mathbf {v})}$ for a line bundle $A \in \mathrm {Pic}(S\times C)$ . The action is free, because the following holds:

$$\begin{align*}\det(p_{C*}(p_{S}^{*}\mathbf{u}\otimes F\otimes p_C^* A)\cong \det(p_{C*}(p_{S}^{*}\mathbf{u}\otimes F)\otimes A,\end{align*}$$

which is due to $\chi ( \mathbf {v} \cdot u)=1$ . The map p is $\Gamma $ -invariant. In particular, we have the induced map

(4.3) $$ \begin{align} \widetilde M_{\mathbf{v},\check{\beta}}(S \times C) /\Gamma \rightarrow M_{\mathbf{v},\check{\beta}}^{0^+}(S\times C). \end{align} $$

Conversely, the inverse of this map is given by a $\Gamma $ -torsor on $M_{\mathbf {v},\check {\beta }}^{0^+}(S\times C)$ which parametrises sheaves F in $M_{\mathbf {v},\check {\beta }}^{0^+}(S\times C)$ together with a ${\mathrm {rk}}(\mathbf {v})^{\mathrm {th}}$ -root of $\det (F)\otimes L^{-1}$ . Using the assumption $h^1(S)=0$ and the identification $ \mathrm {Pic} _0(S\times C)\cong \mathrm {Pic} _0(C)$ , this torsor is constructed as the fiber product,

There exists a $\Gamma $ -equivariant map,

$$\begin{align*}M_{\mathbf{v},\check{\beta}}^{0^+}(S\times C)^{\frac{1}{{\mathrm{rk}}(\mathbf{v})}} \rightarrow \widetilde M_{\mathbf{v},\check{\beta}}(S \times C), \quad F \mapsto F\otimes(\det(F)^{-1}\otimes L)^{\frac{1}{{\mathrm{rk}}(\mathbf{v})}},\end{align*}$$

which induces the inverse of (4.3). We conclude

$$\begin{align*}\widetilde M_{\mathbf{v},\check{\beta}}(S \times C) /\Gamma \cong M_{\mathbf{v},\check{\beta}}^{0^+}(S\times C);\end{align*}$$

in particular, $\widetilde M_{\mathbf {v},\check {\beta }}(S \times C) $ is a $\Gamma $ -torsor over $M_{\mathbf {v},\check {\beta }}^{0^+}(S\times C)$ . A finite-group torsor is étale over the base. The cardinality of $\Gamma $ is ${\mathrm {rk}}(\mathbf {v})^{2g}$ .

Proof. Consider the following diagram:

The trace map

$$\begin{align*}\mathrm{tr}\colon R{\mathcal H} om({\mathcal F}_{r},{\mathcal F}_{r}) \rightarrow {\mathcal O}_{\mathfrak{Coh}_{r}(S,\mathbf{v})}\end{align*}$$

has a section given by the inclusion of the identity morphism ${\mathcal O}_{\mathfrak {Coh}_{r}(S,\mathbf {v})}\rightarrow R{\mathcal H} om({\mathcal F}_{r}, {\mathcal F}_{r})$ ; therefore,

$$\begin{align*}R{\mathcal H} om({\mathcal F}_{r},{\mathcal F}_{r})\cong R{\mathcal H} om({\mathcal F}_{r},{\mathcal F}_{r})_{0}\oplus {\mathcal O}_{\mathfrak{Coh}_{r}(S,\mathbf{v})},\end{align*}$$

and by the moduli interpretation of $\mathfrak {Coh}_{r}(S,\mathbf {v})$ , we get that

Hence, by functoriality of the trace and the splitting above, we obtain that

By the base change theorem,

This proves the claim.

Proof. A sheaf $F \in M^{\epsilon }_{\mathbf {v}, \check {\beta }}(S \times C_{g,N})({\mathbb {C}})$ is perfect since it is a family of sheaves on a smooth surface S over C. A threefold $S\times C$ has at worst normal crossing singularities; its dualizing sheaf is therefore a line bundle. Moreover, by Lemma 4.11, the sheaf F is simple. Using Serre’s duality and simplicity of F, we conclude that

$$\begin{align*}\operatorname{Ext}^i(F,F)_0=0, \quad i\not\in \{1,2,3\}.\end{align*}$$

Hence, $R{\mathcal H} om_{\pi _2}({\mathbb {F}},{\mathbb {F}})_{0}[1]$ is at most of amplitude $[0,2]$ . By Proposition 5.1, the complex is therefore also at most of amplitude $[0,2]$ . We need to show that it is zero in degree $2$ . It is enough to check it over a point $[f\colon (C,\mathbf {p})\rightarrow \mathfrak {Coh}_{r}(S,\mathbf {v})]\in Q^{\epsilon }_{g,n}(M(\mathbf {v}), \beta )({\mathbb {C}})$ . Consider the distinguished triangle

(5.1) $$ \begin{align} \tau_{\leq 0}f^*{\mathbb{T}}^{\mathrm{vir}} \rightarrow f^*{\mathbb{T}}^{\mathrm{vir}} \rightarrow \tau_{\geq 1} f^*{\mathbb{T}}^{\mathrm{vir}}\rightarrow, \end{align} $$

where $\tau _{\dots }$ is the truncation of complex with respect to the standard t-structure. Taking the long exact sequence of cohomologies associated to (5.1), we obtain

$$\begin{align*}\dots \rightarrow H^2(C,\tau_{\leq 0}f^*{\mathbb{T}}^{\mathrm{vir}}) \rightarrow H^2(C,f^*{\mathbb{T}}^{\mathrm{vir}}) \rightarrow H^2(C,\tau_{\geq 1} f^*{\mathbb{T}}^{\mathrm{vir}}) \rightarrow \dots\end{align*}$$

Let us now analyse the terms in the long exact sequence. First, f maps generically to the stable locus $M(\mathbf {v})$ . By the assumption, ${\mathbb {T}}^{\mathrm {vir}}_{|M(\mathbf {v})}$ is a locally free sheaf concentrated in degree 0. Hence, the object $f^*{\mathbb {T}}^{\mathrm {vir}}$ is generically a locally free sheaf concentrated in degree 0. Moreover, ${\mathbb {T}}^{\mathrm {vir}}$ is of perfect amplitude $[-1,1]$ . These two facts imply that $\tau _{\geq 1} f^*{\mathbb {T}}^{\mathrm {vir}}$ is a $0$ -dimensional sheaf in degree 1 supported on the base points of f; therefore,

$$\begin{align*}H^2(C,\tau_{\geq 1} f^*{\mathbb{T}}^{\mathrm{vir}})=0.\end{align*}$$

Since C is a curve,

$$\begin{align*}H^2(C,\tau_{\leq 0}f^*{\mathbb{T}}^{\mathrm{vir}}) =0.\end{align*}$$

By the long exact sequence, we therefore obtain that $H^2(C,f^*{\mathbb {T}}^{\mathrm {vir}})=0$ .

Proof. By [Reference Toën and VaquiéTV07, Reference Toën and VezzosiTV08, Reference LurieLur12] and [Reference Schürg, Toën and VezzosiSTV15], there exists a derived algebraic stack ${\mathbb {R}} \mathfrak {Coh}_{r}(S,\mathbf {v})$ whose classical truncation is $\mathfrak {Coh}_{r}(S,\mathbf {v})$ ,

$$\begin{align*}t_0({\mathbb{R}} \mathfrak{Coh}_{r}(S,\mathbf{v}))\cong \mathfrak{Coh}_{r}(S,\mathbf{v}),\end{align*}$$

and

$$\begin{align*}j^* {\mathbb{T}}_{{\mathbb{R}} \mathfrak{Coh}_{r}(S,\mathbf{v})} \cong {\mathbb{T}}^{\mathrm{vir}}\end{align*}$$

with respect to the natural inclusion $j\colon \mathfrak {Coh}_{r}(S,\mathbf {v}) \hookrightarrow {\mathbb {R}} \mathfrak {Coh}_{r}(S,\mathbf {v})$ .

A derived enhancement gives rise to an obstruction theory of the underlying classical stack; see [Reference Schürg, Toën and VezzosiSTV15, Section 1] for more details. By Proposition 4.5, the stack $Q^{\epsilon }_{g,N}(M(\mathbf {v}), \beta )$ is open in the mapping stack $\mathrm {Map}_{{\mathfrak {M}}_{g,N}}(\mathfrak {C}_{g,N}, \mathfrak {Coh}_r(S,\mathbf {v})\times {\mathfrak {M}}_{g,N})$ . Hence, the obstruction-theory morphism

can be given by the derived mapping stack

$$\begin{align*}{\mathbb{R}}\mathrm{Map}_{{\mathfrak{M}}_{g,N}}(\mathfrak{C}_{g,N}, {\mathbb{R}}\mathfrak{Coh}_r(S,\mathbf{v})\times {\mathfrak{M}}_{g,N}),\end{align*}$$

which is algebraic by [Reference Toën and VezzosiTV08, Reference LurieLur12]. We refer to [Reference Halpern-Leistner and PreygelHLP23, Theorem 5.1.1] for a more recent treatment of derived mapping stacks; the stack ${\mathbb {R}} \mathfrak {Coh}_{r}(S,\mathbf {v})$ satisfies the requirements of [Reference Halpern-Leistner and PreygelHLP23, Theorem 5.1.1]. The obstruction-theory complex is perfect by Proposition 5.2.

By [Reference SiebertSie04], a virtual fundamental class depends only on Chern characters of the corresponding obstruction-theory complex. The second part of the claim therefore follows from Proposition 5.1.

Proof. Using the discussion in Section 6.1 and in the beginning of Section 6.2, the identification of the moduli stacks follows from the same arguments as in Theorem 3.16. The rest follows from the main results of Section 4 and 5.

Proof. The object F is of rank 1, and $\det (F)\cong {\mathcal O}_{S\times C}$ by the choice of the section $s_{\det }$ . Hence, by [Reference TodaTod10, Lemma 3.11], in order to show that F is a stable pair, we have to establish the following properties:

1. (i) ${\mathcal H}^{i}(F)=0$ , for $i\neq 0,1$ ,

2. (ii) ${\mathcal H}^{0}(F)$ is a rank 1 torsion-free sheaf and ${\mathcal H}^{1}(F)$ is 0-dimensional,

3. (iii) $\operatorname {Hom}(Q[-1], F)=0$ for any 0-dimensional sheaf Q,

where ${\mathcal H}^{i}(F)$ is the cohomology of a complex with respect to the standard t-structure. Note that the proof of [Reference TodaTod10, Lemma 3.11 (ii)] easily extends to the case of a threefold with normal crossing singularities, as long as F satisfies the first two properties above and its restriction to the singular locus is an ideal sheaf (i.e., it is admissible over the singular locus). Indeed, in this case, ${\mathcal H}^{0}(F)$ is an ideal sheaf by the discussion in Section 6.1, ${\mathcal H}^1(F)$ is supported away from the singularity, and F fits into the distinguished triangle,

$$\begin{align*}{\mathcal H}^{0}(F) \rightarrow F \rightarrow {\mathcal H}^1(F)[-1] \rightarrow\hspace{-1pt}.\end{align*}$$

Then [Reference TodaTod10, Lemma 3.11 (ii)] follows from the same arguments as in the smooth case.

$\mathbf {(i)}$ Consider two distinguished triangles that the object F fits in,

$$\begin{align*}\tau_{\leq-1}F\rightarrow F\rightarrow \tau_{\geq0}F \rightarrow,\end{align*}$$

$$\begin{align*}\tau_{\leq1}F\rightarrow F \rightarrow \tau_{\geq2}F \rightarrow,\end{align*}$$

where the truncation is taken with respect to the standard t-structure. First, over a dense open subsect $U \subseteq C$ , which contains all nodes, F is a family of rank 1 sheaves. Hence,

$$\begin{align*}(\tau_{\leq -1}F)_{|U}=0, \quad (\tau_{\geq 2}F)_{|U}=0.\end{align*}$$

We now deal with $p \in C\setminus U$ . By using the distinguished triangles above and the associated long exact sequences of sheaf cohomologies (cohomologies with respect to the standard t-structure), we obtain the long exact sequence

$$\begin{align*}\dots \rightarrow {\mathcal H}^{-2}((\tau_{\geq0}F)_p) \rightarrow {\mathcal H}^{-1}((\tau_{\leq -1 }F)_p) \rightarrow {\mathcal H}^{-1}(F_p) \rightarrow \dots \end{align*}$$

Since $C\setminus U$ contains only regular points, fibers $(\tau _{\geq 0}F)_p$ are of amplitude $[-1,a]$ for some a. On the other hand, fibers $F_p$ are of amplitude $[0,1]$ . We conclude that ${\mathcal H}^{-1}((\tau _{\leq -1 }F)_p)=0$ . The same applies to cohomologies of lower degrees; hence,