Proof. We first address the first part. By properness, $c$ is finite over the locus where $c$ has zero-dimensional fibers. Then condition (i) holds over this locus if and only if $c$ is an isomorphism there. Now let $x\in C_T$ be a point where $c$ has a one-dimensional fiber. It follows that $c$ has to be a contraction of components. Since formation of $R^1c_*$ commutes with base change, the vanishing of $R^1c_*\mathcal{O}_{\widetilde {C}}$ around $x$ is equivalent to $c^{-1}(x)$ being of arithmetic genus zero.

We now address the second part. Assume (i) and (i’) hold. If (ii) holds, then the twisted dualizing sheaf must be trivial on any components contracted by $c$ . We already know each such component is rational, so they must have precisely two distinguished points. Since $c^{-1}(x)$ has arithmetic genus zero, its components must be arranged as a chain $R_1\cup \cdots \cup R_j$ . The distinguished points on the end of the chain must come from intersection with a remaining component of $\widetilde {C}$ or from marked points. If there is a marked point on $c^{-1}(x)$ , it must map to a marked point of $C$ (in particular, there can be only one). Otherwise, we must have that $c^{-1}(x)$ intersects the closure of $\widetilde {C}_y\setminus c^{-1}(x)$ at two points, lying on components $S_0,S_1$ (which might coincide) of $\widetilde {C}_y$ not inside $c^{-1}(x)$ . Since $c$ is an isomorphism over a punctured étale neighborhood of $x$ , we must have a node at $x$ with $S_0,S_1$ mapping to the two branches.

Conversely, assume that (ii’) holds. Then use the following standard characterization of the dualizing sheaf $\omega _C$ of a nodal curve $C$ : a section of $\omega _C$ is the same as a meromorphic section of the sheaf of differentials of the normalization $C^{\nu }$ with at most simple poles at the preimages of nodes of $C$ and such that for a node $x\in C$ , the residues at the two preimages of $x$ in $C^{\nu }$ add to zero. Now, say $x\in \Sigma _c$ is a node. Then, working locally around $x$ , we may assume that $\tilde {C}_y^{\nu } = C_y^{\nu }\coprod R_1 \coprod \cdots \coprod R_j$ . We define a map $c^*\omega _{C_y}\to \omega _{\tilde {C}_y}$ by sending a section of $\omega _{C_y}$ to the unique section of $\omega _{\widetilde {C}_y}$ with the same value on $C_y^{\nu }$ . This uses that $R_i\simeq \mathbb{P}^1$ and that for any two points $z_1,z_2$ on $\mathbb{P}^1$ , there is a unique section of $\Omega _{\mathbb{P}^1}(z_1 + z_2)$ with prescribed residue $r$ at $z_1$ (and which then has residue $-r$ at $z_2$ ). The resulting map $c^*\omega _{C_y}\to \omega _{\tilde {C}_y}$ sends a nowhere-vanishing section of $\omega _{C_y}$ to one of $\omega _{\tilde {C}_y}$ and hence defines an isomorphism. In the case of $x$ being a marked point, one proceeds analogously.

Proof. Openness of (i) follows from Tor-independent cohomology and base change [Sta23, Lemma 08IB], using that $\tilde {C}_T$ and $C_T$ are flat over $T$ , and from the fact that the support of a coherent sheaf is closed and commutes with any base change. Now assume that (i) holds for $c$ . Then condition (ii’) is equivalent to the vanishing of $R^1c_*(\omega _{\widetilde {C}_B/B}(\sigma _1'+\ldots + \sigma _n'))^{\otimes 2 }$ , which is open on the base, since the formations of $\omega _{\widetilde {C}_B/B}$ , of $R^1c_*$ and of the support of a coherent sheaf all commute with base change in $B$ .

Proof. We construct a quasi-inverse using flat descent for algebraic spaces (e.g. [Sta23, Lemma 0ADV]). For this, we may assume that $C_B'\to C_B$ is surjective by replacing $C_B'$ with $C_B'\coprod C_B^{\circ }$ , where $C_B^{\circ }\subseteq C_B$ denotes the open subset obtained by removing the nodes and marked points (this leaves our assumptions as well as ${\textrm {Exp}}_{C_B'/B}$ unchanged). Let $T$ be a test scheme over $B$ , and let $\tilde {C}'_T\to C'_T$ be an expansion. Let $C''_T := C'_T\times _{C_T} C_T'$ . Then the diagonal map $C_T'\to C_T''$ is an inclusion of a connected component. By assumption, the other components of $C_T''$ do not contain any nodes or preimages of marked points. Thus, there is a unique structure of marked curve on $C_T''$ such that the diagonal $C_T'\to C_T''$ and the two projections $\operatorname {pr}_i:C_T''\to C_T'$ preserve the markings. Moreover, we have that the expansions $\operatorname {pr}_1^*\tilde {C}_T'$ and $\operatorname {pr}_2^*\tilde {C}_T'$ of $\tilde {C}_T''$ are canonically isomorphic: over the diagonal copy of $C_T'$ inside $C_T''$ they are identified with $\widetilde {C}_T'$ , and over its complement they are trivial expansions. Using similar arguments, one checks that this isomorphism provides a canonical descent datum for $\widetilde {C}_T'$ and thus, by descent, an expansion $\widetilde {C}_T$ of $C_T$ . In this way, we obtain a map ${\textrm {Exp}}_{C_B'/B}\to {\textrm {Exp}}_{C_B/B}$ . A further routine check shows that this construction provides a quasi-inverse to pullback along $\gamma$ .

Proof of Proposition 2.4. Consider the stack $\mathscr{M}$ of all marked nodal curves [Sta23, Tag 0DSX] with universal family $\mathscr{C}$ , and on $\mathscr{M}\times B$ , consider the morphism space ${\textrm {Hom}}_{\mathscr{M}\times B}(\mathscr{C},C_B)$ , which is an algebraic stack over $B$ locally of finite presentation [Sta23, Tag 0DPN]. The locus that preserves the markings is closed. The locus in which (i) and (ii) of Definition 2.1 hold is then open in this closed substack. Thus, we get ${\textrm {Exp}}_{C_B/B}$ as a locally closed substack, and, in particular, it is algebraic and locally of finite presentation over $B$ .

For the remaining properties, we may work locally on $B$ and assume without loss of generality that $B=\textrm {Spec}\,R$ is the spectrum of a Henselian local ring with separably closed residue field. Then the result follows from Lemma 2.6.

Proof. By the étale local structure of nodes combined with Lemma 2.5 and our assumptions on $B$ , we may reduce to the case that $C$ is a disjoint union of open sets of the following form: for each node $q_i$ , there is a morphism $g_i: B\to {\mathbb{A}}^1$ such that the component $U_i$ containing $q_i$ is isomorphic to the pullback along $g_i$ of a standard degeneration ${\mathbb{A}}^2\to {\mathbb{A}}^1$ given by multiplication of the coordinates. Each marked point is contained in a component $V_j\simeq B\times {\mathbb{A}}^1$ , with the marking given by the origin of ${\mathbb{A}}^1$ . It follows that ${\textrm {Exp}}_{C_B/B}$ is a fiber product over $B$ of pullbacks of the stacks ${\textrm {Exp}}_{{\mathbb{A}}^2/{\mathbb{A}}^1}\to {\mathbb{A}}^1$ and ${\textrm {Exp}}_{({\mathbb{A}}^1,0)}\to \textrm {Spec}\,{\mathbb{C}}$ . Following [Reference LiLi01, § 1] and [Reference Abramovich, Cadman, Fantechi and WiseACFW13, § 1 + § 6], we can give explicit descriptions of ${\textrm {Exp}}_{\mathbb{A}^2/\mathbb{A}^1}$ and ${\textrm {Exp}}_{(\mathbb{A}^1,0)}$ from which the result follows.

Marked points. We first make the connection of our setting to theirs. Our Lemma 2.2 exhibits ${\textrm {Exp}}_{({\mathbb{A}}^1,0)}$ as equal to the stack $\mathcal{T}_{na\ddot {\imath }ve}({\mathbb{A}}^1,0)$ of naïve expansions of Definitions 2.1.2 and 2.1.3 in [Reference Abramovich, Cadman, Fantechi and WiseACFW13]. By [Reference Abramovich, Cadman, Fantechi and WiseACFW13, Proposition 3.2.2 and Proposition 6.3.3], we further have an isomorphism $\mathcal{T}_{na\ddot {\imath }ve}({\mathbb{A}}^1,0)\simeq \mathcal{T}_{Li}({\mathbb{A}}^1,0)$ , where the latter is the space of expanded pairs constructed in [Reference LiLi02]. As explained in the proof of Proposition 6.3.3 in [Reference Abramovich, Cadman, Fantechi and WiseACFW13], the stack $\mathcal{T}_{Li}({\mathbb{A}}^1,0)$ (and hence ${\textrm {Exp}}_{({\mathbb{A}}^1,0)}$ ) has a smooth cover by affine spaces $({\mathbb{A}}^n)_{n\geqslant 0}$ . Here, for each $n$ , the map ${\mathbb{A}}^n\to {\textrm {Exp}}_{({\mathbb{A}}^1,0)}$ is induced by an explicit expansion $c[n]:\widetilde {C}[n]_{{\mathbb{A}}^n}\to {\mathbb{A}}^n\times {\mathbb{A}}^1$ in which the fiber over the origin of ${\mathbb{A}}^n$ is the expansion of $({\mathbb{A}}^1,0)$ with the fiber over $0$ being a chain of $n$ copies of ${\mathbb{P}}^1$ . Each coordinate axis of ${\mathbb{A}}^n$ corresponds to a deformation smoothing exactly one of the $n$ nodes in the expansion. The automorphism group of the central fiber is $\mathbb{G}_m^n$ . This group acts on the whole family $\widetilde {C}[n]_{{\mathbb{A}}^n}$ so that $c[n]$ becomes equivariant, where we let $\mathbb{G}_m^n$ act trivially on ${\mathbb{A}}^1$ and by scaling coordinates on ${\mathbb{A}}^n$ . The induced map $[{\mathbb{A}}^n/\mathbb{G}_m^n]\to {\textrm {Exp}}_{({\mathbb{A}}^1,0)}$ is étale.

Nodes. Again by Lemma 2.2, we have an isomorphism ${\textrm {Exp}}_{{\mathbb{A}}^2/{\mathbb{A}}^1}\simeq \mathfrak{T}_{na\ddot {\imath }ve}({\mathbb{A}}^2/{\mathbb{A}}^1,0)$ where the latter is the stack of naïve expanded degenerations defined in [Reference Abramovich, Cadman, Fantechi and WiseACFW13, Definition 2.3.1 and Definition 2.3.2]. Combining their Proposition 3.2.3 and Proposition 6.3.3, we have an isomorphism $\mathfrak{T}_{na\ddot {\imath }ve}({\mathbb{A}}^2/{\mathbb{A}}^1,0) \simeq \mathfrak{T}_{Li}({\mathbb{A}}^2/{\mathbb{A}}^1,0)$ , where the latter is the stack of expanded degenerations defined in [Reference LiLi02]. As explained in [Reference Abramovich, Cadman, Fantechi and WiseACFW13] in the proof of Proposition 6.3.3 and in § 6.1, the stack $\mathfrak{T}_{Li}({\mathbb{A}}^2/{\mathbb{A}}^1,0)$ (and hence ${\textrm {Exp}}_{{\mathbb{A}}^2/{\mathbb{A}}^1}$ ) has a smooth cover by affine spaces $({\mathbb{A}}^{n+1})_{n\geqslant 0}$ . Again, the map ${\mathbb{A}}^{n+1} \to {\textrm {Exp}}_{{\mathbb{A}}^2/{\mathbb{A}}^1}$ is obtained from an explicit standard model. In this case, this map is induced by a standard expanded degeneration $c[n]:\tilde {C}[n]_{{\mathbb{A}}^{n+1}} \to {\mathbb{A}}^{n+1}\times _{{\mathbb{A}}^1} {\mathbb{A}}^2$ , where the map ${\mathbb{A}}^{n+1}\to {\mathbb{A}}^1$ is given by multiplication of all coordinates. The fiber over the origin of ${\mathbb{A}}^{n+1}$ is an expansion of the nodal curve $(xy=0)\subset {\mathbb{A}}^2$ with the preimage of the node being a chain of $n$ copies of ${\mathbb{P}}^1$ . Each coordinate direction of ${\mathbb{A}}^{n+1}$ corresponds to a smoothing of one of the $n+1$ nodes in the expansion. The automorphism group of the central fiber is $\mathbb{G}_{m}^{n}$ , and the action extends to $\tilde {C}[n]$ , which makes the expansion equivariant with respect to an action on ${\mathbb{A}}^{n+1}$ for which the map to ${\mathbb{A}}^1$ is invariant. The induced map $[{\mathbb{A}}^{n+1}/\mathbb{G}_m^{n}]\to {\textrm {Exp}}_{{\mathbb{A}}^2/{\mathbb{A}}^1}$ is étale.

Proof. As in the proof of Lemma 2.6, we may reduce the proof to showing the existence part of the valuative criterion for the standard models ${\textrm {Exp}}_{({\mathbb{A}}^1,0)}\to \textrm {Spec}\,{\mathbb{C}}$ and ${\textrm {Exp}}_{{\mathbb{A}}^2/{\mathbb{A}}^1\to {\mathbb{A}}^1}$ , which we have identified as stacks of naïve expansions. In [Reference Abramovich, Cadman, Fantechi and WiseACFW13, Lemma 3.1.3 and Lemma 3.1.6], these stacks are identified as relative stacks of semistable maps into a projective target. These stacks are well known to satisfy the existence part of the valuative criterion. For example, one may reduce to the case of stable maps by adding suitable sections (this may involve passing to an extension of the DVR) and then use the properness of the space of stable maps (see [Reference Abramovich, Olsson and VistoliAOV11, Theorem 4.3] for a version treating the case of families).

Proof. The first two points are Theorem B in [Reference Drezet and NarasimhanDN89]. The last point is [Reference RamananRam73, Remark 2.9].

Proof. By Riemann–Roch, we have

\begin{align*} \operatorname {rk} R\pi _*(E\otimes V) &= \deg E|_{\operatorname {pt} \times F} \operatorname {rk} V + \operatorname {rk} E \deg V|_{\operatorname {pt}\times F} + \operatorname {rk}E \operatorname {rk} V (1-g) \\ &= d_0 \left (dr + r^2(g-1)- rd +r^2(1-g) \right ) = 0. \end{align*}

From the definition of $L_V(-)$ , the result follows from the following chain of isomorphisms:

\begin{align*} \det R\pi _*(E \otimes \pi ^*N \otimes V)&\simeq \det \left (R\pi _*(E\otimes V) \otimes N \right ) \simeq \det R\pi _*(E\otimes V) \otimes N^{\operatorname {rk}R\pi _*(E\otimes V)}\\ & \simeq \det R\pi _*(E\otimes V). \end{align*}

Here, the first isomorphism is the projection formula and the second follows from the formula for the determinant of a tensor product of vector bundles after choosing a two-term resolution of $R\pi _*(E\otimes V)$ .

Proof. We have that $c_1(E)= d[{\mathbb{P}}^1\times y] +\ell [x\times F]$ for some $\ell \in \mathbb{Z}$ . Thus, $c_1(E)^2=2\ell d$ , and

\begin{align*}k\deg L_V(E) =k d_0r c_2(E) - kd_0(r-1)\ell d\equiv d_0\ell k d \equiv d_0\ell = (L,c_1(E)) \mod d_0 r.\end{align*}

Proof. Since $E$ is torsion-free, it maps injectively to its double dual with zero-dimensional cokernel. We have $\Delta (E)\geqslant \Delta (E^{\vee \vee })$ , with equality if and only if $E$ is locally free. By replacing $E$ with $E^{\vee \vee }$ if necessary, we may therefore assume that $E$ is locally free.

By Langton’s procedure of elementary modifications [Reference LangtonLan75], one may find a locally free subsheaf $E' \subset E$ whose restriction to every fiber over ${\mathbb{P}}^1$ is stable and such that $E'$ is obtained from $E$ through successive elementary modifications along maximally destabilizing quotients of fibers. One checks that $\Delta (E)$ strictly decreases after each such modification, so $\Delta (E')\leqslant \Delta (E)$ , with equality if and only if $E$ is already stable on every fiber. By replacing $E$ with $E'$ if necessary, we may assume that this is the case.

Then, after possibly tensoring $E$ by a line bundle from $\mathbb{P}^1$ , we may assume that $E$ is a pullback of the universal sheaf along a morphism $\nu :\mathbb{P}^1\to M_F(r,L)$ for some $L$ . If $g=1$ , this implies that $\nu$ is constant. Otherwise, if $g\geqslant 2$ , we have $\Delta (E)=2k\deg \nu ^* L_V$ for $L_V$ as in Proposition 2.8. Since $L_V$ is ample, this implies that $\deg \nu ^*L_V\geqslant 0$ , with equality if and only if $\nu$ is constant.

Proof. By Assumption 2.13, the dimension of the tangent space of the Picard scheme $\textrm {Pic}_{X_b}$ at the identity is locally constant. Since we are in characteristic zero, this implies (i). Then (ii) follows as in [Reference KleimanKle05, Proposition 5.20]. Finally, (iii) follows from the same argument as in [Reference KleimanKle05, Remark 5.21] using Assumption 2.13 in place of smoothness (the projectivity there is only used to invoke the GAGA theorem, which holds more generally for a proper morphism).

Proof. By construction, the morphism $\mathcal{NS}_{X/B}\to B$ is locally of finite type and has everywhere-vanishing relative Kähler differentials, which implies (i). The same holds for $ \mathcal{NS}_{C_B/B}$ , which is moreover smooth over $B$ . Hence, it is étale over $B$ , which gives (ii). Point (iii) follows from the local constancy of the Euler characteristic. The morphism in point (iv) is well defined, since the morphism $\varphi :\textrm {Pic}_{C_B/B}\to \textrm {Pic}_{X_B/B}, L\mapsto f^*L$ preserves the identity components. By Lemma 2.18, the map $\varphi$ is a closed embedding and $\varphi ^{-1}\textrm {Pic}_{X_B/B}^0 = \textrm {Pic}_{C_B/B}^0$ . It follows that the induced morphism of quotient groups $\overline {\varphi }:\mathcal{NS}_{C_B/B}\to \mathcal{NS}_{X_B/B}$ is a monomorphism. We claim it is also proper. By the third isomorphism theorem, we identify

\begin{align*}\mathcal{NS}_{X_B/B} = \left [\textrm {Pic}_{X_B/B}\middle / \textrm {Pic}_{X_B/B}^0\right ] = \left [[\textrm {Pic}_{X_B/B}/\textrm {Pic}^0_{C_B/B}]\middle / [\textrm {Pic}_{X_B/B}^0/\textrm {Pic}^0_{C_B/B}]\right ].\end{align*}

Hence, we may factor $\overline {\varphi }$ as

\begin{align*}[\textrm {Pic}_{C_B/B}/\textrm {Pic}_{C_B/B}^0] \to [\textrm {Pic}_{X_B/B}/\textrm {Pic}_{C_B/B}^0]\to \left [[\textrm {Pic}_{X_B/B}/\textrm {Pic}_{C_B/B}^0]\middle /[\textrm {Pic}_{X_B/B}^0/\textrm {Pic}_{C_B/B}^0]\right ].\end{align*}

The first of these is a closed embedding, since $\varphi$ is a closed embedding. Properness of the second map follows from properness of the quotient $[\textrm {Pic}^0_{X_B/B}/\textrm {Pic}^0_{C_B/B}]$ , which is Lemma 2.19 below. Finally, since $\mathcal{NS}_{C_B/B}$ is étale over $B$ and $\mathcal{NS}_{X_B/B}$ is unramified over $B$ , it follows that $\mathcal{NS}_{C_B/B}$ is formally étale, and hence étale over $\mathcal{NS}_{X_B/B}$ . Any étale (even flat) monomorphism is, in particular, an open immersion [Sta23, Theorem 025G].

Proof. A closed immersion is the same as a proper monomorphism. Since $f_*\mathcal{O}_{X_B} = \mathcal{O}_{C_B}$ , we have for any line bundle $L$ on $C_B$ that $f_*f^*L = L$ . The same holds after any base change on $B$ , so pullback indeed gives a monomorphism. For properness, it suffices to check the existence part of the valuative criterion. (The uniqueness part is an immediate consequence of being a monomorphism.) Hence, we may assume $B = \textrm {Spec}\,R$ for a DVR $R$ with generic point $\eta$ and that we are given a line bundle $L$ on $X_B$ so that $L_{\eta }$ is isomorphic to a line bundle pulled back from $C_{\eta }$ or, equivalently, so that $f_*L$ is a line bundle on $C_{\eta }$ and that $f^*f_{*}L\to L$ is an isomorphism over $X_{\eta }$ . We want to exhibit a line bundle $M$ on $C_R$ so that $L\simeq f^*M$ . Viewing $f:X_R\to C_R$ as a family of curves, then, at least over the locus where $f$ is smooth, we can conclude that $f_*L$ is free and $f^*f_*L\to L$ is an isomorphism, since the relative Picard scheme of a family of smooth curves is separated. Thus, these conditions hold, except possibly over a finite set of points $x_1,\ldots ,x_n\in C_{\xi }$ contained in the smooth locus of $C_{\xi }$ . Since $C_R$ is regular of dimension two around each of the $x_i$ , the double dual $M:= (f_*L)^{\vee \vee }$ is then locally free. But, since $X_R$ is Cohen–Macaulay, any locally free sheaf is determined by its restriction away from any codimension-two locus. More precisely, this means that the isomorphism $f^*(f_*L)^{\vee \vee }\xleftarrow {\sim } f^*f_*L\xrightarrow {\sim } L$ , defined away from the fibers over $x_i$ , extends to an isomorphism $f^*M\xrightarrow {\sim } L$ on all of $X_R$ .

To obtain properness, it remains to show that $\textrm {Pic}_{C_B/B}\to \textrm {Pic}_{X_B/B}$ is quasi-compact. For this, we may work locally on $B$ , and even decompose $B$ into locally closed strata [Sta23, Lemma 040W]. In particular, we may assume that $C_B$ has constant topological type over $B$ , that is, that $C_B$ is a union of smooth curves over $B$ . Moreover, we may assume that $B$ is smooth, quasi-compact and irreducible, and that for each component $C_i$ of $C_B$ we have an effective Cartier divisor $D_i\subset f^{-1}C_i$ that is generically finite over $C_i$ . Then the numbers $\chi (f^{-1}C_i,L|_{f^{-1}C_i} \otimes k D_i)$ are locally constant on $\textrm {Pic}_{X_B/B}$ and hence define a decomposition into open and closed subsets. The preimage of each such subset is either empty or consists of line bundles on $C_B$ of a fixed multi-degree which is a quasi-compact set.

For the statement about identity components, it is immediate that $\varphi ^{-1}\textrm {Pic}_{X_B/B}^0$ is an open subspace of $\textrm {Pic}_{C_B/B}$ that contains $\textrm {Pic}^0_{C_B/B}$ . To show equality of these open subspaces, it is enough to work pointwise on $B$ . Thus, we may reduce to the setting of the previous paragraph, where $C_B = \cup C_i$ is a union of smooth irreducible components and where we have chosen effective Cartier divisors $D_i\subset f^{-1}C_i$ generically finite over $C_i$ . Then the identity component $\textrm {Pic}_{X_B/B}^0$ is contained in the open subset defined by $\chi (f^{-1}C_i, L|_{f^{-1}C_i} \otimes kD_i)=0$ for all $i,k$ . The preimage of this subset under $\varphi$ is exactly the subspace of $\textrm {Pic}_{C_B/B}$ of line bundles of multi-degree zero, which is $\textrm {Pic}_{C_B/B}^0$ .

Proof. By Lemma 2.18, the map $\textrm {Pic}^0_{C_B/B}\to \textrm {Pic}^0_{X_B/B}$ is a closed embedding. This implies that $Y_B:=[\textrm {Pic}_{X_B/B}^0/\textrm {Pic}_{C_B/B}^0]$ is separated over $B$ . By Theorem2.16 (ii), it is also of finite type. It remains to check properness. Working locally on $B$ , we may assume $B$ is affine. By Chow’s lemma [Sta23, Lemma 088U], we can find a proper birational map $g:Z\to Y_B$ with $Z$ quasi-projective over $B$ (in particular, a scheme). It suffices to show that $Z$ is proper. By [Reference GrothendieckGro66, Corollary 15.7.11], a separated finite type morphism of schemes $Z\to B$ with geometrically connected fibers is proper provided that it has proper fibers and that – at least after base change along some proper surjective map $B'\to B$ – there exists a section. For the existence of a section, we may take $B'$ to be the preimage of the identity section under $g$ . Properness of the fibers of $g$ is equivalent to properness of fibers of $Y_B\to B$ . Hence, we are reduced to showing that $[\textrm {Pic}_{X}^0/\textrm {Pic}_C^0]$ is proper in the case that $B$ is a point. For this, we can use the existence part of the valuative criterion. Let $C_i$ and $X_i$ be the irreducible components of $C$ and $X$ , respectively. Then, by writing any line bundle on $C$ in terms of line bundles on its components and glueing data, we have an exact sequence

\begin{align*}0\to \mathbb{G}_m^{b}\to \textrm {Pic}^0_C\to \prod \textrm {Pic}^0_{C_i}\to 0,\end{align*}

where $b$ is the first Betti number of the dual graph of $C$ . For $X$ in place of $C$ , we analogously have

\begin{align*}0\to \mathbb{G}_m^b \to \textrm {Pic}_X^0 \to \prod \textrm {Pic}_{X_i}^0 \to \prod \textrm {Pic}^0 _{F_j},\end{align*}

where $F_j$ runs through the components of the singular locus of $X$ . This latter sequence shows that the quotient of $\textrm {Pic}_X^0$ by $\mathbb{G}_m^b$ in $\prod \textrm {Pic}_{X_i}^0$ is closed, and hence it is an abelian subvariety. Pullback along $f$ induces the following commutative diagram.

It follows that we have an isomorphism

\begin{align*}\left [{\textrm {Pic}}_X^0/{\textrm {Pic}}_C^0\right ]\simeq \left [[\textrm {Pic}_X^0/\mathbb{G}_m^b]\middle /[\textrm {Pic}_C^0/\mathbb{G}_m^b]\right ], \end{align*}

which is a quotient of an abelian variety by a subvariety, and hence it is an abelian variety. This shows the desired properness in the case of a point base.

Proof. We treat only the case of $\overline {\mathcal{NS}}_{X_T/T}$ ; the other is similar. Since the pullback map $\textrm {Pic}_{X_T/T}\to \textrm {Pic}_{\widetilde {X}_T/T}$ preserves the identity components, it descends to a morphism $\mathcal{NS}_{X_T/T}\to \mathcal{NS}_{\widetilde {X}_T/T}$ . Since the total degree is preserved under pullback along $\widetilde {C}_T\to C_T$ , this descends to a morphism $\overline {\mathcal{NS}}_{X_T/T}\to \overline {\mathcal{NS}}_{\widetilde {X}_T/T}$ . We claim that this is an isomorphism. It is enough to show that each section of $\overline {\mathcal{NS}}_{\widetilde {X}_T/T}$ has a unique preimage. For this, we may work étale locally on $T$ and assume that $\widetilde {C}_T\to T$ has sections meeting each irreducible component of each fiber. Suppose that $\overline {c_1} \in \overline {\mathcal{NS}}_{\widetilde {X}_T/T}(S)$ for some $T$ -scheme $S$ . Then, locally on $S$ , we may assume that $\overline {c_1}$ is represented by some line bundle $L$ on $\widetilde {X}_S$ . Up to twisting $L$ by a line bundle $N$ pulled back from $\widetilde {C}_S$ of total degree zero, we may assume that $L$ is pulled back from $X_S$ so that $\overline {c_1}$ comes from an element $\overline {c_1'}\in \overline {\mathcal{NS}}_{X_T/T}(S)$ . Moreover, we see that $\overline {c_1'}$ is uniquely determined, since any two possible choices of $N$ differ by a line bundle pulled back from $C_S$ .

Proof. By additivity of the Chern character, we have

\begin{align*} & \Delta (E)/r-\Delta (G_1)/r_1-\Delta (G_2)/r_2 \\ &\quad = (c_1(G_1)+c_1(G_2))^2/(r_1+r_2)-c_1(G_1)^2/r_1 -c_1(G_2)^2/r_2 \\ &\quad =-\frac {1}{r r_1 r_2}\left (r_2c_1(G_1)-r_1c_1(G_2)\right )^2 .\end{align*}

Proof. This is [Reference YoshiokaYos99, Lemma 1.1].

Proof. Let $\mu _t$ denote the slope with respect to $H_t$ . Suppose that there is a change of stability condition at $t_0$ , i.e. there is a sheaf $E$ of the given invariants which is $H_t$ -semistable for $t=t_0$ but not for bigger (resp., smaller) values of $t$ . Then we may find an exact sequence

\begin{align*}0\to E_1 \to E\to E_2 \to 0,\end{align*}

which is destabilizing for values of $t$ slightly bigger (resp., smaller) than $t_0$ , but which consists of semistable objects for $t=t_0$ (take part of a HN filtration). In particular, $\mu _{t_0}(E_1) = \mu _{t_0}(E_2)$ . By Lemma 2.22 and the Bogomolov inequality applied to $E_1$ , $E_2$ , we have

\begin{align*}\Delta (E)\geqslant -\frac {1}{r r_1r_2} (r_2c_1(E_1)-r_1c_1(E_2))^2=-D^2/(r r_1 r_2),\end{align*}

where we set $D:=r_2c_1(E_1)-r_1c_2(E_1)$ and $r_i$ is the rank of $E_i$ . The assumption that $r$ and $(c_1, F)$ are coprime implies that $(D, F)=r_2(c_1(E_1),F)-r_1(c_1(E_2),F)\neq 0$ . We also have $(D,H_{t_0})=r_1r_2(\mu _{t_0}(F_1)-\mu _{t_0}(F_2))=0$ . Therefore, Proposition 2.23 applies to $D$ , and we find that

\begin{align*}\Delta (E)\geqslant \frac {1}{r r_1 r_2 (H,F)^2}(H^2+2t_0 (H,F)) \geqslant \frac {H^2+2t_0(H,F)}{r^3 (H,F)^2}.\end{align*}

This shows that the values that $t_0$ can take are bounded above, so (1) follows.

To address (2), let $\eta \in C$ denote the generic point and $F_{\eta }:= f^{-1}(\eta )$ . For any coherent sheaf $G$ on $X$ of rank $r_G\gt 0$ , the slope with respect to $H_t$ is

\begin{align*}\mu _t(G) = ((c_1(G)),H)+t (c_1(G),F))/r_G.\end{align*}

Suppose that $E$ is semistable beyond the last wall. Then for any subsheaf $E'$ , we have

\begin{align*}\mu _t(E')\leqslant \mu _t(E)\end{align*}

for sufficiently large $t$ . Dividing by $t$ and taking the limit as $t\to \infty$ gives

\begin{align*}\mu (E'|_{F_{\eta }}) = (c_1(E'),F)/r' \leqslant (c_1(E),F)/r = \mu (E|_F).\end{align*}

By the coprimeness assumption, we have strict inequality, and therefore the restriction of $E$ to $F_{\eta }$ is stable.

Conversely, assume that the restriction of $E$ to the generic fiber of $f$ is stable. Then for any subsheaf $E'$ , we know that $(E',F)/r'\lt (E,F)/r$ . Let $\mu _{0,max}(E)$ be the maximum value of $\mu _0$ of a lower-rank subsheaf of $E$ , and let $\mu _{max}(E|_F)\lt \mu (E|_F)$ be the least slope of a non-zero lower-rank subsheaf of $E|_F$ . Let $E'\subset E$ be an arbitrary lower-rank subsheaf. Then we have

\begin{align*}\mu _t(E') =(E',H)/r'+t (E',F)/r'\leqslant \mu _{0,max}(E) + t\mu _{max}(E) \end{align*}

and the right-hand side is strictly smaller than $\mu _t(E) = \mu _0(E) +t\mu (E|_F)$ for sufficiently large $t$ , independent of $E'$ . Therefore, there is no destabilizing subsheaf for sufficiently large $t$ .

It follows from what we have shown that $H_t$ -semistability of $E$ for $t\gg 0$ is equivalent to $H_t$ -stability and stability of $E|_{F_{\eta }}$ , as desired.

Proof. We work over the relative stack of expansions ${\textrm {Exp}}_{C_B/B}$ . Let $c:\mathcal{C}\to C_B$ denote the universal expansion, and let $\mathcal{X}:=X_B\times _{C_B}\mathcal{C} \to \mathcal{C}$ denote the induced family of fibered surfaces over ${\textrm {Exp}}_{C_B/B}$ . Let $\mathscr{M}(r,d)$ denote the stack of all torsion-free coherent sheaves of rank $r$ and fiber degree $d$ on $\mathcal{X}$ over ${\textrm {Exp}}_{C_B/B}$ . Then, since being locally free and being slope stable are open conditions, one can see that the locus of those sheaves satisfying (i)-(iii) of Definition 3.1 is open. Thus, we get $\mathcal{M}_{X_B/C_B}(r,d)\subseteq \mathscr{M}(r,d)$ as the open locus of $f$ -stable sheaves.

Proof. We may assume that $B = \textrm {Spec}\,R$ for a DVR $R$ with generic point $\eta$ , that $\widetilde {C}_{\eta }\to C_{\eta }$ is a given expansion of $C_B$ over $\eta$ and that $E_{\eta }$ is an $f$ -stable sheaf on $\widetilde {X}_{\eta }:=\widetilde {C}_{\eta }\times _C X$ . Our goal is to show that we can extend this to a family of $f$ -stable sheaves, possibly after passing to some extension of $R$ . By Corollary 2.7, we can at least extend $\widetilde {C}_{\eta }$ to some expansion $\widetilde {C}_B$ over $B=\textrm {Spec}\,R$ . For the proof, we may as well replace $C_{B}$ by $\widetilde {C}_B$ and hence assume that $\widetilde {C}_{\eta }\to C_{\eta }$ is an isomorphism. Let $\xi$ denote the closed point of $\textrm {Spec}\,R$ .

Case 1: $C_{\eta }$ is smooth. In this case, the desired result follows from Proposition 3.6 below.

Case 2: $C_{\eta }$ is of compact type. We proceed by induction on the number of components of $C_{\eta }$ . If there is only one component, we are in Case 1. Otherwise, we may decompose $C_{B} = C^1_{B}\cup _q C^2_{B}$ along any chosen node $q$ that extends over the generic fiber. We add a marked point $q_i$ on each $C^i_{B}$ where the node was. By inductive assumption, for each $i=1,2$ , we can find an expansion $\widetilde {C}^i_B\to C^i_B$ that is an isomorphism over the generic fiber together with an $f$ -stable sheaf $E^i$ on $\tilde {X}^i_B:= X_B\times _{C_B}\widetilde {C}^i_B$ extending the restriction of $E_{\eta }$ . We denote the marked point on $\widetilde {C}^i_B$ lying over $q_i$ again by $q_i$ . In order to glue the total family back together over $\textrm {Spec}\,R$ , we need to extend the isomorphism of $E^1|_{q_1}$ and $E^1|_{q_2}$ from $\eta$ over all of $\textrm {Spec}\,R$ . But since these are fiberwise isomorphic families of stable sheaves, there exists such an extension, possibly after twisting one of $E^1$ or $E^2$ by a multiple of $\widetilde {X}_{\xi }^i$ . Here, we use Knudsen’s clutching construction [Reference KnudsenKnu83, Theorem 3.4] to glue the expansions $\widetilde {C}^i_B$ along the marked points.

Case 3: $C_{\eta }$ is arbitrary. We do an induction on the first Betti number of the dual graph of $C_{\eta }$ . If it is zero, we are in Case 2. Otherwise, we may choose a non-separating node $q$ on $C_B$ that extends over the generic fiber and take a partial normalization $\nu :C_B'\to C_B$ around $q$ , while remembering the preimage of the node through adding in markings $q_{1}$ and $q_{2}$ . Then, by our inductive hypothesis, we can find some expansion $\widetilde {C}'_{B}\to C_B'$ which is an isomorphism over $\eta$ and an extension $E'$ of $\nu ^*E_{\eta }$ to $\tilde {X}'_B:=X_B\times _{C_B} \widetilde {C}_B'$ . We again denote the lifts of $q_1,q_2$ to $\tilde {C}_B'$ by $q_1,q_2$ . In order to glue together the total family along the markings $q_1$ and $q_2$ , we need to produce an isomorphism $E'|{q_{1}}\simeq E'|_{q_2}$ extending the given one over the generic fiber. As in the previous case, such an isomorphism exists after twisting, say, $E'|_{q_1}$ along the preimage of $\xi$ . However, since $\widetilde {C}'_B$ is connected, it is a priori not clear how to do this without simultaneously changing $E'|_{q_2}$ . To get around this issue, we may pass to a further expansion by blowing up $\widetilde {C}'_B$ along the intersection of $q_1$ and the special fiber $\widetilde {C}'_{\xi }$ . Then we can achieve the desired modification by twisting the pullback of $E'$ by an integer multiple of the exceptional component.

Proof. We may modify $C_R$ by repeatedly blowing up singular points to obtain a family with regular total space [Sta23, Tag 0CDE], which will automatically be an expansion of $C_R$ . Without loss of generality, we may therefore assume that $C_R$ is nonsingular.

Let $\xi$ denote the closed point of $\textrm {Spec}\,R$ . By the argument in [Reference Gieseker and LiGL94, proof of the last statement of Proposition 3.3], after passing to some extension of $R$ and further expanding $C_R$ over the special fiber, we can assume that $E_{\eta }$ extends to a torsion-free coherent sheaf $E_R$ on $X_R$ with the following properties.

1. (a) The restriction of $E_R$ to $X_\xi$ is torsion-free.

2. (b) The sheaf $E_R$ is locally free along the fibers of $X_\xi \to C_{\xi }$ over singular and marked points.

Let $P(C_{\xi })$ denote the collection of singular, marked and generic points of $C_{\xi }$ . Unless $E_R$ is $f$ -stable, there exists $x \in P(C_{\xi })$ such that the restriction $E_x$ of $E_R$ to the curve $f^{-1}(x)$ is unstable. Let $a$ be the minimum of the values $\mu _{min}(E_x)$ , where $\mu _{min}$ denotes the minimal slope in a Harder–Narasimhan filtration. Let $\rho$ be the maximum rank of a maximally destabilizing quotient of $E_{x}$ , ranging over those $x\in P(C_{\xi })$ for which $\mu _{min}(E_x) = a$ . We claim that after taking a base change on $R$ and a further expansion of $C_R$ , we may find a different completion of $E_{\eta }$ such that either $a$ increases or $a$ stays the same and $\rho$ decreases. Since there are only finitely many possible values for $\rho$ , and since the set of possible values of $a$ lies in ${\mathbb{Z}}/r$ and is bounded above by $d/r$ , we find that after doing so finitely many times, we end up with an $f$ -stable extension of $E_{\eta }$ .

To prove this claim, consider the relative Quot-scheme

\begin{align*}q:\operatorname {Quot}_{E_R,X_R/C_R}(\rho ,a)\to C_R\end{align*}

parametrizing quotients of $E_R$ of rank $\rho$ and slope $a$ on the fibers of $X_R\to C_R$ . By assumption, the image of $q$ contains some points of $P(C_{\xi })$ . From the minimal choice of $a,\rho$ , it follows from an analysis of the relative deformation space that at each such point, the map $q$ is locally a closed embedding (see the proof of Theorem 5 in [Reference NitsureNit11]).

We claim that, after passing to an extension of $R$ and further expanding $C_{R}$ along the special fiber, we can arrange that around each point in $P(C_{\xi })$ , the set-theoretic image of $q$ is supported on $C_{\xi }$ . This is equivalent to asking that the closure in $C_R$ of the set-theoretic image $\operatorname {Im} q|_{\eta }$ is disjoint from $P(C_{\xi })$ . By $f$ -stability, $\operatorname {Im} q|_{\eta }$ is a finite collection of closed points of $C_{\eta }$ not meeting any nodes or marked points. After possibly passing to a finite extension of $R$ , we may add new marked points $\rho _{i,\eta }$ so that $\operatorname {Im}q|_{\eta }$ is the union of the images of the $\rho _{i,\eta }$ . Now the claim follows from the semistable reduction theorem for marked nodal curves. It follows that there is a closed subscheme $Z\subset C_R$ whose associated points are all in $P(C_{\xi })$ and such that around each of its associated points it agrees with the closed subscheme defined by the Quot-scheme.

If $Z$ contains a component $C_i$ of $C_{\xi }$ , we may replace $E_R$ by the elementary modification of $E_R$ along a maximally destabilizing quotient on that component. By the minimal choice of $a$ and $\rho$ , the resulting sheaf will still be locally free at fibers over marked points and nodes. This has the effect of dividing the ideal sheaf of $Z$ by the uniformizer of that component. Thus, if $Z$ is locally principal, one can do so until $Z$ becomes empty, in which case there is no point left in $P(C_{\xi })$ with maximally destabilizing subsheaf of slope $a$ and rank $\rho$ . If $Z$ is not locally principal, one can use Lemma 3.7 to find an extension $R'$ of $R$ and an expansion $c:\widetilde {C}_{R'}\to C_{R'}$ over $R'$ which is trivial over the generic fiber such that the scheme-theoretic preimage of $Z$ in $\widetilde {C}_{R'}$ is principal. Since the relative Quot-scheme is compatible with pullback, this reduces to the case that $Z$ is principal, which we already treated.

Proof. We first argue that by repeatedly blowing up at marked points, one can achieve that the preimage of $Z$ is principal around each marked point. Working étale locally around a marked point $x$ , we may assume that $Z$ is supported at $x$ and that the family is locally given by $\textrm {Spec}\,R[t]\to \textrm {Spec}\,R$ , with the section given by $t=0$ . Let $\pi$ be a uniformizer of $R$ . Let $Z_{\pi }$ and $Z_t$ be the intersections of $Z$ with the loci $(\pi=0)$ and $(t=0)$ respectively. We claim that the invariant $\ell (Z_{\pi })+\ell (Z_{t})$ decreases for the preimage of $Z$ on the blowup. Indeed, the new marked point on the blowup is cut out by coordinates $\pi , u$ , where $t= \pi u$ . There exist elements $g_1 = t^a+\pi f_1$ and $g_2 = \pi ^b +t f_2$ in the defining ideal $I_Z$ of $Z$ , where $a=\ell (Z_{\pi })$ and $b = \ell (Z_{t})$ . Let also $k$ be maximal so that $I_Z\subseteq (\pi ,t)^k$ . Thus, $q^{-1}Z$ contains the exceptional divisor to order at least $k$ . Note that $1\leqslant k\leqslant a,b$ . Let $\widetilde {Z}$ be the non-principal part of $q^{-1 }Z$ at the marked point. Then by a direct computation, one has

\begin{align*} \ell (\widetilde {Z}_\pi ) & \leqslant k,\\ \ell (\widetilde {Z}_u) & \leqslant b - k. \end{align*}

In particular, $\ell (\widetilde {Z}_\pi ) +\ell (\widetilde {Z}_\pi ) \leqslant b \lt \ell (Z_\pi ) +\ell (Z_t)$ . Essentially the same argument works for $x$ a node in $C_{\xi }$ , where one has parameters $s,t$ locally cutting out the components of $C_{\xi }$ at $x$ . Here, one needs to repeatedly blow up the nodes in the reduced preimage of $C_{\xi }$ . This yields a modification $\widehat {C}_R\to C_{R}$ , which principalizes $Z$ and is an isomorphism over the generic fiber, but where the fiber $\widehat {C}_\xi$ may have non-reduced components. After taking a ramified extension $R'$ of $R$ with sufficiently divisible degree, taking the normalization of $\widehat {C}_{R'}$ and resolving the singularities through repeated blowups, we obtain an expansion of $\widetilde {C}_{R'} \to C_{R'}$ with the desired properties.

Proof. Let $R$ be a DVR with generic and closed points $\eta$ and $\xi$ and with a given morphism $\textrm {Spec}\,R\to B$ . Let $E_{\eta }$ be a sheaf on an expansion $\widetilde {X}_{\eta }\to \widetilde {C}_{\eta }$ of $X_{\eta }\to C_{\eta }$ such that $E_{\eta }$ is $f$ -stable and $\alpha$ -balanced. By Proposition 3.6, we can find some extension $\widetilde {X}_R\to \widetilde {C}_R$ of this data to an $f$ -stable family $E_R$ (possibly after replacing $R$ by an extension). We claim that we can modify this data to obtain an $\alpha$ -balanced bundle on a minimal expansion.

For this, pick a line bundle $N_0$ on $\widetilde {C}_R$ that has degree zero on exceptional components of $\widetilde {C}_{\xi }$ and such that $ M(E_R)\otimes N_0$ has degree a multiple of $rd_0$ on each component of $C_{\xi }$ (this may require further extending $R$ ). This is possible by Remark 3.11 (ii). Then, further extending $R$ if needed, we can pick $N_1$ on $\widetilde {C}_R$ such that $\underline {\deg }\, N_1|_{\widetilde {C}_\xi }^{\otimes rd_0} = \underline {\deg } \,(M(E|_{Z})\otimes N_0)|_{\widetilde {C}_{\xi }}$ . By changing $N_0$ , we may in fact assume without loss of generality that $N_1^{\otimes r d_0} \cong M(E) \otimes N_0$ on $\widetilde {C}_R$ .

Now consider the stability condition $\alpha' := \alpha + \underline {\textrm {deg}} \, N_0 / (r d_0)$ . Then we have the following lemma.

Lemma 3.16. A sheaf $E$ is $\alpha$ -balanced if and only if $N_1$ is $\alpha'$ -semistable in the following sense: $N_1$ has non-negative degree on exceptional components, and for every subcurve $\emptyset \subsetneq Z \subsetneq \widetilde {C}$ , we have

\begin{align*}\left | \deg N_1|_Z - \sum _{D\subset Z} \alpha'(D) \right | \leqslant \frac {\# (Z\cap Z^c)}{2}.\end{align*}

Proof. Let $\emptyset \subsetneq Z \subsetneq \widetilde {C}$ be a subcurve. Then

\begin{align*}\deg N_1|_{Z} = \frac {\deg M(E)|_Z}{rd_0} + \frac {\deg N_0|_Z}{rd_0}\end{align*}

and

\begin{align*}\sum _{D\subset Z} \alpha '(D) = \sum _{D\subset Z} \alpha (D) + \frac {\deg N_0|_D}{r d_0} = \sum _{D\subset Z} \alpha (D) + \frac {\deg N_0|_{Z}}{rd_0},\end{align*}

so

\begin{align*}\left | \deg N_1|_Z - \sum _{D\subset Z} \alpha '(D) \right | = \left | \frac {\deg M(E)|_Z}{rd_0 } - \sum _{D\subset Z} \alpha (D) \right |.\end{align*}

Since also the degree of $N_1$ on exceptional components is equal to a positive multiple of the degree of $M(E)$ , the lemma follows.

We consider the coherent sheaf $N_1':=c_*N_1$ on $C_R$ with adjunction map $\psi :c^*N_1'\to N_1$ . If this is surjective, we obtain an induced map $P:\widetilde {C}_{R}\to \mathbb{P}(N_1')$ .

As in [Reference Kass and PaganiKP19, Corollary 4.3], it follows from Simpson stability that any $\alpha '$ -semistable torsion-free sheaf on the generic fiber has an $\alpha '$ -semistable limit. Let $N'_2$ be an $\alpha '$ -semistable limit that agrees with $N_1'$ on the generic fiber. After possibly further expanding $\widetilde {C}_{\xi }$ , we can assume that we have a morphism $P_2:\widetilde {C}_R\to \mathbb{P}(N_2')$ and denote by $N_2$ the pullback of the universal line bundle along $P_2$ . Note that this implies that $N_1$ and $N_2$ are isomorphic over $\eta$ . We consider the line bundle $N_E:=N_2\otimes N_1^{\vee }$ on $\widetilde {C}_R$ .

Claim: The sheaf $E_1:= E\otimes f^*N_E$ is $\alpha$ -balanced.