The Geometry of Break Divisors

I’d like to continue this discussion of break divisors on graphs & tropical curves by describing an interesting connection to algebraic geometry.  In this post, I’ll explain a beautiful connection to the theory of compactified Jacobians discovered by Tif Shen, a recent Ph.D. student of Sam Payne at Yale.

Review of break divisors

Let $\Gamma$ be a tropical curve of genus $g$, i.e., a finite graph $G$ of genus $g$ together with an identification of each edge $e \in E(G)$ with a line segment of some length $\ell(e)>0$.  We call $G$ a model for $\Gamma$, and say that $G$ is a regular model if $\ell(e)=1$ for all $e$.

Recall that a break divisor on $\Gamma$ is an effective divisor $D$ of degree $g$ such that for every closed connected subgraph $H$ of $\Gamma$, the degree of $D$ restricted to $H$ is at least the genus of $H$.  A break divisor on $G$ is a break divisor on $\Gamma$ supported on the vertices of $G$.

Recall also that there is a canonical polyhedral decomposition ${\mathbb B} = {\mathbb B}_G$ of the $g$-dimensional real torus ${\rm Pic}^g(\Gamma)$ associated to each model $G$ for $\Gamma$.  The maximal faces of ${\mathbb B}_G$ are naturally in bijection with spanning trees of $G$, and the minimal faces (i.e., vertices) of ${\mathbb B}_G$ are naturally in bijection with break divisors on $G$.  We will refer to ${\mathbb B}$ as the ABKS decomposition.

Compactified Jacobians

Suppose $X$ is a proper reduced algebraic curve over an algebraically closed field $k$.  If $X$ is smooth and irreducible, then the Jacobian ${\rm Pic}^0(X)$ is an Abelian variety, and in particular ${\rm Pic}^0(X)$ is proper.  However, if $X$ is not necessarily smooth or irreducible then ${\rm Pic}^0(X)$ (which parametrizes isomorphism classes of line bundles whose restriction to each irreducible component of $X$ has degree zero) is no longer proper.  For example, if $X$ is an irreducible nodal conic then ${\rm Pic}^0(X)$ is isomorphic to the multiplicative group ${\mathbb G}_m$.  Finding “nice” compactifications of ${\rm Pic}^0(X)$ when $X$ is singular and/or reducible is an important problem in algebraic geometry.   Among other things, this is important because if you have a (flat) family of smooth curves degenerating to such an $X$, one would like to be able to keep track of how line bundles on the generic fiber degenerate.  In our nodal conic example, if you take a family $X_t$ of smooth conics degenerating at $t=0$ to $X=X_0$, together with a family of points $P_t$ on $X_t$ converging to the node $P_0$ on $X_0$, the line bundles $L_t = {\mathcal O}_{X_t}(-P_t)$ will converge to the ideal sheaf of $P_0$ on $X_0$, but this is not a line bundle.  Morally speaking, this explains why ${\rm Pic}^0(X)$ fails to be compact.  It also suggests a solution: look at a moduli space which parametrizes objects more general than line bundles.  In our example, although the ideal sheaf of $P_0$ is not a line bundle, it is still a coherent sheaf which is “pure of rank one” (in a suitable technical sense).

A number of mathematicians have pursued the idea of constructing a moduli space of (suitably restricted) coherent sheaves in this and much more general situations.  In most cases the key idea is to use Geometric Invariant Theory.  This involves introducing the notions of stable and semistable coherent sheaves, which we briefly review in the context of arbitrary projective schemes, though we will eventually specialize to the case of nodal curves.

Simpson compactifications

Let $X$ be a projective scheme of pure dimension $d$ over $k$.  Let $L$ be an ample line bundle on $X$ (called a polarization), and let ${\mathcal F}$ be a coherent sheaf on $X$.  We say that ${\mathcal F}$ is pure (of dimension $d$) if the dimension of the support of every nonzero subsheaf ${\mathcal E}$ of ${\mathcal F}$ is $d$. The Hilbert polynomial of a pure coherent sheaf ${\mathcal F}$ with respect to $L$ is by definition $P_{\mathcal F}(n) := \chi({\mathcal F} \otimes L^n)$.  We write $p_{\mathcal F}(n)$ for the unique monic scalar multiple of $P_{\mathcal F}(n)$.   We say that ${\mathcal F}$ is semistable (resp. stable) if it is pure and for every nonzero proper subsheaf ${\mathcal E}$ of ${\mathcal F}$ we have $p_{\mathcal E}(n) \leq p_{\mathcal F}(n)$ for $n \gg 0$ (resp. $<$ instead of $\leq$).  More generally, if ${\mathfrak X} / S$ is a projective $S$-scheme and ${\mathfrak L}$ is a relatively ample line bundle on ${\mathfrak X}$, a coherent sheaf ${\mathcal F}$ on ${\mathfrak X}$ is called semistable (resp. stable) if it is flat over $S$ and the restriction of ${\mathfrak X}$ to $X_s$ is semistable (resp. stable) for each geometric point $s \in S$.

Theorem (Simpson): Given a polynomial $P \in {\mathbb Z}[n]$, there is a projective scheme $M_P$ corepresenting the functor ${\mathcal M}_P$ taking an $S$-scheme $S'$ to the set of semistable sheaves on ${\mathfrak X}' := {\mathfrak X} \times_S S'$ with Hilbert polynomial $P$.

Compactified Jacobians of nodal curves and “independence of $L$

Now let’s specialize to the case where $X$ is a nodal curve.  A coherent sheaf ${\mathcal F}$ on $X$ is said to have rank one if it has rank one at every generic point of $X$.  Such a sheaf is pure iff it does not contain a skyscraper sheaf, and the Hilbert polynomial of ${\mathcal F}$ is $P_{\mathcal F}(n) = {\rm deg}(L) n + \chi({\mathcal F})$.  The degree of a coherent sheaf ${\mathcal F}$ is $\chi({\mathcal F}) - \chi({\mathcal O}_X)$. For each degree $d$, there is a projective Simpson coarse moduli space, denoted $\widehat{{\rm Pic}}_L^d(X)$, of semistable (with respect to $L$) rank one coherent sheaves of degree $d$ on $X$.

An interesting question which immediately arises is whether the scheme $\widehat{{\rm Pic}}_L^d(X)$ really depends on the choice of $L$.  In general it does.  However, it turns out that there are certain choices of $d$ for which it does not.  Alexeev was the first to observe that when $d=g-1$, the scheme $\widehat{{\rm Pic}}_L^d(X)$ is independent of $L$ and thus canonically defined.  The reason for this is fairly straightforward.  Following Oda and Seshadri, one shows that for each connected projective subcurve $Y \subset X$, there is a unique maximal subsheaf ${\mathcal F}_Y$ of ${\mathcal F}$ supported on $Y$, and to check the semistability of a pure coherent sheaf ${\mathcal F}$ of rank one it suffices to check the inequality defining semistability for the finitely many subsheaves ${\mathcal E} = {\mathcal F}_Y$.  In other words, ${\mathcal F}$ is semistable iff $\frac{{\rm deg}({\mathcal F}_Y) + \chi({\mathcal O}_Y)}{{\rm deg}(L|_Y)} \leq \frac{{\rm deg}(F) + \chi({\mathcal O}_X)}{{\rm deg}(L)}$ for every subcurve $Y \subset X$.

To each subcurve $Y$ there is a closely related sheaf ${\mathcal F}^Y$ on $X$, which is the maximal torsion-free quotient of the restriction of ${\mathcal F}$ to $Y$.  Comparing ${\mathcal F}_Y$ and ${\mathcal F}^Y$, and letting $Y'$ denote the closure of the complement of $Y$ in $X$, one finds (setting $d := {\rm deg}({\mathcal F})$) that ${\mathcal F}$ is semistable if and only if ${\rm deg}({\mathcal F}^Y) \leq \# Y \cap Y' + (g_Y - 1) +\frac{{\rm deg}(L|_Y)}{{\rm deg}(L)}(d + 1 - g)$ for all subcurves $Y$ as above.  This is visibly independent of $L$ if $d = g-1$, explaining Alexeev’s observation.

Much has been written (by Alexeev, Caporaso, Kass, Melo, Viviani…) about the “canonical” compactified Jacobian $\widehat{{\rm Pic}}^{g-1}(X)$.  However, there is another canonical choice for $d$ that yields a compactified Jacobian $\widehat{{\rm Pic}}_L^d(X)$ which is independent of $L$ (and this seems to have flown under the radar of algebraic geometers).  Namely, if we take $d = g$ then the above semistability condition is also independent of $L$ whenever $Y \subsetneq X$, since in that case $0 < \frac{{\rm deg}(L|_Y)}{{\rm deg}(L)}(d + 1 - g) = \frac{{\rm deg}(L|_Y)}{{\rm deg}(L)} < 1$.  Moreover, this shows that when $d=g$, $F$ is semistable if and only if it is stable.  It follows that the canonical compactified Jacobian $\widehat{{\rm Pic}}^{g}(X)$ is in fact a fine moduli space, and in particular has some desirable properties which $\widehat{{\rm Pic}}^{g-1}(X)$ lacks.

Tropicalization of pure rank 1 sheaves

This leads us back, as promised, to break divisors.  Let $G$ be the dual graph of $X$, so that $G$ has a vertex $v$ for each irreducible component $X_v$ of $X$, and an edge $e = vw$ for each node $x_e \in X_v \cap X_w$.  We enhance $G$ to a vertex-weighted graph by assigning the vertex $v$ a weight equal to the geometric genus $g_v$ of $X_v$.  Let $\Gamma$ be any tropical curve (metric graph) with $G$ as a model, and enhance $G$ to a vertex-weighted tropical curve in the same way as above.  By definition, we say that a break divisor on a vertex-weighted tropical curve $\Gamma$ is a divisor $D$ on $\Gamma$ such that $D - \sum_v g_v (v)$ is a break divisor on the corresponding unweighted tropical curve.

Given a pure coherent sheaf ${\mathcal F}$ of degree $d$ and rank 1 on $X$, we define its tropicalization ${\rm trop}({\mathcal F}) \in {\rm Div}^d(\Gamma)$ as follows.  Let ${\rm NF}({\mathcal F})$ be the set of nodes at which ${\mathcal F}$ is not locally free, and identify ${\rm NF}({\mathcal F})$ with a subset of $E(G)$.  Define ${\rm trop}({\mathcal F}) := \sum_{v \in V(G)} {\rm deg}({\mathcal F}|_{X_v})(v) + \sum_{e \in {\rm NF}({\mathcal F})} (p_e),$ where $p_e \in \Gamma$ is the midpoint of $e \subset \Gamma.$  (When ${\mathcal F}$ is a line bundle, this is just the “usual” multidegree or tropicalization map.) It is not difficult to see that $D \in {\rm trop}({\mathcal F})$ is an effective divisor of degree $d$.

Theorem (Shen): The sheaf ${\mathcal F}$ is semistable iff ${\mathcal F}$ is stable iff ${\rm trop}({\mathcal F})$ is a break divisor on the vertex-weighted tropical curve $\Gamma$.

The proof of this theorem is straightforward given the discussion above.  The main points are that (a) for each subcurve $Y \neq X$ we have ${\rm deg}({\mathcal F}^Y) = {\rm deg}(D|_{\Gamma_Y})$ for any $D \in {\rm trop}({\mathcal F})$, where $\Gamma_Y$ is the (geometric realization of the) induced subgraph $G_Y$ of $G$ corresponding to $Y$; and (b) a divisor $D$ on an unweighted graph $G$ is a break divisor if and only if for every proper connected induced subgraph $H$, we have ${\rm deg}(D|_H) \leq g_H - 1 + {\rm outdeg}_H(G).$  (Apply the definition of break divisors to the complement of $H$ in $G$.)

The local type stratification(s)

Shen then goes on to establish several interesting connections between the geometry of the Simpson compactified Jacobian $\widehat{{\rm Pic}}^{g}(X)$ and the combinatorics of the break divisor decomposition ${\mathbb B}_G$ of ${\rm Pic}^g(\Gamma)$.  For example, Jesse Kass has defined a stratification of $\widehat{{\rm Pic}}_L^d(X)$ which he calls the local type stratification. [Added 10/6/17: Jesse tells me that Melo and Viviani independently came to the same idea, and that the definition is essentially already present in the earlier paper of Oda and Seshadri.] In the case $d=g$ of interest to us, one can describe the poset structure on the local type stratification of $\bar{J}_g := \widehat{{\rm Pic}}^g(X)$ as follows. For each subset $S \subseteq E(G)$, corresponding to a subset of nodes of $X$, define $J_g^S$ to be the subset of $\bar{J}_g$ corresponding to coherent sheaves ${\mathcal F}$ with ${\rm NF}({\mathcal F})=S$.  The local type stratification in this case is $\bar{J}_g = \coprod_{S} J_g^S.$ We put a partial order on the strata in the usual way by declaring that $J_g^{S'} \leq_{\rm LTS} J_g^S$ iff $J_g^{S'}$ is contained in the Zariski closure of  $J_g^S$.

The strata $J_g^S$ are not in general connected.  Shen shows that the connected components of $J_g^S$ are in one-to-one-correspondence with break divisors $D$ on $\Gamma$ such that $D - \sum_{e \in S} p_e$ is supported on the vertices of $G$.  Let $J_g^D =\{ {\mathcal F} \; : \; {\rm trop}({\mathcal F}) = D \}$ be the corresponding stratum of $\bar{J}_g$.  These strata are indexed by the finite collection $\Lambda$ of break divisors of $\Gamma$ supported on $V(G')$, where $G'$ is the first barycentric subdivision of $G$.   We call $\bar{J}_g = \coprod_{D \in \Lambda} J_g^D$ the refined local type stratification.  The partial order $\leq_{\rm LTS}$ induces a partial order $\leq_{\rm RLTS}$ on $\Lambda$ in a natural way.

The map which associated to a break divisor $D \in \Lambda$ the unique face $C_D$ of the ABKS decomposition ${\mathbb B}$ such that $[D] \in {\rm relint}(C_D)$ is a bijection from $\Lambda$ to the set of faces of ${\mathbb B}$.

Theorem (Shen): For $D,D' \in \Lambda$, we have $C_{D'} \subseteq C_{D}$ iff $D \leq_{\rm RLTS} D'$.  Thus there is a canonical order-reversing bijection $J_g^D \leftrightarrow C_D$ between the strata of the refined local type decomposition of $\bar{J}_g$ and the faces of the ABKS decomposition of ${\rm Pic}^g(\Gamma)$.  Moreover, the codimension of $J_g^D$ is equal to the dimension of $C_D$ (which is the number of edges of $G$ on whose relative interior $D$ is non-integral).

Coda: Mumford models and relative Simpson compactifications

Let $R$ be a discrete valuation ring with residue field $k$, and let ${\mathfrak X} / R$ be a regular strictly semistable curve with smooth and proper generic fiber $X_\eta$ of genus $g$ and totally degenerate special fiber $X_0$.  By a construction due to Mumford, the ABKS decomposition ${\mathbb B}$ of the real torus ${\rm Pic}^g(\Gamma)$ gives rise in a natural way to an $R$-model $J_{\mathbb B}$ of ${\rm Jac}(X_\eta)$.  Shen proves that $J_{\mathbb B}$ is isomorphic over $R$ to the degree $g$ Simpson relative compactified Jacobian $\bar{\mathcal J}_g.$  Moreover, the smooth locus of $\bar{\mathcal J}_g$ is isomorphic to the Néron model of ${\rm Jac}(X_\eta)$.

Concluding remarks

(1) The midpoint $p_e$ of $e \in E(G)$ is not really a distinguished point in any natural sense.  It would be more natural to define ${\rm trop}({\mathcal F})$ as a subset, rather than an element, of $\Gamma$ by allowing each $p_e$ to be any point in the relative interior of $e$.

(2) In his thesis, Shen does not actually make full use of Theorem 2 from loc. cit. Our presentation in this post represents a streamlined treatment of Shen’s arguments.

(3) It follows from the work of Alexeev that a pure sheaf ${\mathcal F}$ of degree $g-1$ is semistable iff ${\rm trop}({\mathcal F})$ is orientable, meaning that it’s equal to ${\rm div}({\mathcal O})$ for some orientation ${\mathcal O}$ of $G'$ (see this post for the definition of ${\rm div}({\mathcal O})$).  The work of Alexeev, Oda, and Seshadri implies that the canonical Simpson compactified Jacobian $\bar{J}_{g-1}$ is isomorphic to the Mumford model of ${\rm Jac}(X_\eta)$ associated to the Voronoi decomposition for the tropical theta function.  By the tropical version of Riemann’s theorem (proved by Mikhalkin and Zharkov), this is also isomorphic to the Mumford model of ${\rm Jac}(X_\eta)$ associated to $W_{g-1}(\Gamma) \subset {\rm Pic}^{g-1}(\Gamma)$.

(4) What we have simply called “semistable” is usually called p-semistable in the literature.  There is also a closely related notion of slope semistability, which coincides with p-semistability when $X$ has dimension 1.

(5) The same argument as above shows that $\widehat{{\rm Pic}}_L^{g-2}(X)$ is also independent of $L$.  We are not aware of a non-trivial description of the resulting canonical compactified Jacobian in the literature.

(6) [Added 10/6/17, h/t Jesse Kass] Alexeev proves much more than we’ve alluded to here about degree $g-1$ Simpson compactified Jacobians.  For example, he shows that every such space carries a canonical polarization that makes it into a stable pair, which this characterizes the moduli space up to isomorphism.  This is one way to see that $\bar{J}_{g-1}$ is isomorphic to the Mumford model of ${\rm Jac}(X_\eta)$ associated to the Voronoi decomposition (since the latter can also be shown to define a stable pair).