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 be a tropical curve of genus , i.e., a finite graph of genus together with an identification of each edge with a line segment of some length . We call a model for , and say that is a regular model if for all .
Recall that a break divisor on is an effective divisor of degree such that for every closed connected subgraph of , the degree of restricted to is at least the genus of . A break divisor on is a break divisor on supported on the vertices of .
Recall also that there is a canonical polyhedral decomposition of the -dimensional real torus associated to each model for . The maximal faces of are naturally in bijection with spanning trees of , and the minimal faces (i.e., vertices) of are naturally in bijection with break divisors on . We will refer to as the ABKS decomposition.
Suppose is a proper reduced algebraic curve over an algebraically closed field . If is smooth and irreducible, then the Jacobian is an Abelian variety, and in particular is proper. However, if is not necessarily smooth or irreducible then (which parametrizes isomorphism classes of line bundles whose restriction to each irreducible component of has degree zero) is no longer proper. For example, if is an irreducible nodal conic then is isomorphic to the multiplicative group . Finding “nice” compactifications of when 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 , 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 of smooth conics degenerating at to , together with a family of points on converging to the node on , the line bundles will converge to the ideal sheaf of on , but this is not a line bundle. Morally speaking, this explains why 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 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.
Let be a projective scheme of pure dimension over . Let be an ample line bundle on (called a polarization), and let be a coherent sheaf on . We say that is pure (of dimension ) if the dimension of the support of every nonzero subsheaf of is . The Hilbert polynomial of a pure coherent sheaf with respect to is by definition . We write for the unique monic scalar multiple of . We say that is semistable (resp. stable) if it is pure and for every nonzero proper subsheaf of we have for (resp. instead of ). More generally, if is a projective -scheme and is a relatively ample line bundle on , a coherent sheaf on is called semistable (resp. stable) if it is flat over and the restriction of to is semistable (resp. stable) for each geometric point .
Theorem (Simpson): Given a polynomial , there is a projective scheme corepresenting the functor taking an -scheme to the set of semistable sheaves on with Hilbert polynomial .
Compactified Jacobians of nodal curves and “independence of ”
Now let’s specialize to the case where is a nodal curve. A coherent sheaf on is said to have rank one if it has rank one at every generic point of . Such a sheaf is pure iff it does not contain a skyscraper sheaf, and the Hilbert polynomial of is . The degree of a coherent sheaf is . For each degree , there is a projective Simpson coarse moduli space, denoted , of semistable (with respect to ) rank one coherent sheaves of degree on .
An interesting question which immediately arises is whether the scheme really depends on the choice of . In general it does. However, it turns out that there are certain choices of for which it does not. Alexeev was the first to observe that when , the scheme is independent of and thus canonically defined. The reason for this is fairly straightforward. Following Oda and Seshadri, one shows that for each connected projective subcurve , there is a unique maximal subsheaf of supported on , and to check the semistability of a pure coherent sheaf of rank one it suffices to check the inequality defining semistability for the finitely many subsheaves . In other words, is semistable iff for every subcurve .
To each subcurve there is a closely related sheaf on , which is the maximal torsion-free quotient of the restriction of to . Comparing and , and letting denote the closure of the complement of in , one finds (setting ) that is semistable if and only if for all subcurves as above. This is visibly independent of if , explaining Alexeev’s observation.
Much has been written (by Alexeev, Caporaso, Kass, Melo, Viviani…) about the “canonical” compactified Jacobian . However, there is another canonical choice for that yields a compactified Jacobian which is independent of (and this seems to have flown under the radar of algebraic geometers). Namely, if we take then the above semistability condition is also independent of whenever , since in that case . Moreover, this shows that when , is semistable if and only if it is stable. It follows that the canonical compactified Jacobian is in fact a fine moduli space, and in particular has some desirable properties which lacks.
Tropicalization of pure rank 1 sheaves
This leads us back, as promised, to break divisors. Let be the dual graph of , so that has a vertex for each irreducible component of , and an edge for each node . We enhance to a vertex-weighted graph by assigning the vertex a weight equal to the geometric genus of . Let be any tropical curve (metric graph) with as a model, and enhance 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 is a divisor on such that is a break divisor on the corresponding unweighted tropical curve.
Given a pure coherent sheaf of degree and rank 1 on , we define its tropicalization as follows. Let be the set of nodes at which is not locally free, and identify with a subset of . Define where is the midpoint of (When is a line bundle, this is just the “usual” multidegree or tropicalization map.) It is not difficult to see that is an effective divisor of degree .
Theorem (Shen): The sheaf is semistable iff is stable iff is a break divisor on the vertex-weighted tropical curve .
The proof of this theorem is straightforward given the discussion above. The main points are that (a) for each subcurve we have for any , where is the (geometric realization of the) induced subgraph of corresponding to ; and (b) a divisor on an unweighted graph is a break divisor if and only if for every proper connected induced subgraph , we have (Apply the definition of break divisors to the complement of in .)
The local type stratification(s)
Shen then goes on to establish several interesting connections between the geometry of the Simpson compactified Jacobian and the combinatorics of the break divisor decomposition of . For example, Jesse Kass has defined a stratification of 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 of interest to us, one can describe the poset structure on the local type stratification of as follows. For each subset , corresponding to a subset of nodes of , define to be the subset of corresponding to coherent sheaves with . The local type stratification in this case is We put a partial order on the strata in the usual way by declaring that iff is contained in the Zariski closure of .
The strata are not in general connected. Shen shows that the connected components of are in one-to-one-correspondence with break divisors on such that is supported on the vertices of . Let be the corresponding stratum of . These strata are indexed by the finite collection of break divisors of supported on , where is the first barycentric subdivision of . We call the refined local type stratification. The partial order induces a partial order on in a natural way.
The map which associated to a break divisor the unique face of the ABKS decomposition such that is a bijection from to the set of faces of .
Theorem (Shen): For , we have iff . Thus there is a canonical order-reversing bijection between the strata of the refined local type decomposition of and the faces of the ABKS decomposition of . Moreover, the codimension of is equal to the dimension of (which is the number of edges of on whose relative interior is non-integral).
Coda: Mumford models and relative Simpson compactifications
Let be a discrete valuation ring with residue field , and let be a regular strictly semistable curve with smooth and proper generic fiber of genus and totally degenerate special fiber . By a construction due to Mumford, the ABKS decomposition of the real torus gives rise in a natural way to an -model of . Shen proves that is isomorphic over to the degree Simpson relative compactified Jacobian Moreover, the smooth locus of is isomorphic to the Néron model of .
(1) The midpoint of is not really a distinguished point in any natural sense. It would be more natural to define as a subset, rather than an element, of by allowing each to be any point in the relative interior of .
(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 of degree is semistable iff is orientable, meaning that it’s equal to for some orientation of (see this post for the definition of ). The work of Alexeev, Oda, and Seshadri implies that the canonical Simpson compactified Jacobian is isomorphic to the Mumford model of 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 associated to .
(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 has dimension 1.
(5) The same argument as above shows that is also independent of . 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 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 is isomorphic to the Mumford model of associated to the Voronoi decomposition (since the latter can also be shown to define a stable pair).