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Moduli Spaces of Curves: Classical and Tropical

Melody Chan

Communicated by Notices Associate Editor Steven Sam

Article cover

In this article, I’ll try to do two things. First, I’ll give an informal and elementary introduction to the idea of a moduli space, and then to moduli spaces of Riemann surfaces and their Deligne-Mumford-Knudsen compactifications. There is nothing new in this first section of the article, but I hope some people will enjoy it anyway.

Second, I’ll discuss some tropical geometry, assuming no prior knowledge of the subject, and build up to some recent results on moduli spaces that were obtained using tropical techniques. We’ll encounter tropical curves, weight filtrations, graph complexes, and more on the way. Those who know the usual story of moduli spaces can start at page 1 for the second part of the article. The discussion of new results begins on page 1.

In case you are interested, I put some exercises, ranging from elementary to not so elementary, at the end of the article.

What is a Moduli Space?

A moduli space is a parameter space—usually, a parameter space for classes of geometric objects of interest. Think of a moduli space like a mail-order catalog. Pointing to the catalog conjures up a geometric object, off in a warehouse somewhere.

Furthermore, the catalog should be nicely organized. Nearby points of the catalog specify “nearby” geometric objects. Instead of a precise definition, let’s start with a toy example to get some intuition. We ask:

(1) What is a moduli space of lines in ?

Actually, let’s back up even further, and start with a warm-up question:

(0) What is a moduli space of nonvertical lines in ?

An answer to question (0) is provided by the usual from high school—or if you’re from the UK, or probably yet other conventions. A separate copy of , with coordinates called and , will do to answer question (0). In other words, by associating a point with the line

we regard this -plane as a “mail-order catalog” for nonvertical lines. Look at Figure 1.

Next, how about question (1): how can we “glue in” a space that parametrizes all vertical lines in the plane?

Elementary projective geometry provides an answer. Regard the original as sitting at height inside . Then a line in sweeps out a plane in through . Conversely, any plane in through , except for the plane , uniquely determines a line in .

Next, the space of planes in through may be identified with the space of lines in (a dual copy of) through , by associating to a plane the line through and .

Figure 1.

The -plane of nonvertical lines in .

Graphic for Figure 1.  without alt text

Finally, the space of lines in through is exactly the real projective plane , obtained as , where is antipodal identification on the -sphere. One way to picture is as a closed Northern hemisphere of the unit sphere in , with each antipodal pair of equatorial points identified. Just picture a line through in piercing the unit sphere : it does so in two antipodal points. Either exactly one of the two is in the open Northern hemisphere, or both of them are equatorial.

Working backwards, we conclude that the moduli space of lines in is minus a point—which is an (open) Möbius strip. The -plane of nonvertical lines inside it is then obtained by deleting a “line’s worth of lines,” cutting the Möbius strip.

Onwards to a second toy example:

(2) What is a moduli space of triangles, i.e., for isomorphism classes of Euclidean triangles?

Here we mean the familiar notion of triangles in the Euclidean plane, with isomorphisms being isometries. This is an excellent example to soak up at this point, as was apparently suggested by M. Artin. It is an opportunity to probe what we really mean by a moduli space, and shows some of the limitations of moduli spaces and the necessity of moduli stacks in certain situations. Here, the added subtlety is that triangles can have automorphisms, unlike our example above of lines together with embedding in , which don’t.

I haven’t given myself enough space here to take up moduli of triangles very much, but there is a really nice article by K. Behrend Beh14 introducing algebraic stacks through this lens, and which is written to be accessible, at least in part, to undergraduates. I’ll follow that article in this section.

What do we really mean by a moduli space of triangles? The most desirable situation would be to have a topological space which is a moduli space of triangles in the following strong sense. Not only do

the points of correspond bijectively to isomorphism classes of triangles,

but also, for an arbitrary topological space ,

families of triangles over , up to isomorphism, should correspond bijectively to continuous maps ,

where the bijection is required to take a family of triangles over to the natural continuous map sending to the point of corresponding to the triangle over .

Here, one has to have a robust notion of what a family of triangles over is, and what an isomorphism between two such is. You are invited to come up with your own precise definition of a family of triangles over ! Intuitively, it should be a triangle sitting above each point of , with a coherent notion of how edge lengths vary continuously as one walks around .

No such can exist! Suppose instead it did exist. Look at the family of triangles, over a line segment, drawn in Figure 2. (Let’s just say all the triangles involved are equilateral of equal side length .) No contradiction so far… but now imagine all the ways of gluing the family over the endpoints of the segment to produce a family of triangles over . There are six ways to do this, corresponding to the bijections . Yet all six families of triangles over yield the same constant map .

Figure 2.

There are six isomorphism classes of families of (equilateral, side-length-1) triangles over a circle.

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At this point, rather than throwing up our hands and giving up, there are two ways to proceed. Option 1: work with the coarse moduli space, which is a space that, in a way that can be made precise, comes closest among all spaces to satisfying our two desired conditions. Option 2: pass to an appropriate category of stacks, living with, and eventually embracing, the fact that one is no longer working with an actual space.

A word on Option 2: a stack is a category together with a functor from to the category of topological spaces—or to or whatever kind of geometry you’re interested in—satisfying some precise extra conditions we won’t get into. (One of them is called being fibered in groupoids.)

For example, continuing to work over , just as an example, there is a category whose objects are families of triangles , whose morphisms are pullback squares, and whose functor to sends to .

This truly enlarges the category in the following sense: given a fixed topological space , one can soup it up into a category (fibered in groupoids) over . The objects of the category are continuous maps , and the morphisms are commuting triangles . The functor to sends to .

This discussion is painfully brief, but I mention it in case it is helpful. It used to bug me to no end to hear people in graduate school say confidently, and cryptically, “Okay, so this isn’t really a space, but let’s just pretend it is a space” and I was like, “Pretend what is a space?” What is it? Well, it’s a certain category, which generalizes the notion of topological space, or scheme, or whatever, essentially by a Yoneda embedding.

In any case, we shall be primarily interested in this article in moduli spaces, and their rational cohomology in particular, that are Deligne-Mumford stacks—very roughly, locally admitting a scheme covering space. In this situation, the rational cohomology of the Deligne-Mumford stack coincides with that of the coarse moduli space. So, having made my excuses, I will now do the thing that bugged me in graduate school, which is to refer to the coarse moduli spaces and the stacks interchangeably and ambiguously whenever it suits us to do so.

Again, I refer enthusiastically to Beh14 for further reading, as well as to Fantechi’s short article Fan01.

Why moduli spaces?

Many spaces of classical interest in algebraic geometry may naturally be regarded as moduli spaces: Grassmannians and flag varieties, Hilbert schemes, moduli spaces of vector bundles, moduli spaces of abelian varieties… Moduli spaces are interesting! Things that might be simple when studied individually, such as a line in , are richer when studied in continuous families. (See Exercise 1.)

In fact, sometimes one is forced to study families even if one is solely interested only in the behavior of single objects: the geometry of the moduli space itself can tell you about the individual objects being parametrized.

Moduli spaces are also a natural setting in which intersection theory is useful. This is one of the reasons that compact moduli spaces are paramount. Otherwise, intersections can “escape to infinity.” Therefore, if our moduli space is not compact, then we seek a compactification, ideally a modular compactification. By this we mean an embedding of the original space into a compact space which is itself a moduli space, for some geometrically natural class of objects that enlarges the original. For example, compactifies the moduli space of lines in by adding a single point: the “line at infinity.” Technically, of course, every space is tautologically a moduli space—for its own (functor of) points.

What’s in this article

We will now put lines and triangles aside, and turn to the main characters of this article: the moduli spaces and of Riemann surfaces, and their Deligne-Mumford-Knudsen compactifications and . I will describe these spaces, as well as some recent work, joint with S. Galatius and S. Payne. In doing so, I shall attempt to illustrate another reason that compactifications are useful: suitable compactifications can provide insight into the topology of the space being compactified. The chain of reasoning we shall illustrate has many main characters: Riemann surfaces and their moduli, tropical curves and their moduli, dual complexes, mixed Hodge structures, graph complexesturn.

To close this section, let me also recommend D. Ben-Zvi’s 2008 survey on moduli spaces BZ08, which I discovered after drafting this article. That article takes a very similar expository at the outset, including a similar warm-up example of . But it also discusses a range of interesting but still accessible examples of moduli spaces.

Riemann Surfaces

The first main character of this article is the moduli space of -marked Riemann surfaces of genus .

A Riemann surface is a compact, connected complex manifold of complex dimension 1.

(In this article, we build “compact” and “connected” into the definition of Riemann surface.) An -marking of a Riemann surface is simply a choice of an ordered -tuple of distinct points on it.

The most basic invariant of a Riemann surface is its genus. That is, a Riemann surface is a compact, connected oriented 2-manifold; the orientation comes from its complex structure. Therefore it is homeomorphic to a -handled torus”—that is, the connect sum of tori—for some integer . This number is its genus.

Here are just a few examples. First, we assert that there is, up to isomorphism, just one Riemann surface of genus : the Riemann sphere, also known as the (complex) projective line . Note that itself has a natural description as a moduli space: it is the space of lines in through the origin. (See Exercise 2.)

How about Riemann surfaces of genus ? One may obtain examples of the form , where is a lattice—that is, is a discrete, finitely generated additive subgroup of of rank . So is homeomorphic to a parallelogram that has its two pairs of opposite sides glued appropriately. Thus is a topological torus, of genus , with its complex structure inherited from that on itself. Moreover, these are all possible examples in genus : a Riemann surface of genus is, after choice of a basepoint, identified isomorphically with its Jacobian variety via the Abel-Jacobi map. So after choosing the basepoint, it is an abelian variety of dimension , and hence of the form .

How about Riemann surfaces of arbitrary genera? At least in principle, one way to access them all is to exhibit them as branched covers of . Indeed, this approach plays a prominent role historically. More specifically, suppose you fix the following data arbitrarily:

a number ,

distinct points , …, on , and

permutations , …, such that

Now pick a basepoint on your distinct from the ’s, together with based loops around the points whose concatenation is topologically trivial. Then the Riemann existence theorem implies that there is a unique Riemann surface with a degree branched cover to , branched at the with monodromy around as specified by the permutations .

The case is already nice to consider. Given an even number, say , of points on , there is a unique branched cover of of degree , branched exactly over these points: here each for each . A Riemann surface obtained in this way is called hyperelliptic, and an Euler characteristic check—more precisely, the Riemann-Hurwitz formula—shows that it has genus .

We have now seen the definition of a Riemann surface. However, an algebraic geometer might offer the following definition instead:

A Riemann surface is a smooth, projective, connected algebraic curve over .

Here, we have to live with an unfortunate terminology clash: algebraic geometers call them curves, since they have dimension over ; but topologically they are surfaces, of dimension over . Sorry about that! I will tend to stick to the terminology of curves as we go further.

The equivalence of the two definitions of Riemann surface offered in this section—algebraic vs. analytic—is not at all obvious. The proof relies on finding enough nonconstant meromorphic functions on a Riemann surface, via the Riemann-Roch theorem.

There is much more to say about Riemann surfaces, but I will forge ahead to a discussion of their moduli spaces.

The Moduli Spaces

Fix numbers with . We now give a rough definition:

denotes the moduli space of genus , -marked Riemann surfaces.

We write . This definition is really a theorem, which says that there exists a complex variety—more precisely, a Deligne-Mumford stack—that can rightly be called a moduli space, as discussed in the introduction. First and foremost is the fact that the complex points of correspond to isomorphism classes of genus , -marked Riemann surfaces. For a detailed history of the construction of , you can see the interesting survey Ji15.

Example ().

Let us begin with some examples, starting with . First, we have asserted that every Riemann surface of genus is isomorphic to . Furthermore, Exercise 2, which appears at the end of the article, implies that for any genus Riemann surface and distinct points , , and on , there is a unique isomorphism taking , , to any fixed ordered triple of distinct points of . In other words, is a single point. And the uniqueness mentioned above implies that really is a point as a stack, with no automorphisms in its stack structure.

By the way, the discussion above hints at why we started with when , and more generally why we required . Namely, this numerical condition ensures that an -marked Riemann surface of genus has finitely many automorphisms. That finiteness condition is necessary for the requirement that Deligne-Mumford stacks admit what I roughly called a “scheme covering space”: an étale, surjective morphism from a scheme.

Example ().

Next: what are the points of ? We are really asking: what are all configurations of four distinct points on the projective line , up to isomorphism? This is a very good example to understand completely.

Let us say the same answer in two ways. First, by Exercise 2, we may assume that . Then may vary freely. Thus is isomorphic to

Here is another way of saying the same thing. There is a classical algebraic invariant of ordered quadruples of points on called the cross ratio, which, by rewriting the points of for short as and , is expressed

The formula above should be interpreted as the appropriate limit if one of is . There are varying conventions for the exact expression; I chose one which has the property that

Moreover, the cross ratio is a coordinate for , as you may show in Exercise 3. The general case of is in a similar vein; see Exercise 4.

Example ().

What about a moduli space of elliptic curves? When do two lattices and produce isomorphic Riemann surfaces of genus ?

It is necessary and sufficient that there is a biholomorphism, i.e., complex-linear map, of to itself taking to . Using such a biholomorphism we may first assume that and for in the complex upper half-plane. Then a computation, which I omit but can be found in standard sources, shows that if and only if and are in the same -orbit. Here, acts on the upper half-plane by Möbius transformations.

So is a quotient of the upper half-plane by . (There is an oft-drawn picture of a fundamental domain of this quotient; see, e.g., Har77, Figure 16.) But algebraically, a coarse moduli space for is given by an affine line, parametrizing the -invariant of the elliptic curve. The precise relationship between these two descriptions is not at all obvious. In particular, it is not elementary, and is quite beautiful, to describe how may be calculated from . But this is beyond the scope of this article.

Example ().

Just one more example. It is a standard fact that every genus Riemann surface is hyperelliptic, a property that we previously discussed. Such curves admit a unique degree morphism to ramified at points, called Weierstrass points. Thus , at least as a variety, is a quotient : the moduli space of six distinct, unlabelled, points on .

Okay—these pleasant explicit descriptions of can’t go on forever. For one thing, for larger , is not even a rational variety; for even larger , it is not even unirational. So, eventually, no nice descriptions (as rational varieties) like what we have seen can possibly exist!

A Brief, Biased Survey of the Cohomology of

The space was already known to Riemann, who coined the term “moduli” in his 1857 paper. The construction of , as a stack over having the appropriate moduli functor, was obtained a century later, thanks to Deligne and Mumford. Yet the topology of moduli spaces of curves remains largely a mystery, despite the fact that is so well-studied, and inhabits several different flavors of geometry, topology, and physics.

In this article, we shall not focus on the constructions of and —one can’t do everything! Very briefly, though, there are a few ways of going about it. One of them is constructing it as a quotient of a Hilbert scheme. Basically, one finds all genus curves as embedded in a projective space by a suitable power of the canonical bundle. All such embedded curves have the same Hilbert polynomial; then is obtained as the quotient of a subvariety of the relevant Hilbert scheme. The quotient is simply by automorphisms of .

Another (nonalgebraic) perspective that we are giving no time to, sadly, is the Teichmüller approach to , namely realizing as the quotient of Teichmüller space by a properly discontinuous action of the mapping class group . From this, though, it follows that the rational cohomology of is the same thing as the rational cohomology of .

A few things we do know: is a connected, indeed irreducible, variety of complex dimension . This number was already known to Riemann. Another thing we know: Harer-Zagier proved that the orbifold Euler characteristic of is

where denotes the th Bernoulli number. They also show that the (ordinary) Euler characteristic of is asymptotically the same.

This tantalizing result suggests yet-to-be-uncovered structure. It also shows that there is, asymptotically, lots of cohomology! Indeed, asymptotically as , it follows from 1 that

grows superexponentially.

On the other hand, the rational cohomology of is largely a mystery. It is entirely known only for .

Geometers have (rightly) devoted lots of attention to the stable rational cohomology of . Harer, in 1985, proved that is in fact independent of for sufficiently large. Subsequently, Mumford conjectured, and Madsen-Weiss eventually proved, that the cohomology ring , regarded as a graded -algebra, is isomorphic in degrees up to to the graded polynomial algebra

Here, denotes the th Miller-Morita-Mumford kappa class. This polynomial algebra had already been shown to be contained in the stable cohomology of by Miller and Morita at the time of Mumford’s conjecture.

Here is a humbling realization. As you may check, the stable cohomology of grows only like . (A little more precisely, we are asserting that the vector space dimension of the degree at most part of , with in degree , is bounded by .)

Therefore, the stable cohomology in fact occupies a vanishingly small proportion of the rational cohomology of . By the end of this article we shall get our hands on a newly discovered source of exponentially many of these unstable cohomology classes.

Here is another humbling realization. Consider again Harer-Zagier’s Euler characteristic of . Notice that when is even, is a negative number with magnitude growing superexponentially in . That is to say: when is even and very large, must have lots of rational cohomology in odd degree.

On the other hand, almost no explicit nonzero groups for odd are known to this day. In fact, as remarked by Harer-Zagier, at the time of their paper, none were known. In 2005, O. Tommasi found one, showing, en route to her calculation of the cohomology of in the category of rational Hodge structures, that Tom05.

By the end of this article we shall get our hands on a few more nonzero, odd-degree rational cohomology groups of . But we are definitely far from the end of the road here.

B. Farb refers to the problem of explicit unstable cohomology classes of as the “dark matter” problem for : we know there is a lot of it, but we don’t know explicitly where. (Cohomology classes in odd degree are all examples of unstable classes: the stable cohomology of , being generated by -classes, is entirely in even degree.) The dark matter problem for is yet another manifestation of the usual difficulty in mathematics in overcoming the gap between existence and construction.

The Deligne-Mumford-Knudsen Compactification

Other than , the spaces are not compact. A beautiful modular compactification of and was obtained by Deligne-Mumford in 1969 and by Knudsen (in the case of marked points), called the compactification by stable curves, which we now discuss.

The insight of Deligne and Mumford was to enlarge the notion of a smooth, proper curve, to allow nodal singularities—the mildest possible singularities—with only finitely many automorphisms. Such curves are called stable. To peek ahead at some pictures, see Figure 3.

Definition 1.

A nodal curve of genus is a proper, connected algebraic curve over with arithmetic genus whose only singularities, if any, are nodes. A node is a complex point with analytic-local equation : two branches meeting transversely.

Definition 2 (Marked points and stability).

A nodal, -marked curve is a nodal curve as above with , …, distinct smooth complex points of . Simply put, you are forbidden from marking a node.

Say is stable if its automorphism group is finite. That is, only finitely many automorphisms of fix the pointwise. (Exercise 5.)

Definition 3.

Fix with . Then denotes the moduli space of -marked stable curves of genus .

Again, this definition is really a remarkable theorem, that a space (or Deligne-Mumford stack) that deserves to be called a moduli space for stable curves really exists.

Figure 3.

Above, the five strata of . Below, the five stable graphs of type , corresponding to the strata depicted above. Marked points are drawn as labelled “half-edges.”

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In fact, the boundary admits a stratification in which the strata are assembled in a combinatorial way from smaller moduli spaces . To describe the stratification in a way that highlights its combinatorial nature, we shall define the marked, vertex-weighted dual graph of a stable curve. If is a stable curve, its dual graph is a triple

as follows:

(1)

is a (multi)graph, with a vertex corresponding to each irreducible component of , and with an edge between and for every node of on .

(2)

The marking function sends when lies on .

(3)

The weight function is given by setting to be the genus of , the normalization of .

Let’s call a triple arising in this way a stable graph. See Figure 3 for an example of the topological types that arise when , together with the corresponding stable graphs. (Exercise 6.) Incidentally, stability amounts to the following purely combinatorial condition on : for every vertex ,

where is the number of half-edges and marked points at .

Now fix a stable graph , and ask: what is a moduli space of stable curves with dual graph ? Our reference here is ACG11, by the way.

Informally speaking, a curve with dual graph may be specified by naming, for each , an -marked, genus curve; in other words, a point of . But this is an overspecification, exactly by the action of the automorphisms of the combinatorial datum in its action on .

Thus the claim is as follows. Let

Then

is a moduli space of stable curves of dual graph . To be precise, the brackets here denote the quotient stack, which is again a Deligne-Mumford stack. See Beh14 for an elementary explanation of quotient stacks.

For example, let be a graph with two vertices and four parallel edges between them, with no markings or weights. Then

A curve with dual graph may be specified by four points on each of two ’s, that is, choosing two cross ratios , and gluing. See Figure 4. Note that has orbifold points, e.g., along the diagonal . Indeed, such a curve is in the closure of the hyperelliptic locus: it admits a nontrivial automorphism exchanging the two ’s.

Figure 4.

A stable graph on the left, and the corresponding stable curve in .

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The previous example demonstrates, by the way, that considering moduli spaces of marked curves is essential even just to describe the boundary strata of moduli spaces of unmarked curves .

There are many nice surveys of from various perspectives, including in previous issues of the Notices Vak03.

Tropical Curves

I want to (seemingly) switch gears and discuss the tropical moduli space of curves, assuming no background in tropical geometry.

But first: what is tropical geometry? Well, it depends on whom you ask. Tropical geometry has connections to many areas of mathematics: nonarchimedean geometry, mirror symmetry, combinatorics, optimization, even economics. But I shall center our discussion primarily on its connection with algebraic geometry, and in particular its historical antecedents in the form of degeneration techniques in algebraic geometry.

What do we mean by degeneration? The basic idea is to get information on the generic behavior of a smooth algebraic curve, say, by studying a one-parameter family of curves, which degenerates in the limit to a singular curve, instead. In general, the singular curve could have many irreducible components, giving rise to a rich combinatorial structure. The idea is then that properties of the smooth fibers may be deduced from properties of the singular fiber.

For example, consider the family of projective plane quartics , parametrized by , defined by the equation

When is nonzero but close to , the plane curve is smooth. When , the curve degenerates to the zero locus of

Thus the curve over is the union of four projective lines, with each pair meeting transversely at a point.

What, then, is tropical geometry? The following is a slogan:

Tropical geometry is a very drastic degeneration technique in algebraic geometry in which the limiting object is entirely combinatorial.

To glimpse this principle, let us follow the example above all the way into the tropical realm. One may associate a tropical curve to the family of curves in 2. It is , the complete graph on four vertices, equipped with edge lengths of . The edge lengths will not be justified here. (But, roughly speaking, the edge lengths measure the “speed of formation,” relative to , of the six nodes in the fiber over .) The reason that the tropicalization of the family 2 is a is that there are four irreducible components in the fiber over , with each pair of them meeting transversely once.

Figure 5.

The special fiber of the family of curves defined in equation 2, consisting of four ’s, on the left; and the associated tropical curve, on the right.

Graphic for Figure 5.  without alt text

Now observe, for example, that the arithmetic genus, , of the smooth curves is still visible in , which is a tropical curve of genus . By the genus of a graph we mean the number

In summary, we have seen a small example of an invariant of a smooth curve that can be detected by its tropicalization. Of course, there are more novel applications of the tropical point of view than rederiving the degree-genus formula for plane curves. For further reading, see the survey BJ16 on degenerations of linear series, and the references therein.

We now give the precise definition of a tropical curve.

A tropical curve is a pair , where

is a stable graph, in the precise sense of the previous section, and

is a function on the edge set of .

A tropical curve is, more or less, a metric graph: think of it as a combinatorial, or nonarchimedean, analogue of a Riemann surface.

Given a one-parameter family of smooth curves over a neighborhood of , there is a precise way to associate a tropical curve. It goes roughly as in the above example. But see, e.g., Cha17 and the references therein for the details, especially regarding edge lengths.

Let me pause to explain something that might be mystifying if you have seen talks on tropical geometry in which tropical curves are drawn very differently, perhaps more like the pictures in the bottom right of Table 1. If you have seen no such talks, then skip to the next section.

Table 1.

Cartoons of abstract/embedded algebraic/tropical curves of genus 3.

Graphic for Table 1.  without alt text

Morally, the two different definitions of tropical curves, on display in the bottom row of Table 1, arise in parallel to the two different ways to think of algebraic curves in classical algebraic geometry, on display in the top row of the figure. Algebraic curves can arise as subvarieties of projective spaces, given as the vanishing locus of homogeneous polynomials. Or, they can be given as Riemann surfaces, equipped with complex structure by specifying an appropriate sheaf of functions. (The original sin of drawing Riemann surfaces as complex 1-dimensional manifolds, but embedded curves as if they were real 1-dimensional manifolds, is on full display in the top row of Table 1.)

It is interesting to study all four squares in Table 1 and their relationships with each other. For example, going from the top left picture to top right “is” Brill-Noether theory: the theory of embedding curves into projective space. And the corresponding theory of tropical linear series, going from the bottom left box to the bottom right, is very interesting too; see BJ16.

Going from the top right to the bottom right box is the theory of “embedded” tropicalization of subvarieties of toric varieties. This is the usual setting of introductions to tropical geometry, e.g., MS15.

But the focus of this article is about “abstract” tropicalization, i.e., getting from the top left box in Table 1 to the bottom left box. I hope this helps explain the bigger picture.

Moduli Spaces of Tropical Curves

The tropical moduli space parametrizes isomorphism classes of -marked tropical curves of genus . Roughly speaking, it is a combinatorial space, glued from polyhedral cones, each cone parametrizing all possible ways to “metrize” a stable graph. To peek ahead at a picture, see Figure 6.

Tropical moduli spaces of curves were constructed in this form by Brannetti-Melo-Viviani, building on work of Caporaso and Mikhalkin, and with antecedents in related constructions of Gathmann-Markwig. Actually, many of the ideas can be traced back even further to the work of Culler-Vogtmann on Outer Space , a space of marked metric graphs of genus on which the outer automorphism group acts. There have been some results on the precise connection between Outer Space and tropical moduli space and in bringing techniques from geometric group theory to play, but more attention is certainly needed.

To define precisely, fix a stable graph . (For example, if , then pick one of the five stable graphs in Figure 3.) What is a parameter space for all isomorphism classes of tropical curves of this combinatorial type?

Our first guess might be : that is, we specify a tropical curve of type by assigning a positive real number, interpreted as a length, to each edge. But this overcounts because of automorphisms of . For example, for as in the top left of Figure 6, the two tropical curves given by edge lengths and are isomorphic for any . Thus a better parameter space would be

where the automorphism group acts by permuting coordinates.

Now we need to glue these spaces together. With foresight, let us allow edge lengths to go to zero, with the understanding that such a point in the moduli space shall be identified with the tropical curve obtained by contracting—shrinking away—edges of length zero. (The weight of the vertex resulting from contracting an edge is the sum of the weights of the endpoints—or if the contracted edge was a loop based at .) Note, then, that contraction defines a partial ordering on the stable graphs of type ; when , the Hasse diagram of the resulting poset is shown in Figure 3.

In other words, define

where the equivalence relation is generated by contracting zero-length edges, as described above. A picture of is shown in Figure 6.

Figure 6.

The tropical moduli space . Compare with Figure 3.

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To end this section, notice that is always contractible. Indeed, it is an instance of a generalized cone complex, i.e., glued from polyhedral cones via face morphisms. And all connected generalized cone complexes, being glued from cones, deformation retract to the cone point.

On the other hand, the link of at its cone point is extremely interesting. Denote this link . The link is a “cross-section” of . Concretely, may be identified with the subspace of parametrizing tropical curves having total edge length .

The space plays a starring role in this article. To adequately explain our interest in , we will stop to discuss mixed Hodge structures on cohomology groups of complex varieties.

The Weight Filtration

One of Deligne’s many significant contributions was the theory of mixed Hodge structures developed in the 1970s. At its most basic level, this is some extra structure on the rational cohomology of any complex variety, not just smooth projective ones, where classical Hodge theory applies. It is something that can really depend on the complex structure of a variety, and not just on the homeomorphism type of its underlying topological space.

First, recall the definition of a pure Hodge structure. A pure Hodge structure of weight is a finitely generated free abelian group together with a decomposition of

such that .

Now suppose is a complex variety. No other requirements on are yet imposed; in particular we don’t require that it be smooth or compact. A reasonable first example to keep in mind is , the algebraic torus, which is smooth but isn’t compact.

Deligne defines a weight filtration on the rational singular cohomology of

in such a way that the weight graded piece, namely

is equipped with a pure Hodge structure of weight . (In fact, this pure Hodge structure is induced by a single Hodge filtration on simultaneously for all the graded pieces.) Thus it becomes interesting to study the weight filtration on the rational cohomology of , as a finer invariant than singular cohomology. For example, when , in which weights is the superexponential growth of Euler characteristic hiding? We really don’t know.

Though there are many things to say regarding mixed Hodge theory, I shall specifically seek to promote the following “combinatorialist’s view” of the (associated graded pieces of the) weight filtration—which was already present in Deligne’s original work—as follows.

Suppose is smooth and is a simple normal crossings compactification of . What this means is:

(1)

is a smooth variety that is complete, i.e., it is compact.

(2)

The irreducible components of the boundary are smooth and intersect transversely.

Transverse means that analytically-locally at any point of , the boundary looks like the transverse intersection of some number of hyperplanes inside an affine space. A good example is the compactification of , whose boundary is hyperplanes meeting transversely.

Say , …, are the irreducible components of the boundary, and let denote the complex dimension of . Then the weight graded piece can be completely understood from the following data:

(1)

The rational cohomology groups

for all ; that is, the rational cohomology of all possible intersections of irreducible components. (I allow the empty intersection and interpret it to be itself.)

(2)

The natural maps between these cohomology groups