The Tour Ahead
The basic objects of algebraic geometry, such as subvarieties of a projective space, are defined by polynomial equations. The seemingly innocuous observation that one can vary the coefficients of these equations leads at once to unexpectedly deep questions:
When are objects with distinct coefficients equivalent?
What types of geometric objects appear if those coefficients move “towards infinity”?
Can we make sense of “equivalence classes at infinity”?
Searching for answers leads to the discovery of moduli spaces and their compactifications, parametrizing equivalence classes of said objects.
The construction of compact moduli spaces and the study of their geometry amounts nowadays to a busy and central neighborhood of algebraic geometry. Any vibrant district in an old city, of course, has too many landmarks to visit, and the first job of a tour guide is to curate a selection of sites and routes — including multiple routes to the same site for the different perspectives they afford. Our tour today has three main stops: elliptic curves, Picard curves (together with “points on a line,” their alter ego), and a brief panoramic glimpse of the general theory.
As for the routes, we first approach the elliptic curve example along the unswerving path that compactifies algebro-geometric moduli spaces with “limiting” algebro-geometric invariants. The way is straight, but entails scaling a brick wall to discover what is meant by “limits.” Our subsequent turn down 19th-century vennels will unveil a connection as old as algebraic geometry itself: associated to our algebraic varieties are analytic periods, producing a period map from our moduli space to a classifying space for periods. While lacking the ideological consistency of the former route, the “limits” of this one are more conceptually straightforward — with Calculus providing a door in the wall.
For the Picard curve example, we reverse the order of these two approaches; this example is important because it is the simplest one where there are multiple natural compactifications of both sorts. Both examples have two very nice features, besides involving objects one can draw on paper. First, the period map is close to being an isomorphism and is inverted by modular forms, an observation going back in the elliptic case to work of Weierstraß. Second, the period map extends to isomorphisms of the various algebraic and analytic compactifications.
These examples will illustrate techniques and methods in moduli theory, preparing the stage for our last stop, high above the city. From there we shall be able to see a vague outline of the modern definition of moduli spaces, as well as various algebro-geometric (GIT, KSBA, K-stable) and Hodge-theoretic (Baily-Borel, toroidal, etc.) compactifications of the same moduli space. The aim of our brief journey is to travel towards understanding their differences — and especially their spectacular coincidences.
1. Elliptic Curves
With a rich history going back to Abel, Jacobi, and Weierstraß in their guise as complex 1-tori (think of the surface of a donut), elliptic curves are central objects in many areas of mathematics, from cryptography to complex analysis. At this first stop on our tour, we’ll use their moduli space to illustrate constructions such as geometric quotients and the period map (and its inversion).
1.1. Algebraic perspective
Our starting point is the fact, first hinted at by Jacobi in 1834, that any complex 1-torus can be realized as a smooth plane cubic — that is, an algebraic curve defined as the zero-locus in of a homogeneous polynomial of degree three in 3 variables
with certain conditions on the coefficients to guarantee the smoothness of Conversely, for an algebraic geometer, it is natural to approach the set of elliptic curves via a suitable quotient of the set of all such cubics. .
Without the smoothness requirement, the coefficients of such equations comprise all ordered 10-tuples of complex numbers not all zero, defined up to scaling (by , Since an equation ). is determined uniquely by its coefficients up to scaling, we conclude that the set of all plane cubics can be identified with the set of complex points of the projective space (Outside the open set parametrizing smooth cubics, there are . different flavors of singular cubics as displayed in Figure 1.)
The fact that the set of all plane cubics is itself a complex algebraic variety is not a coincidence! Instead, it is our first encounter with one of the most important objects in moduli theory: the Hilbert scheme. Indeed, to keep track of the complex solutions of polynomial equations within projective space, we need to fix an invariant known as the Hilbert polynomial. This polynomial records geometric information about our solutions such as their dimension and degree. It was shown in 1961 by Grothendieck that there exists an algebro-geometric space (that is, a projective scheme) which parametrizes all the closed complex solutions of polynomial equations in with Hilbert polynomial In our particular example, all plane cubics have Hilbert polynomial equal to . and the associated Hilbert scheme is , Every point in . corresponds to a curve with this Hilbert polynomial and vice versa.
At this juncture the reader will point out that is certainly not the sought-for “moduli space of elliptic curves,” because it includes singular cubics. But the open subset parametrizing smooth cubics is not the solution either. The reason is that given an elliptic curve defined by the equation we can use an invertible linear change of coordinates , with to obtain another equation , The elliptic curve defined by this second equation is isomorphic as a complex variety to the first one, and yet the Hilbert scheme tells us that they are different objects. .
The critical observation here is that if we are parametrizing varieties with a fixed Hilbert polynomial, then we want to account for the automorphisms of the ambient projective space. In our example the ambient space of elliptic curves is and the group associated to linear automorphisms has dimension 8. Therefore, of the 9 degrees of freedom associated with the coefficients of the equation only one is intrinsic to the geometry of the elliptic curve, while the other eight are related to linear changes of coordinates in the ambient space. ,
The relation between elliptic curves and is tight; two plane cubics -orbits and represent isomorphic elliptic curves (algebraically or complex analytically) if and only if for some Therefore, the set of elliptic curves up to isomorphism can be identified with . of smooth plane cubics. Although the set of such orbits exists as a topological space, it is not at all obvious that this space is itself a complex variety. We arrive at one of the most delicate problems in algebraic geometry: Given the action of a linear group -orbits on a variety does there exist an algebro-geometric space parametrizing the , in -orbits?
The correct framework for constructing quotients within algebraic geometry is given by Geometric Invariant Theory (GIT), initiated by Mumford in 1969 17. One of the key results from GIT is the existence of an open locus called the stable locus and a well-defined geometric quotient, which in our case is
One of the first exercises in GIT is to show that is the locus parametrizing smooth plane cubics, see 17, Example 7.12. The fact that the quotient is geometric means that every point of corresponds to a unique of a smooth cubic. Therefore, we arrive at our first (and almost correct!) example of a moduli space: -orbit is the “moduli space” of elliptic curves up to isomorphism.
We also arrive at the crux of our problem: is a non-compact variety. Is there a natural compact (projective) variety that contains it and that parametrizes a larger class of algebraic varieties? It will be tempting to consider a naive quotient of by for constructing a compactification of However, these naive quotients are usually either of the wrong dimension or yield a non-Hausdorff topological space. Here we use a second key result from GIT: There exists a (usually larger) open set . called the semistable locus, containing , and admitting a well-defined “categorical quotient.” In our case this is
a complex projective variety compactifying .
To understand the geometry of the above compactification, we recall that parametrizes all possible plane cubics. This includes our large open set parametrizing the smooth plane cubics and smaller loci parametrizing degenerations such as nodal cubics, the union of a conic and a line, etc. It is a non trivial fact that our semistable locus parametrizes all curves with at worst a singularity locally of the form these curves are represented at the top left section of Figure ;1.
The categorical quotient is not the naive one: the points of are not in bijection with the of curves parametrized by the semistable locus -orbits Indeed, all of the orbits within the eight-dimensional locus . are identified with a single point in even though they represent non-isomorphic curves. However, there is a unique “minimal closed” associated to the point -orbit namely, the orbit of the “triangle” cubic :.
1.2. Analytic perspective
Turning to a complex-analytic perspective, we recall that an elliptic curve can be viewed as the cartesian product of two circles with a complex structure. By considering these two circles we obtain a homology basis (oriented so that ).
Up to scale, there is a unique holomorphic form with period ratio , in the upper-half plane This . is well-defined modulo the action of through fractional-linear transformations induced by changing the homology basis. We claim that the resulting (analytic) invariant captures the (algebraic) isomorphism class of .
This is closely related to the classical theory of modular forms. Here “modular” refers to the moduli of complex 1-tori or equivalently of the lattice , and the “forms” are essentially functions on this moduli space (i.e., of the lattice ; with certain transformation properties depending on a weight ) The (biperiodic) Weierstraß . associated to the lattice -function,
satisfies where , and are modular forms of weights resp. That is, they transform by the automorphy factor . under pullback by which makes , into a well-defined function. Evidently, the image of the map sending is a Weierstraß cubic
with period ratio .
The key point is that any can be brought into this form (without changing through the action of ) on coordinates. Fix a flex point (i.e., since the dual curve ); has degree there are 3 more tangent lines , passing through The . are collinear, since otherwise one could construct a degree-1 map So we may choose coordinates to have . , , and , which puts us in the above form ,1. In fact, if then rescaling yields a member of the family ,
over whose period map , composed with extends to the identity Hence . yields the claimed equivalence of analytic and algebraic “moduli,” and is an isomorphism.
While the choice of does not refine the moduli problem, keeping track of the ordered 2-torsion subgroup does. In 1, this preserves the ordering of the which are parametrized by the weight-2 modular forms , with respect to The roles of . and 2 are played by and the Legendre family with , describing the 6:1 covering Notice that . parametrizes the cross-ratio of 4 (ordered) points on .
For any we can let act on by and take the quotient to produce the universal elliptic curve with level- structure (marked over the modular curve -torsion) To produce an algebraic realization, we can use Jacobi resp. modular forms . to embed then in a suitable projective space. (Indeed, and already did this for and The “modular compactification” .) of so obtained adds points called cusps, over which the elliptic fiber degenerates to a cycle of in fact, we have ’s; Going around a cusp subjects a basis of integral homology to a transformation conjugate to ..
Now consider an algebraic realization of e.g., for ; the Hesse pencil , over has a marked subgroup as base-locus (where the curves meet the coordinate axes). The monodromy group generated by all loops in (acting on of some fiber) is tautologically So the period ratio . yields a well-defined period map inverted by modular forms (as for and exchanging algebraic and analytic moduli of smooth objects. The other moral here is that refining the moduli problem (e.g., level structure) produces smaller monodromy group, hence more boundary components (in this case, cusps) in the compactification. ),
1.3. First spectacular coincidence
In Section 1.1, we used plane cubics and GIT techniques to construct a compactification of the moduli space of smooth elliptic curves On the other hand, by using periods and modular forms . in Section 1.2 we constructed the moduli space of elliptic curves with a level as well as their compactifications -structure with We recall that a level structure on an elliptic curve is the additional finite information arising from the choice of ..
We arrive then to a natural question: Are any of the Hodge theoretic compactifications isomorphic to The answer turns out to be yes! From our previous discussion about the invariant ? it is possible to conclude that
Moreover, isomorphisms of this kind (that is, between Hodge theoretic and geometric compactifications) also exist among other geometric realizations of the elliptic curves. For instance, by keeping track of the ordered 2-torsion subgroup and the Legendre family, one can show that is isomorphic to a GIT compactification of the space of 4-tuples of points in And with that remark, we turn the corner en route to the next stop on our tour. .
2. Picard Curves and Points in a Line
As we begin to dig into this second example (did we mention that the tour includes amateur archaeological activities?), we shall unearth several ideas that are central for constructing compact moduli spaces. They include the use of finite covers to associate periods to varieties without periods, the use of limits of periods to refine compactifications, and the first example of “stable pairs.”
2.1. Analytic perspective
We begin this time with the complex-analytic point of view. Though ordered collections of points in do not themselves have periods, we can consider covers branched over such collections, generalizing the case of Legendre elliptic curves.
Let denote For . compactifying ,
to yields a genus-3 curve with cubic automorphism and (up to scale) unique holomorphic differential with The moduli spaces .
and parametrize ordered 5-tuples in (namely and resp. unordered 4-tuples in ) (as roots of the polynomial determined by ).
Fix To define period maps, we first describe the monodromy group . through which acts on As it sends symplectic bases to symplectic bases and must be compatible with . it should be plausible that this takes the form ,Footnote1
is a lattice in a unitary group of signature whose elements can be represented by matrices with entries in the Eisenstein integers. ,
and that for this is replaced by the subgroup Now given a basis . the period vector , ( lies in a 2-ball ) (by Riemann’s bilinear relation), on which acts via
with So we get period maps . and whose images omit 1 resp. 6 disk-quotients. Writing , resp. for
and its these maps extend to isomorphisms -quotient, and .
The Picard modular formsFootnote2 describing their inverses are none other than the and in 3. Indeed, the resulting “modular compactifications” of the ball quotients add only (4 resp. 1) points, extending (say) to To understand the meaning of . notice that colliding two , in 3 and normalizing yields a genus 2 curve with cubic automorphism, whose (single) period ratio is parametrized by one of the disk-quotients previously omitted. When 3 collide in 3, the normalization has genus 0 and thus no moduli, which explains the 4 boundary points in The unnormalized scenarios are depicted in the left and middle degenerations in Fig. .2.
Here comprises holomorphic functions on satisfying