Examples of the moduli space of X giving facts about a certain X

Question

What is a good example of a fact about the moduli space of some object telling us something useful about a specific one of the objects?

I am currently learning about moduli spaces (in the context of the moduli space of elliptic curves). While moduli spaces do seem to be fascinating objects in themselves, I am after examples in which facts about a moduli space tell us something interesting about the specific objects that they parametrise. For example, does the study of $\mathbb RP^n$ tell us anything we don't already know about some given line through the origin (say, the $x_1$-axis) in $\mathbb R^{n+1}$?

Answer 1

The easiest example I can think of is the natural incidence correspondence between $\mathbb{P}^3$ and the parameter space of cubic surfaces. This can be used to show that every cubic surface contains a line; from this it follows easily that every smooth cubic surface contains exactly $27$ lines.

Another example is the moduli space of stable maps constructed by Kontsevich; this parametrizes certain maps from curves to (to stick with a simple case) $\mathbb{P}^2$. It can be used to answer the following question: given $3d-1$ points in $\mathbb{P}^2$ in general position, compute the number $N_d$ of rational curves of degree $d$ passing through these points. It turns out that the values $N_d$ satisfy a certain recursive relation which allows you to compute all these numbers starting from the obvious $N_1 = 1$ (through $2$ points passes exactly one line). You can find the formula at the entry Kontsevich's formula on Rigorous trivialities; it yields for instance $N_2 = 1$ and $N_3 = 12$.

Yet another example, again more elementary is the following. The Grassmannian $G = \operatorname{Gr}(1, \mathbb{P}^3)$ parametrizes lines in $\mathbb{P}^3$. The computation of the cohomology of $G$ allows you to compute the number of lines which are incident to $4$ fixed lines in general position (it turns out this number is $2$).

Answer 2

Here is an example. The generic Riemann surface of genus $g>2$ has no automorphisms.
The proof uses the following ingredients:

• Dimension of the moduli space of curves is $3g-3$;

• If a automorphism of a non-hyperelliptic Riemann surface $C$ fixes the Weierstrass points, then it is the identity;

• Riemann-Hurwitz formula.

Now, keep in mind that every Riemann surface can be embedded into $\mathbb P^3$, however the theorem says that in a generic case, not even a automorphism of $\mathbb P^3$ (which is linear change of variables) induces a automorphism of the curve! (genus >2).

Answer 3

You have chosen an example where the moduli space (a projective space) is a homogeneous space. So, geometrically, all the objects it parametrises are "the same", and the nature of the moduli space merely confirms that.

Perhaps it should be said first, therefore, that moduli spaces are not always homogeneous spaces. Not all points on the moduli space look the same, and therefore questions arise. This is seen classically for elliptic curves, where the typical automorphism group (preserving the identity) of an elliptic curve is of order 2, but in a few cases it may be of order 4 or order 6. Does this show up in the moduli space? Yes, when you construct it in the classical way from a fundamental domain in the upper half-plane. Moral: if there are "special" points in the moduli space, there is a geometrical reason they are special.

There are actually three levels to look at: the structure of the moduli space qua space (manifold-like, let's say, for complex geometry); for sophisticates using scheme theory the so-called infinitesimal structure in the sheaf given on the space; and the "moduli" themselves, such as the classical j-invariant, namely the parameters used to describe the space. It depends from what direction you are coming, but certainly for arithmetic special values of the moduli read back in an interesting way.

Answer 4

Earlier today I thought of something that may be an example of this. If not, at least it's very elementary and has something of the same spirit. I don't really know what any kind of moduli space is in any technical sense.

Suppose you know the formula for volume of a sphere, but you do not know the formula for surface area. Parameterize the set of all spheres in $ \mathbf{R}^3 $ at the origin by their radius $ r $, forming the space $ X = ( 0, \infty ) $. Assignment of volume $ V(r) $ to the sphere with radius $ r $ defines a differentiable function $ V $ on $ X $. It is clear from geometry that if $ r $ is changed by $ \Delta r $ then its volume should change by approximately $ A ( r ) \cdot \Delta r $, for $ A ( r ) $ the surface area of the sphere with radius $ r $. In other words, $ A ( r ) = V^{\prime} ( r ) = 4 \pi r^2 $.

I guess the point is that even if you are interested in only one specific sphere S, you can compute the number that you want by forming a continuum of spheres and using analysis on that parameter space. Until you have the space, there is no derivative. Does that make any sense?

Answer 5

Here is one example.

Let $M^3$ be a hyperbolic manifold. Consider the moduli space curvature $-1$ metrics on $M^3$ modulo $\operatorname{Diff}_0(M^3)$. This is a point. Conclusion: every diffeomorphism of $M^3\to M^3$ is homotopic to an isometry.

Answer 6

Looks like I'm 12 years late to the answer-party, but here goes...

By parallel transporting around loops (based at p) in the moduli space, you can construct diffeomorphisms of the object X corresponding to p. This is how we get many symplectic Dehn twists in symplectic geometry. If your moduli space is fine, you really get a map from it to the classifying space of the diffeomorphism group of X so you can also use it to study higher homotopy groups of Diff(X) (or Symp(X)).

On a more basic level, if your moduli space is connected then any two objects X and Y it parameterises will be diffeomorphic.

In another direction, there is an action of the mapping class group of a 2-dimensional surface on Teichmüller space (not quite the moduli space, but something like its universal cover). By knowing that Teichmüller space is a disc (and understanding a certain compactification) and looking at fixed points for this action, Thurston was able to prove his classification theorem for surface diffeomorphisms (i.e. self-diffeomorphisms of a single surface).

Answer 7

Another example also in the flavor of enumerative geometry: by considering the (Deligne–Mumford compactification of) the moduli space of elliptic curves, we see that the point at infinity represents the genus one curve with one node, while each other point represents curves with fixed $j$-invariant. The moduli space now looks somewhat like $\mathbb{P}^1$ (except that it is not, for some stacky reasons, but for current purpose, we can pretend that it is) and we can view the nodal curve as having $j = \infty$. Since any two points on $\mathbb{P}^1$ are equivalent as divisors, the nodal curve is “equivalent” to any other curve with fixed $j$-invariant. (Except when $j$ is $0$, $1$, $1728$.)

This property translates well into the Kontsevich moduli space of stable maps of elliptic curves to say, $\mathbb{P}^2$. The locus of maps of nodal curves are equivalent to the locus of maps of any curves with fixed $j$-invariant as divisors, hence they should have the same enumerative invariants, because after all, enumerative invariants are just intersection numbers on the moduli space of maps. Thus, the problem of counting elliptic curves with fixed $j$-invariant is the same as counting nodal curves, and in $\mathbb{P}^2$, nodal curves are the same as rational curves (as rational plane curves necessarily have nodes if the degree is more than 2). This argument was used by Pandharipande to compute the characteristic numbers of plane elliptic curves with fixed $j$-invariant.

Answer 8

There exists a projective $K3$ surface that is not a quartic.

Proof. The moduli space of quartics is irreducible, while the moduli space of projective $K3$s has countably many irreducible components.

Answer 9

$\DeclareMathOperator\Pic{Pic}$Another example: If $X$ is a smooth projective variety over a field $k$, then the relative Picard functor is representable by a smooth group scheme $\Pic_{X/k}$. This is the moduli space of line bundles on $X$, in particular $\Pic_{X/k}(k)$ (Edit: If $X(k)\neq \emptyset$) is the usual Picard group. The geometry of the Picard scheme tells us a lot about the line bundles; for example one can look at the connected component $\Pic_{X/k}^0$ of the identity of $\Pic_{X/k}$, and the points $L\in \Pic_{X/k}^0(k)$ are precisely the line bundles belonging to divisors algebraically equivalent to $0$.