These are my notes, and are only a rough approximation of the actual talk:

1. General Torelli and Schottky Problems

Take a class of smooth projective varieties with the same Hodge numbers. We can immediately ask two questions about the map from the set of these objects to the period domain:

  1. Torelli: Is the map injective? If not, how does injectivity fail?
  2. Schottky: What is the image? That is, which Hodge structures can be achieved?

We’ll look at a specific case, the one that’s usually the simplest: smooth curves of genus {g}.

In this case, the map associates to each curve its Jacobian: {\mathcal{J}:\mathcal{M}_g\rightarrow\mathcal{A}_g}. It is well known, in this case, that {\mathcal{J}} is injective. However, determining the closure of the image, call it {\mathcal{J}_g}, is much more difficult.

2. Moduli and Maps

We start by defining some moduli spaces (we’ll actually need some compactifications, but we won’t worry about those details):

  1. {\mathcal{RM}_g=\{(C,\mu)|C\in\mathcal{M}_g,0\neq\mu\in J(C)[2]\}=} {\{(C,\tilde{C})|\tilde{C}\rightarrow C}\mbox{ etale double cover } {\}}
  2. {\mathcal{RA}_g=\{(A,\mu)|A\in\mathcal{A}_g,0\neq\mu\in A[2]\}}

Also, we get maps {\alpha:\mathcal{A}_g\rightarrow \mathbb{P}(U_g)} and {\beta:\mathcal{RA}_g\rightarrow \mathbb{P}(U_{g-1})} by using the classical second order theta functions, which are related to symmetric divisors in the principal polarization, and {U_g} is a vector space of dimension {2^g}. (It is actually a specific one built out of a representation of a Heisenberg group)

Finally, we define {\mathcal{P}:\mathcal{RM}_g\rightarrow \mathcal{A}_{g-1}} to be the Prym map, which takes a double cover {\tilde{C}\rightarrow C} of curves to {\ker^0\mathrm{Nm}}, which can be shown to be a ppav.

3. Schottky-Jung

In 1888, Schottky wrote down a modular form for genus 4, which he claimed vanished precisely on {\mathcal{J}_4}. Igusa announced a proof in 1968 and published it in 1981.

In 1909, in a joint paper with Jung, the Schottky-Jung relation was proved

\displaystyle \begin{array}{ccc}\mathcal{RM}_g & \stackrel{\mathcal{J}}{\to} & \mathcal{RA}_g \\ \mathcal{P} \downarrow  & & \downarrow \beta \\ \mathcal{A}_{g-1} & \stackrel{\to}{\alpha} & \mathbb{P}(U_g)\end{array}

In that paper, Schottky and Jung conjectured that if we set {\mathcal{RS}_g=\beta^{-1}(\mathrm{im} \alpha)}, and {\mathcal{S}_g} the image of {\mathcal{RS}_g} in {\mathcal{A}_g}, then {\mathcal{S}_g=\mathcal{J}_g}. This would allow us to actually write down equations for {\mathcal{J}_g} in the natural coordinates on {\mathcal{A}_g}.

4. Non-Jacobians and big Schottky

It turns out that the SJ conjecture is false. There is a trick called the tetragonal construction which shows that the intermediate Jacobians of cubic threefolds are in {\mathcal{S}_5}. This was shown by Donagi in 1987, and in a second paper that year, he offered a solution:

Let {S_g^{\mathrm{big}}} be what was earlier called {\mathcal{S}_g}, and we define { \mathcal{S}_g^{\mathrm{small}}} to be the image of the intersection of the translates of {\mathcal{RS}_g} under the points of order 2, that is, {\mathcal{S}_g^{\mathrm{small}}} is the locus of abelian varieties whose fiber from {\mathcal{RS}_g} is everything. The new conjecture then becomes {\mathcal{S}_g^{\mathrm{small}}=\mathcal{J}_g}, and this is still believed to be true.

5. Genus 0-3

Genus {g\leq 3} is fairly uninteresting, as {\mathcal{J}} turns out to be dominant, as is shown by a simple dimension count for {\mathcal{M}_g} and {\mathcal{A}_g}.

6. Genus 4

Our approach is different from Igusa’s. His was hands on, ours is based on the fibers of {\mathcal{P}} and {\beta}. First, we set {\mathcal{C}\subset\mathcal{A}_5} to be the locus of intermediate Jacobians of cubic threefolds, and this breaks into {\mathcal{RC}^0} and {\mathcal{RC}^1} in {\mathcal{RA}_5}, the even and odd parts (odd and even refers to the dimension of a certain cohomology group). It turns out that there is a map {\kappa:\mathcal{A}_4\rightarrow \mathcal{RC}^0} which is birational.

Now, if {A\in\mathcal{A}_4}, the fiber over {A} of {{\mathcal{P}} is {\widetilde{F(\kappa(A))}}, a double cover of the Fano surface of lines in {\kappa(A)}. This turns out to let us prove that {\beta^{-1}(J(C))} is, away from the boundary (which we ignore in this case), just {K(J(C))} for {C\in\mathcal{M}_3}, and so is irreducible. As {\mathrm{im}\alpha} is as well, we know that {\beta^{-1}(\mathrm{im}\alpha)} is.

7. Genus 5

Genus 5 is current work inprogress. First we look at {\mathcal{P}:\mathcal{RM}_6\rightarrow\mathcal{A}_5}. It turns out that {\deg\mathcal{P}=27}. As it happens, the tetragonal construction furnishes an incidence relation on the fibers, and it is equivalent to that of lines on a cubic surface.

Using this, we can show that {\beta} is also finite, and of degree 119.

Claim (Almost a theorem): {\mathcal{RS}_5=\mathcal{RJ}_5\cup (\mathcal{A}_4\times\mathcal{RA}_1)\cup\mathcal{RC}^0\cup \partial^I\mathcal{RA}_5} in the toroidal partial compactification. (Here, {\partial^I} denotes the component of the boundary where the vanishing cycle is {\mod 2} orthogonal to the point of order 2)

Partial Proof: We can compute the degree of {\beta} on each component. {\deg\beta|_{\mathcal{RJ}_5}=54=2*27}, {\deg\beta|_{\mathcal{RC}^0}=1}, {\deg \beta|_{\mathcal{A}_4\times\mathcal{RA}_1}=0} and {\deg\beta|_{\partial^I\mathcal{RA}_5}=64}, so all that remains is a computation to show that nothing else appears which contributes 0.

This claim then implies {\mathcal{S}_5^{\mathrm{small}}=\mathcal{J}_5}.

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