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In [N], there is a nice and very explicit description of the Torelli space ${\rm Tor}_{1,n}$ of $n$-pointed elliptic curves, for any $n\geq 1$:

$$ {\rm Tor}_{1,n}=\left\{ \big(\tau, (z_1,\ldots,z_n)\big) \in \mathbb H_1 \times \mathbb C^{n} \; \bigg\lvert \; \; z_1=0, z_i-z_j\not \in \mathbb Z\oplus \tau \mathbb Z , \; {\rm for } \, i,j=1,\ldots,n, \, i\neq j \right\} $$ where $\mathbb H_1$ stands for the upper half-plane $\mathbb H_1=\{ z \in \mathbb C\, \lvert {\rm Im}(z)>0\, \}$.

Question (1): is there a similar explicit description of ${\rm Tor}_{2,n}$ for $n\geq 1$?


Technical details, some questions and a possible candidate:
It is known (cf. [H]) that ${\rm Tor}_{2,0}$ is $${\rm Tor}_{2,0}=\mathbb H_2 \setminus \bigcup_{\phi \in {\rm Sp}_2(\mathbb Z)} \phi(\mathbb H_1 \times \mathbb H_1) $$ where $\mathbb H_2$ stands for Siegel upper half space of $2\times 2$ symmetric complex matrices whose imaginary part is positive definite: $ \mathbb H_2=\left\{ Z\in {\rm M}_2(\mathbb C)\, \big\lvert\, {}^tZ=Z\, , \; {\rm Im} (Z)>0\, \right\}$.

For $\tau\in \mathbb H_2$, denote by $J(\tau)=\mathbb C^2/(\mathbb Z^2\oplus \tau \mathbb Z^2)$ the associated 2-torus and let $\theta(\cdot,\tau)$ be the associated Riemann's theta function defined by $ \theta(z,\tau)=\sum_{N\in \mathbb Z^g} \exp\big(2i\pi \big(\frac{1}{2} {}^tN\cdot \tau \cdot N+{}^tN\cdot z\big) \big)%\, , \qquad z\in \mathbb C^2. $ for any $z\in \mathbb C^2$.

For $\tau\in {\rm Tor}_{2,0}$ it follows from Riemann's theorem that there exists a constant $\kappa(\tau)$ such that the equation $$ \theta(z+\kappa(\tau)\lvert \tau)=0 \qquad \qquad (\star) $$ cut out in $J(\tau)$ a Riemann surface $C(\tau)$ whose matrix of periods (for a certain symplectic basis of its homology) is precisely $\tau$.

Preliminary questions:
(2) Can the map $\tau\mapsto \kappa(\tau)$ be made explicit?
(3) For $\tau \in \mathbb H_2$ fixed, what is the topology of the subset of $\mathbb C^2$ cut out by equation $(\star)$?

Let us consider the set ${\rm Tor}'_{2,n}$ formed by the elements $\big(\tau, (z_1,\ldots,z_n)\big) $ of $ \mathbb {\rm Tor}_{2} \times {(\mathbb C^2)}^{n}$ satisfying the two following conditions:

(i) $\theta(z_i+\kappa(\tau) \, \lvert \tau)=0$ for $ i=1,\ldots,n $;

(ii) $z_i-z_j\not \in \mathbb Z^2\oplus \tau \mathbb Z^2 $ for $i,j=1,\ldots,n, \, i\neq j$.

That is: $$ {\rm Tor}'_{2,n}=\left\{ \big(\tau, (z_1,\ldots,z_n)\big) \in {\rm Tor}_{2} \times {(\mathbb C^2)}^{n} \; \bigg\lvert \; \substack{ (i) \, z_i-z_j\not \in \mathbb Z^2\oplus \tau \mathbb Z^2 \\ (ii) \; \;\theta(z_i+\kappa(\tau) \, \lvert \tau)=0 } \; {\rm for } \, i,j=1,\ldots,n, \, i\neq j \right\} $$

The definition of ${\rm Tor}'_{2,n}$ mimics the one of ${\rm Tor}_{1,n}$ given at the beginning. Maybe it is too naive, but ${\rm Tor}'_{2,n}$ appears as a possible candidate for the Torelli space ${\rm Tor}_{2,n}$ (at least to me).

Precised question (4): how are related ${\rm Tor}'_{2,n}$ and ${\rm Tor}_{2,n}$?


References

[H] R. Hain, Finiteness and Torelli spaces.
Problems on mapping class groups and related topics, 57–70,
Proc. Sympos. Pure Math., 74
http://arxiv.org/abs/math/0508541

[NAG] S. Nag, The torelli spaces of punctured tori and spheres.
Duke Math. Journal 48 (1981), p. 359-388
http://projecteuclid.org/download/pdf_1/euclid.dmj/1077314655

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