2
$\begingroup$

Let $X$ be a compact, genus $g$ Riemann surface (given as the desingularization and compactification of a plane algebraic curve), $J(X)$ its Jacobian, and $A : X \to J(X)$ the Abel map

$$A(P) = \left( \int_{P_0}^P \omega_1, \ldots, \int_{P_0}^P \omega_g \right),$$

where $P_0 \in X$ is some fixed place and $\{\omega_1, \ldots, \omega_g\}$ are the normalized basis of holomorphic differentials on $X$.

My goal is to solve the following equation for $K$, the Riemann constant vector:

Theorem: For any canonical divisor $\mathcal{C} \in \text{Div}(X)$, $$2K \equiv - A(\mathcal{C})$$

In general, there are $2^{2g}$ possible solutions to this equation: one for each half period $h_j = \alpha_j + \Omega \beta_j$ where $[I \; \Omega]$ is the period matrix of $X$ (i.e. $J(X) = \mathbb{C}^g / \Lambda$ where $\Lambda = \mathbb{Z}^g + \Omega \mathbb{Z}^g$) and the vectors $\alpha_j,\beta_j \in \{0, 1/2\}^g$. Given a canonical divisor, my current approach is to construct $2^{2g}$ vectors

$$K_j := h_j - \tfrac{1}{2}A(\mathcal{C})$$

and test them against the following theorem:

Theorem: Let $\theta : J(X) \times \mathfrak{h}_g \to \mathbb{C}$ be the Riemann theta function. Then $$\theta(W, \Omega) = 0$$ if and only if $$W = A(\mathcal{D}) + K$$ for some degree $g-1$ effective divisor $\mathcal{D}$.

(That is, evaluate $\theta(A(\mathcal{D}) + K_j)$ for all $j$.) However, in my computational experiments I've found that the choice of correct $h_j$ appears to be unique! That is, only one of the $2^{2g}$ half periods leads to the correct Riemann constant vector.

Question: What sort of techniques / theorems are out there that would help in proving this uniqueness statement?

Follow-Up: Are there specific techniques that would lend themselves toward an algorithm for computing $K$ that does not scale exponentially with the genus of the Riemann surface? (Such as in this approach?)

I've done some reading on holomorphic line bundles and spin bundles and I think I may have found some necessary, but not sufficient conditions for the uniqueness of a solution to this problem. For example, I conjecture that the half period $h_j$ must be odd.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.