Let \(C\) be a curve of genus \(g\) over the complex field. It is known that the maps \(\psi_n : \text{Sym}^n H^0(K_C) \to H^0(K_C^n)\) where \(n \geq 2\) and \(K_C\) is the canonical line bundle, are surjective when \(g=2\) or \(C\) is non hyperelliptic. The kernels of these maps hence describe equations for the canonical embedding of \(C\).
Let \(N_n = \dim_{\mathbb{C}} \text{Sym}^n H^0(K_C)\) and \(M_n= \dim_{\mathbb{C}} H^0(K_C^n)\) and \(K_n=M_n-N_n\). In the first part of the article, for each \(n\), the authors exhibit holomorphic global sections \(s_{i_1,\ldots,i_{K_n}}\), for \(i_1,\ldots,i_{K_n} \in \{1,\ldots,M_n\}\), of certain vector bundles over the moduli space of curves of genus \(g\). If \(\omega_{i_1}, \ldots, \omega_{i_{N_n}}\) are the images under \(\psi_n\) of a subset of the canonical basis of \(\text{Sym}^n H^0(K_C)\), the section \(s_{i_{N_n+1},\ldots,i_{M_n}}\) vanishes when \(\omega_{i_1}, \ldots, \omega_{i_{N_n}}\) is not a basis (Proposition 2.5). Applications to super-string theory are developed in a recent paper of the first author [“Extending the Belavin-Knizhnik ‘wonderful formula’ by the characterization of the Jacobian”, arxiv:1208.5994].
The second part of the article explores an application of this proposition when \(g=4\) and \(n=2\). In this case, a non hyperelliptic curve canonically embedded in \(\mathbb{P}^3\) is supported by a unique quadric \(Q\), whose equation is given by \[ Q : \sum_{i,j=1}^{4} \frac{1+\delta_{ij}}{2} s_{ij}(\tau) \omega_i \omega_j=0 \] where \(\delta_{ij}\) is the Kronecker symbol and the subscript \(i,j\) of \(s\) must be understood as an element \((ij)\) of \(\{1,\ldots,10\}\) such that \(\omega_{(ij)}\) is the image under \(\psi_2\) of \(\omega_i \omega_j\). On the other hand, a similar equation can be derived from the Shottky-Igusa form \(F_4\), expressing the discriminant \(\Delta_4\) of the quadric \(Q\) in two different ways (Lemma 3.2). Moreover, since the quadric has rank \(3\) if and only if an even Thetanullwert is zero, the authors deduce that \(\Delta_4\) is up to a constant equal to the square root of the Siegel modular form \(\chi_{68}\) defined as the product of the even Thetanullwerte (Theorem.3.4). Finally, they also find a functional relation between the singular component of the theta divisor and the Riemann period matrix (Theorem 3.5).
In a finale remark, the authors compare their result with a formula found by F. Klein [Math. Ann. XXXVI, 1–83 (1890 ; JFM 22.0498.01)] and mentioned in [Math. Res. Lett. 17, No. 2, 323–333 (2010; Zbl 1228.14028)]. This formula expresses \(\chi_{68}\) up to a constant as the product of (a different normalization) of the square of \(\Delta_4\) with the eighth power of the tact invariant [G. Salmon, Paris. Gauthier-Villars et Fils. [J. de Math. spéc. (3) V. 280.] (1891; JFM 23.0716.02)]. Although the present work enlightens Klein’s formula, it seems to the reviewer that it does not explain the tact invariant factor by itself.