Singularities of \(2\Theta\)-divisors in the jacobian (Q2773558)

Let \(C\) be a non-hyperelliptic smooth projective curve of genus \(g\geq 3\) over the complex field. Let \(\Theta\) denote the Riemann theta divisor on \(\text{Pic}^{g-1}(C)\) and \(\Theta_0\) the symmetric theta divisor on the Jacobian variety \(JC\) , obtained translating \(\Theta\) by a theta characteristic. \textit{J. Fay} [``Theta functions on Riemann surfaces'', Lect. Notes Math. 352 (1973; Zbl 0281.30013)] observed that, if \(D\) is a divisor in the linear system \(|2\Theta_0|\), then mult\(_0D\geq 4\) if and only if \(D\) contains the surface (difference) \(C-C\). Moreover, if we denote by \(\Gamma_0\) the hyperplane in \(H^0(JC, {\mathcal O}(2\Theta_0))\) of 2\(\theta\)-divisors containing \({\mathcal O}\), and \(\Gamma_{00}\) its subspace of those having multiplicity at least 4 at \({\mathcal O}\), then \(\Gamma_0/\Gamma_{00}\simeq \text{Sym}^2H^0(K)\). NEWLINENEWLINENEWLINEIn the paper under review, the authors study the following subseries of \({\mathbb P}\Gamma_{00}\): \({\mathbb P}\Gamma_{11} =\{D\in{\mathbb P}\Gamma_{00} \mid C_2-C_2\subset D\}\), where \(C_2\) is the second symmetric power of \(C\); \({\mathbb P}\Gamma_{000}\) formed by divisors having multiplicity \(\geq 6\) at \({\mathcal O}\); \({\mathbb P}\Gamma_{00}^{(2)}\) formed by divisors \(D\) that are singular along \(C-C\). In particular they prove the existence of a filtration NEWLINE\[NEWLINE\Gamma_{11} \subset \Gamma_{000} \subset \Gamma_{00}^{(2)} \subset \Gamma_{00} \subset \Gamma_0NEWLINE\]NEWLINE whose quotients are as follows: \(\Gamma_{00}/\Gamma_{00}^{(2)}\simeq \wedge^3H^0(K)\); if \(C\) is non-trigonal, then \(\Gamma_{00}^{(2)}/\Gamma_{11} \simeq \text{Sym}^2I(2)\), where \(I(2)\) is the space of the quadrics in the ideal of the canonical image of \(C\); \(\Gamma_{000}/\Gamma_{11} \simeq \operatorname {Ker}m\), where \(m\) is the multiplication map \(m: \text{Sym}^2I(2)\to I(4)\).