Singularities of the theta divisor and families of secant spaces to a canonical curve
Let \(C\) be a non-hyperelliptic curve of genus \(g \geq 5\) over an algebraically closed field \(k\). Let \(S^{g - 1} C\) be the \((g - 1)\)- symmetric product and \(\alpha_{g - 1} : S^{g - 1} C \to \text{Pic}^{g - 1} (C)\) the Abel map, \(\Theta = \text{Im} (\alpha_{g - 1})\) the theta divisor and \(\text{Sing} (\Theta)\) its singular locus. Let \(\Sigma\) be the open subset of \(\text{Sing} (\Theta)\) defined by the double points \(L \in \text{Sing} (\Theta)\) whose projectivized tangent cone, \(Q_L\), has rank 4 and \(S = \alpha_{g - 1}^{- 1} (\Sigma)\). The authors consider the natural family parametrized by \(S\) \[ \begin{matrix} \bigwedge & \subset S \times \mathbb{P} \\ \pi \downarrow & \;\\ S & \;\end{matrix} \] of \((g - 3)\)-spaces of \(\mathbb{P}^{g - 1}\) \((g - 1)\)-secant to the canonical model of \(C\), \(K(C) \subset \mathbb{P}^{g - 1}\). They show that this family can be reconstructed only in terms of the polarization \(\Theta\) and study the geometry of this family: they define the focal scheme \(F_s \subset \bigwedge_s\) of a point \(s \in S\) and give a criterion for \(F_s\) to be a rational normal curve. As a corollary of the results proved in the paper the authors prove in theorem 4.4 what they claim to be a refinement of Torelli's theorem. In fact this could be a refinement of \textit{A. Andreotti}'s version of this theorem [cf. Am. J. Math. 80, 801-828 (1958; Zbl 0084.173)] but not of Torelli's theorem as was stated by \textit{T. Masusaka} [Am. J. Math. 80, 784-800 (1958; Zbl 0100.356)]. The problem of reconstructing the immersion of the curve into its Jacobian is not considered by the authors; for this question and its connection with Andreotti's version of Torelli's theorem see the paper by the reviewer [Compos. Math. 61, 369- 381 (1987; Zbl 0624.14020)].
scientific article; zbMATH DE number 735329
scientific article