On modular properties of odd theta-characteristics (Q2752191)

A general canonical curve \(X\subset{\mathbb P}^{g-1}\) determines a finite set \(\theta(X)\) of hyperplanes that are tangent to \(X\) at \(g-1\) points. The definition of this set can be extended to a class of singular non-hyperelliptic curves, which include stable curves in the sense of Deligne-Mumford. The author conjectures that the set \(\theta(X)\) of theta-hyperplanes determines uniquely the curve \(X\). In a previous paper [\textit{L. Caporaso} and \textit{E. Sernesi} (preprint Alg-Geom AG/0008239)], it is proved that this is true for curves of arithmetic genus 3. In the present paper, the author studies the problem for curves of any genus. She proves the validity of the conjecture for a split curve, that is, a general nodal canonical curve which is the union of two rational normal curves meeting transversally at \(g+1\) points. She also proves the conjecture for general singular irreducible curves of arithmetic genus 4, under an additional weak restriction. The key for the proof of these results is the determination of the configuration of the set of theta-hyperplanes.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00024].