On Modular Properties of Odd Theta-characteristics

A general canonical curve X ⊂ P determines a finite set θ(X) of hyperplanes that are tangent to X at g − 1 points. Such a set is in bijective correspondence with the set of odd theta-characteristics of X. We generalize the definition of θ(X) to certain singular curves (including Deligne-Mumford stable curves) in a way that is compatible with degenerations. We give the explicit description of θ(X), with its enumerative and local data. This is applied to show that some singular curves can be recovered from θ(X). This paper generalizes the preliminary steps used in [CS] to show that, if g = 3, θ(X) uniquely determines X.