Deformation of theta characteristics

Any non-hyperelliptic curve of genus g admits an embedding into the projective space of dimension g-1 via its space of holomorphic differentials. This is the only embedding that can be consistently defined over every family of non-hyperelliptic curves and is deservedly called the canonical embedding.

The general curve of genus g, canonically embedded in $P^{g-1}$, will admit exactly $2^{g-1}*(2^g - 1)$ hyperplanes tangential to the curve at g-1 points. The configuration of these hyperplanes and their points of tangency admit numerous geometric questions, most of which are essentially impossible to answer directly.

We will demonstrate how such questions can be answered by degenerating a smooth curve to a nodal curve and by following these points of tangency all the way to the nodal curve. Theta characteristics provide us with a convenient language and the means to study infinitesimal perturbations of such degenerations.