Canonical systems and their limits on stable curves

We propose an object called 'sepcanonical system' on a stable curve X0 which is to serve as limiting object - distinct from other such limits introduced previously - for the canonical system, as a smooth curve degenerates to X0. First, for curves which cannot be separated by 2 or fewer nodes (the so-called '2-inseparable' curves), the sepcanonical system consists of the sections of the dualizing sheaf, and fails to be very ample iff X0 is a limit of smooth hyperelliptic curves (such X0 are called 2-inseparable hyperelliptics). For general, 2-separable curves X0, this assertion is false, leading us to introduce the sepcanonical system, which is a collection of linear systems on the '2-inseparable parts' of X0, each associated to a different twisted limit of the canonical system, where the entire collection varies smoothly with X0. To define sepcanonical system, we must endow the curve with extra structure called an 'azimuthal structure'. We show (Theorem 6.5) that the sepcanonical system is 'essentially very ample' unless the curve is a tree-like arrangement of 2-inseparable hyperelliptics. In a subsequent paper [11] we will show that the latter property is equivalent to the curve being a limit of smooth hyperelliptics, and will essentially give defining equation for the closure of the locus of smooth hyperelliptic curves in the moduli space of stable curves.