50,862 research outputs found
Canonical systems and their limits on stable curves
We propose an object called 'sepcanonical system' on a stable curve
which is to serve as limiting object- distinct from other such limits
introduced previously- for the canonical system, as a smooth curve degenerates
to . First for curves which cannot be separated by 2 or fewer nodes, the
so-called '2-inseparable' curves, the sepcanonical system is just the sections
of the dualizing sheaf, which is not very ample iff is a limit of smooth
hyperelliptic curves (such are called 2-inseparable hyperelliptics).
For general, 2-separable curves this assertion is false, leading us to
introduce the sepcanonical system, which is a collection of linear systems on
the '2-inseparable parts' of , each associated to a different twisted
limit of the canonical system, where the entire collection varies smoothly with
. To define sepcanonical system, we must endow the curve with extra
structure called an 'azimuthal structure'. We show that the sepcanonical system
is 'essentially very ample' unless the curve is a tree-like arrangement of
2-inseparable hyperelliptics. In a subsequent paper, 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.
The current version includes additional references to, among others, Catanese,
Maino, Esteves and Caporaso.Comment: arXiv admin note: substantial text overlap with arXiv:1011.0406; to
appear in J. Algebr
Incidence stratifications on Hilbert schemes of smooth surfaces, and an application to Poisson structures
Given a smooth curve on a smooth surface, the Hilbert scheme of the surface
is stratified according to the length of the intersection with the curve. The
strata are highly singular. We show that this stratification admits a natural
log-resolution, namely the stratified blowup. As a consequence, the induced
Poisson structure on the Hilbert scheme of a Poisson surface has unobstructed
deformations.Comment: To appear in Int. J. Mat
- β¦