49,291 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
Comment on "Orientational Distribution of Free O-H Groups of Interfacial Water is Exponential"
In a recent letter (PRL,121,246101,2018), Sun et al. reported that combined
MD simulation and sum frequency generation vibrational spectroscopy (SFG-VS)
measurements led to conclusions of a broad and exponentially decaying
orientational distribution, and the presence of the free O-H group pointing
down to the bulk at the air/water interface. In this comment, we show that
their main conclusions are based on questionable interpretation of the SFG-VS
data presented in the letter [1], and are also contrary to the established data
analysis and interpretations in the literature [2-5].Comment: 2 pages, 0 figure
Discrete diffraction managed solitons: Threshold phenomena and rapid decay for general nonlinearities
We prove a threshold phenomenon for the existence/non-existence of energy
minimizing solitary solutions of the diffraction management equation for
strictly positive and zero average diffraction. Our methods allow for a large
class of nonlinearities, they are, for example, allowed to change sign, and the
weakest possible condition, it only has to be locally integrable, on the local
diffraction profile. The solutions are found as minimizers of a nonlinear and
nonlocal variational problem which is translation invariant. There exists a
critical threshold ?cr such that minimizers for this variational problem exist
if their power is bigger than ?cr and no minimizers exist with power less than
the critical threshold. We also give simple criteria for the finiteness and
strict positivity of the critical threshold. Our proof of existence of
minimizers is rather direct and avoids the use of Lions' concentration
compactness argument.
Furthermore, we give precise quantitative lower bounds on the exponential
decay rate of the diffraction management solitons, which confirm the physical
heuristic prediction for the asymptotic decay rate. Moreover, for ground state
solutions, these bounds give a quantitative lower bound for the divergence of
the exponential decay rate in the limit of vanishing average diffraction. For
zero average diffraction, we prove quantitative bounds which show that the
solitons decay much faster than exponentially. Our results considerably extend
and strengthen the results of [15] and [16].Comment: 49 pages, no figure
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