Vector bundles with a fixed determinant on an irreducible nodal curve

Let $M$ be the moduli space of generalized parabolic bundles (GPBs) of rank $r$ and degree $d$ on a smooth curve $X$. Let $M_{\bar L}$ be the closure of its subset consisting of GPBs with fixed determinant ${\bar L}$. We define a moduli functor for which $M_{\bar L}$ is the coarse moduli scheme. Using the correspondence between GPBs on $X$ and torsion-free sheaves on a nodal curve $Y$ of which $X$ is a desingularization, we show that $M_{\bar L}$ can be regarded as the compactified moduli scheme of vector bundles on $Y$ with fixed determinant. We get a natural scheme structure on the closure of the subset consisting of torsion-free sheaves with a fixed determinant in the moduli space of torsion-free sheaves on $Y$. The relation to Seshadri--Nagaraj conjecture is studied.Comment: 7 page

Picard groups of the moduli spaces of semistable sheaves I

We compute the Picard group of the moduli space $U'$ of semistable vector bundles of rank $n$ and degree $d$ on an irreducible nodal curve $Y$ and show that $U'$ is locally factorial. We determine the canonical line bundles of $U'$ and $U'_L$, the subvariety consisting of vector bundles with a fixed determinant. For rank 2, we compute the Picard group of other strata in the compactification of $U'$.Comment: 16 pages, no figures, no table

Moduli spaces of vector bundles on a singular rational ruled surface

We study moduli spaces $M_X(r,c_1,c_2)$ parametrizing slope semistable vector bundles of rank $r$ and fixed Chern classes $c_1, c_2$ on a ruled surface whose base is a rational nodal curve. We show that under certain conditions, these moduli spaces are irreducible, smooth and rational (when non-empty). We also prove that they are non-empty in some cases. We show that for a rational ruled surface defined over real numbers, the moduli space $M_X(r,c_1,c_2)$ is rational as a variety defined over $\mathbb R$.Comment: Final versio

Grassmannian framed bundles and generalized parabolic structures

We build compact moduli spaces of Grassmannian framed bundles over a Riemann surface, essentially replacing a group by its bi-invariant compactification. We do this both in the algebraic and symplectic settings, and prove a Hitchin-Kobayashi correspondence between the two. The spaces are universal spaces for parabolic bundles, and the reduction to parabolic bundles commutes with the correspondence. An analogous correspondence is outlined for the generalized parabolic bundles of Bhosle.Comment: 45 pages; revised version, with additional explanations and some corrections; final version; final version with typos correcte

The Boden-Hu conjecture holds precisely up to rank eight

Consider moduli schemes of vector bundles over a smooth projective curve endowed with parabolic structures over a marked point. Boden and Hu observed that a slight variation of the weights leads to a desingularisation of the moduli scheme, and they conjectured that one can always obtain a small resolution this way. The present text proves this conjecture in some cases (including all bundles of rank up to eight) and gives counterexamples in all other cases (in particular in every rank beyond eight). The main tool is a generalisation of Ext-groups involving more than two quasiparabolic bundles.Comment: 17 page

Characterization of Acoustic Resonance in a High Pressure Sodium Lamp

With the last decades, the high pressure sodium (HPS) lamp has been supplied in high frequency in order to increase the efficacy of the lamp/ballast system. However, at some given frequencies, standing acoustic waves, namely acoustic resonance (AR), might develop in the burner and cause lamp luminous fluctuation, extinction and destruction in the most serious case. As we seek for a control method to detect and avoid the lamp AR some main characteristics of the acoustic resonances in a 150W HPS lamp are presented in this paper,. The first one is the characteristic of the lamp AR threshold power, the second one is the differences between forward and backward frequency scanning effects during lamp open loop operation. Thirdly, lamp AR behaviour in closed loop operation with an LCC half bridge inverter will be presented and leads to a new point of view and a change in the choice of the AR detection method. These characteristics allow us to further understand the AR and to better control the lamp

NANOTEXTURED TITANIUM SURFACES FOR IMPLANTS: MANUFACTURING AND PACKAGING ASPECTS

It has been shown that nanotexturing the surface of otherwise smooth titanium orthopedic materials increases osteoblast proliferation in vitro, and the bone-implant contact area and pullout force in vivo. However, this prior work has not focused on the requirements for scale-up to industrial processes. This dissertation reports on titanium surface modifications by electrochemical anodization using a benign NH4F electrolyte, and a hybrid electrolyte also containing AgF, rather than hazardous hydrofluoric acid used elsewhere. Nanotube fabrication of Ti6Al4V foils, rods, thermal plasma sprayed commercial implants, and laser and e-beam melted powder materials was demonstrated. It was found that the nanotextured morphology depends on electrolyte composition, and dimensional variation depends on anodization conditions using different NH4F and ethylene glycol electrolytes. The fluorine concentration was found to be the most influential factor affecting formation of porous nanostructures. Recognizing the importance of packaged implant storage, the wetting behavior of nanotube surfaces was investigated. It was found that increased surface hydrophobicity due to aging in air can be restored by annealing, and the release of residual fluorine from the surface was measured. The kinetics of the amorphous to crystalline anatase transformation of nanotubes was quantified with isochronal and isothermal experiments by X-ray diffraction and transmission electron microscopy. The anatase phase transformation of TiO2 nanotubes was achieved in as little as 5 minutes at 350C, in contrast to reports of higher temperature and for hours. The fluorine consumed by the formation of the nanotubes during anodization was analyzed and sources of fluorine consumption were identified. Fluorine from the electrolyte is removed and retained in the nanotubes and by the metal removed to form the nanotubes. A metric describing the fluorine removed from the electrolyte per anodized area was developed to help quality control in manufacturing scale-up. A single-step anodization with controlled nanosilver deposition within and among the nanotubes, using a new hybrid electrolyte of NH4F and AgF was demonstrated. Successful fabrication of potentially antibacterial nanotubes on foils, rods and thermal plasma sprayed surfaces was demonstrated and nanosilver concentration was quantified. These new understandings led to improved manufacturing and storage technologies needed for regulatory approvals of nanotextured titanium surfaces for better orthopedic implants