On automorphisms of moduli spaces of parabolic vector bundles

Fix $n\geq 5$ general points $p_1, \dots, p_n\in\mathbb{P}^1$, and a weight vector $\mathcal{A} = (a_{1}, \dots, a_{n})$ of real numbers $0 \leq a_{i} \leq 1$. Consider the moduli space $\mathcal{M}_{\mathcal{A}}$ parametrizing rank two parabolic vector bundles with trivial determinant on $\big(\mathbb{P}^1, p_1,\dots , p_n\big)$ which are semistable with respect to $\mathcal{A}$. Under some conditions on the weights, we determine and give a modular interpretation for the automorphism group of the moduli space $\mathcal{M}_{\mathcal{A}}$. It is isomorphic to $\left(\frac{\mathbb{Z}}{2\mathbb{Z}}\right)^{k}$ for some $k\in \{0,\dots, n-1\}$, and is generated by admissible elementary transformations of parabolic vector bundles. The largest of these automorphism groups, with $k=n-1$, occurs for the central weight $\mathcal{A}_{F}= \left(\frac{1}{2},\dots,\frac{1}{2}\right)$. The corresponding moduli space ${\mathcal M}_{\mathcal{A}_F}$ is a Fano variety of dimension $n-3$, which is smooth if $n$ is odd, and has isolated singularities if $n$ is even.Comment: 13 page

Understanding the surface chemistry of ceria nanoparticles using a multi-method approach

Ceria nanoparticles (NPs), due to their widespread applications, have attracted a lot of concern about their toxic effects on both human health and environment. Cerium occurs in two oxidation states, Ce (III) and Ce (IV), and has the unique ability to readily switch between these two states. There is a lot of speculation on the redox behaviour of cerium oxide being related to its toxicity but there are large gaps in knowledge of whether Ce (III) or Ce (IV) is responsible for such toxic behaviours, their toxicological mechanism and safety assessment. The aim of this study is to accurately quantify the ratio of Ce (III) and Ce (IV) in synthesised ceria samples using a multi-method approach thus providing an insight in understanding their surface chemistry and hence biological behaviour. Ceria NPs of different shapes and sizes were produced with different strengths of interaction between core and capping agent/no capping agent and with both steric and charge stabilization. The oxidation state of the samples was determined using STEM-EELS and XPS. Later in the study, we investigated the uptake and internalisation of different shapes and sizes of ceria NPs in lung-derived A549 cell lines (Adenocarcinomic human alveolar basal epithelial cells, A549)

Semi-homogeneous vector bundles on abelian varieties: moduli spaces and their tropicalization

We construct a moduli space of semi-homogeneous vector bundles with a fixed N\'eron-Severi class $H$ on an abelian variety $A$ over an algebraically closed field of characteristic zero. When $A$ has totally degenerate reduction over a non-Archimedean field, we describe our moduli space from the perspective of non-Archimedean uniformization and show that the essential skeleton may be identified with a tropical analogue of this moduli space. For $H=0$ our moduli space may be identified with the moduli space $M_{0,r}(A)$ of semistable vector bundles with vanishing Chern classes on $A$. In this case we construct a surjective analytic morphism from the character variety of the analytic fundamental group of $A$ onto $M_{0,r}(A)$, which naturally tropicalizes. One may view this construction as a non-Archimedean uniformization of $M_{0,r}(A)$.Comment: v2: added further explanation to Section 1, 31 pages, comments very welcome