The Fine Moduli Space of Representations of Clifford Algebras

Given a fixed binary form $f(u,v)$ of degree $d$ over a field $k$, the associated \emph{Clifford algebra} is the $k$-algebra $C_f=k\{u,v\}/I$, where $I$ is the two-sided ideal generated by elements of the form $(\alpha u+\beta v)^{d}-f(\alpha,\beta)$ with $\alpha$ and $\beta$ arbitrary elements in $k$. All representations of $C_f$ have dimensions that are multiples of $d$, and occur in families. In this article we construct fine moduli spaces $U=U_{f,r}$ for the irreducible $rd$-dimensional representations of $C_f$ for each $r \geq 2$. Our construction starts with the projective curve $C \subset \mathbb{P}^{2}_{k}$ defined by the equation $w^d=f(u,v)$, and produces $U_{f,r}$ as a quasiprojective variety in the moduli space $\mathcal{M}(r,d_r)$ of stable vector bundles over $C$ with rank $r$ and degree $d_r=r(d+g-1)$, where $g$ denotes the genus of $C$.Comment: Final version. To appear in Int. Math. Res. Not. IMR

Ulrich Bundles on Quartic Surfaces with Picard Number 1

In this note, we prove that there exist stable Ulrich bundles of every even rank on a smooth quartic surface $X \subset \mathbb{P}^3$ with Picard number 1.Comment: Final version. To appear in Comptes Rendus Mathematiqu

The Period-Index Problem of the Canonical Gerbe of Symplectic and Orthogonal Bundles

We consider regularly stable parabolic symplectic and orthogonal bundles over an irreducible smooth projective curve over an algebraically closed field of characteristic zero. The morphism from the moduli stack of such bundles to its coarse moduli space is a $\mu_2$-gerbe. We study the period and index of this gerbe, and solve the corresponding period-index problem.Comment: 19 pages. Complete rewrite of the previous version, including expanded results on the moduli of parabolic G-bundles. To appear in the Journal of Algebra. Comments welcom

On Nori's Obstruction to Universal Bundles

Let $G$ be $Sl_n, Sp(2n)$ or SO(2n). We consider the moduli space $M$ of semistable principal $G$-bundles over a curve $X$. Our main result is that if $U$ is a Zariski open subset of $M$ then there is no universal bundle on $U\times X$

On representations of Clifford algebras of ternary cubic forms

In this article, we provide an overview of a one-to-one correspondence between representations of the generalized Clifford algebra $C_f$ of a ternary cubic form $f$ and certain vector bundles (called Ulrich bundles) on a cubic surface $X$. We study general properties of Ulrich bundles, and using a recent classification of Casanellas and Hartshorne, deduce the existence of irreducible representations of $C_f$ of every possible dimension.Comment: 9 pages, to appear in proceedings for the conference "New Trends in Noncommutative Algebra: A Conference in Honor of Ken Goodearl's 65th Birthday

Pfaffian quartic surfaces and representations of Clifford algebras

Given a nondegenerate ternary form $f=f(x_1,x_2,x_3)$ of degree 4 over an algebraically closed field of characteristic zero, we use the geometry of K3 surfaces and van den Bergh's correspondence between representations of the generalized Clifford algebra $C_f$ associated to $f$ and Ulrich bundles on the surface $X_f:=\{w^{4}=f(x_1,x_2,x_3)\} \subseteq \mathbb{P}^3$ to construct a positive-dimensional family of irreducible representations of $C_f.$ The main part of our construction, which is of independent interest, uses recent work of Aprodu-Farkas on Green's Conjecture together with a result of Basili on complete intersection curves in $\mathbb{P}^{3}$ to produce simple Ulrich bundles of rank 2 on a smooth quartic surface $X \subseteq \mathbb{P}^3$ with determinant $\mathcal{O}_X(3).$ This implies that every smooth quartic surface in $\mathbb{P}^3$ is the zerolocus of a linear Pfaffian, strengthening a result of Beauville-Schreyer on general quartic surfaces.Comment: This paper contains a proof of the main result claimed in the erroneous preprint arXiv:1103.0529. We also extend this result to all smooth quartic surface

A Quantitative Analysis of Ownership-Induced Quality Gaps in The Long-Term Care Sector: Influences of Ownership Conversions, Self-Reporting, Regulatory Reforms, and the Covid-19 Pandemic

This dissertation presents a quantitative analysis of the association between ownership types and quality of services in the long-term care sector in the United States. The study employs dynamic difference-in-differences models to investigate the effects of for-profit ownership conversions on nursing home quality indicators by drawing on national-level panel data for the years between 2013 and 2021. Additionally, the adverse effects of information asymmetries are examined by comparing changes in government-inspected quality measures with changes in self-reported quality measures following a for-profit conversion of a nursing home. Furthermore, the impact of the recent regulatory changes implemented at the end of 2016 in the nursing home sector and the facility-level factors associated with the COVID-19 pandemic outcomes in nursing homes are examined with respect to the quality trends and differences in quality by ownership types. Lastly, this study explores the relationship between ownership and quality in assisted living facilities in the State of Georgia using state inspection data. Overall, this dissertation finds that for-profit ownership status is associated with worse quality outcomes among nursing homes and assisted living facilities, including adverse outcomes of the COVID-19 pandemic. Moreover, the analyses show that the recent regulatory reforms had little to no effect on improving the quality of nursing homes over time. The findings are discussed to help policymakers formulate new policies and effective regulations to improve the quality of long-term care

A QUANTITATIVE ANALYSIS OF OWNERSHIP-INDUCED QUALITY GAPS IN THE LONG-TERM CARE SECTOR: INFLUENCES OF OWNERSHIP CONVERSIONS, SELF-REPORTING, REGULATORY REFORMS, AND THE COVID-19 PANDEMIC

This dissertation presents a quantitative analysis of the association between ownership types and quality of services in the long-term care sector in the United States. The study employs dynamic difference-in-differences models to investigate the effects of for-profit ownership conversions on nursing home quality indicators by drawing on national-level panel data for the years between 2013 and 2021. Additionally, the adverse effects of information asymmetries are examined by comparing changes in government-inspected quality measures with changes in self-reported quality measures following a for-profit conversion of a nursing home. Furthermore, the impact of the recent regulatory changes implemented at the end of 2016 in the nursing home sector and the facility-level factors associated with the COVID-19 pandemic outcomes in nursing homes are examined with respect to the quality trends and differences in quality by ownership types. Lastly, this study explores the relationship between ownership and quality in assisted living facilities in the State of Georgia using state inspection data. Overall, this dissertation finds that for-profit ownership status is associated with worse quality outcomes among nursing homes and assisted living facilities, including adverse outcomes of the COVID-19 pandemic. Moreover, the analyses show that the recent regulatory reforms had little to no effect on improving the quality of nursing homes over time. The findings are discussed to help policymakers formulate new policies and effective regulations to improve the quality of long-term care.Ph.D