Motivic and $$\ell $$-adic realizations of the category of singularities of the zero locus of a global section of a vector bundle

We study the motivic and $$\ell $$ ℓ -adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D. Orlov and J. Burke–M. Walker to give a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n.

The strength for line bundles

We introduce the strength for sections of a line bundle on an algebraic variety. This generalizes the strength of homogeneous polynomials that has been recently introduced to resolve Stillman's conjecture, an important problem in commutative algebra. We establish the first properties of this notion and give some tool to obtain upper bounds on the strength in this framework. Moreover, we show some results on the usual strength such as the reducibility of the set of strength two homogeneous polynomials.

Impact of Central Line Bundle Care on Reduction of Central Line Associated-Infections: A Scoping Review

Purpose/Background Central venous catheters (CVC) are typical for critically ill patients in the intensive care unit (ICU). Due to the invasiveness of this procedure, there is a high risk for central line-associated bloodstream infection (CLABSI). These infections have been known to increase mortality and morbidity, medical costs, and reduce hospital reimbursements. Evidenced-based interventions were grouped to assemble a central line bundle to decrease the number of CLABSIs and improve patient outcomes. This scoping review will evaluate the literature and examine the association between reduced CLABSI rates and central line bundle care implementation or current use. Methods A literature review was completed of nine critically appraised articles from the years 2010-2021. The association of the use of central line bundles and CLABSI rates was examined. These relationships were investigated to determine if the adherence to a central line bundle directly reduced the number of CLABSI rates in critically ill adult patients. A summary evaluation table was composed to determine the associations related to the implementation or current central line bundle care use. Results Of the study sample (N=9), all but one demonstrated a significant decrease in CLABSI rates when a central line bundle was in place. A trend towards reducing CLABSI was noted in the remaining article, a randomized controlled study, but the results were not significantly different. In all the other studies, a meta-analysis, randomized controlled trial, control trial, cohort or case-control studies, and quality improvement project, there was a significant improvement in CLABSI rates when utilizing a central line bundle. The extensive use of different levels of evidence provided an excellent synopsis that implementing a central line bundle care would directly affect decreasing CLABSI rates. Implications for Nursing Practice Results provided in this scoping review afforded the authors a diverse level of evidence that using a central line bundle has a direct outcome on reducing CLABSI rates. This practice can be implemented within the hospital setting as suggested by the literature review to prevent or reduce CLABSI rates. Implementing a standard central line bundle care hospital-wide helps avoid this hospital-acquired infection.

aCM vector bundles on projective surfaces of nonnegative Kodaira dimension

In this paper, we contribute to the construction of families of arithmetically Cohen–Macaulay (aCM) indecomposable vector bundles on a wide range of polarized surfaces [Formula: see text] for [Formula: see text] an ample line bundle. In many cases, we show that for every positive integer [Formula: see text] there exists a family of indecomposable aCM vector bundles of rank [Formula: see text], depending roughly on [Formula: see text] parameters, and in particular they are of wild representation type. We also introduce a general setting to study the complexity of a polarized variety [Formula: see text] with respect to its category of aCM vector bundles. In many cases we construct indecomposable vector bundles on [Formula: see text] which are aCM for all ample line bundles on [Formula: see text].

Non-Archimedean volumes of metrized nef line bundles

Let $L$ be a line bundle on a proper, geometrically reduced scheme $X$ over a non-trivially valued non-Archimedean field $K$. Roughly speaking, the non-Archimedean volume of a continuous metric on the Berkovich analytification of $L$ measures the asymptotic growth of the space of small sections of tensor powers of $L$. For a continuous semipositive metric on $L$ in the sense of Zhang, we show first that the non-Archimedean volume agrees with the energy. The existence of such a semipositive metric yields that $L$ is nef. A second result is that the non-Archimedean volume is differentiable at any semipositive continuous metric. These results are known when $L$ is ample, and the purpose of this paper is to generalize them to the nef case. The method is based on a detailed study of the content and the volume of a finitely presented torsion module over the (possibly non-noetherian) valuation ring of $K$.

Fiberwise linear differential operators

We define a new notion of fiberwise linear differential operator on the total space of a vector bundle E. Our main result is that fiberwise linear differential operators on E are equivalent to (polynomial) derivations of an appropriate line bundle over E ∗ {E^{\ast}} . We believe this might represent a first step towards a definition of multiplicative (resp. infinitesimally multiplicative) differential operators on a Lie groupoid (resp. a Lie algebroid). We also discuss the linearization of a differential operator around a submanifold.