Projective Space Research Articles

Stable sheaves on bielliptic surfaces: from the classical to the modern

Bielliptic surfaces are the last family of Kodaira dimension zero algebraic surfaces without a classification result for the Chern characters of stable sheaves. We rectify this and prove such a classification using a combination of classical techniques, on the one hand, and derived category and Bridgeland stability techniques, on the other. Along the way, we prove the existence of projective coarse moduli spaces of objects in the derived category of a bielliptic surface that Bridgeland semistable with respect to a generic stability condition. By systematically studying the connection between Bridgeland wall-crossing and birational geometry, we show that for any two generic stability conditions τ,σ, the two moduli spaces Mτ(v) and Mσ(v) of objects of Chern character v that are semistable with respect to τ (resp. σ) are birational. As a consequence, we show that for primitive v, the moduli space of stable sheaves of class v is birational to a moduli space of stable sheaves whose Chern character has one of finitely many easily understood “shapes".

Line Spreads That Produce Projective Planes

We explicitly classify those line spreads of projective 5-space over a field that have the property that the given spread induces a spread in the 3-space generated by any pair of spread lines. We determine their fix groups and conclude that there exist such spreads with a trivial fix group. Also, we characterise regular line spreads among all line spreads of projective 3-space by their projectivity group and also by a weakening of the regularity condition.

Mechanisms of alignment of lamellar-forming block copolymer under shear flow.

The potential applications of block copolymer thin films, utilising their self-assembly capabilities, are enhanced when achieving long-range ordering. In this study we explain the experimental alignment of lamellae under shear flow findings [S. Pujari et al. Soft Matter, 2012, 8, 5258] and classify the alignment mechanisms based on shear rate and segregation, uncovering similarities to the systems subjected to electric fields, suggesting a common pathway of lamellae orientations. However, the presence of thin films surfaces introduces distinct features in the lamellae orientation under shear compared to electric fields. Notably, we observe the emergence of a three-dimensional rotation alongside the conventional two-dimensional rotation. Furthermore, a transient regime has been identified within the melting mechanism, which confirms the existence of the checkboard pattern proposed by Schneider et al. [Macromolecules, 2018, 51, 4642]. These findings significantly enhance our understanding of block copolymer alignments and shed light on the intricate interplay between external fields and the lamellar structure.

A quantitative version of Northcott's theorem on points of bounded height: The function field case

AbstractLet be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let denote an algebraic closure of . For given integers and we count points in projective space with absolute logarithmic height and generating an extension of degree over . Specifically, we derive an asymptotic estimate for the number of such points as when and orders of growth for the number of such points when .

КОМФОРТНАЯ ГОРОДСКАЯ СРЕДА В КОНТЕКСТЕ РАЗВИТИЯ РЕКРЕАЦИОННОЙ СИСТЕМЫ РЕГИОНОВ

The article selects and analyzes scientific literary sources on the improvement of public spaces. The analysis of the world experience of landscaping and modern specifics of urban development of public spaces from the point of view of the relationship of green areas with urban development is carried out. The main planning problems of public spaces and the problems of their reconstruction are investigated. The best techniques and features of public space projects have been identified.

Movable cones of complete intersections of multidegree one on products of projective spaces

We study Calabi–Yau manifolds which are complete intersections of hypersurfaces of multidegree 1 in an m-fold product of n-dimensional projective spaces. Using the theory of Coxeter groups, we show that the birational automorphism group of such a Calabi–Yau manifold X is infinite and a free product of copies of Z. Moreover, we give an explicit description of the boundary of the movable cone Mov¯(X). In the end, we consider examples for the general and non-general case and picture the movable cone and the fundamental domain for the action of Bir(X).

On (i)-Curves in Blowups of Pr

In this paper, we study (i)-curves with i∈{−1,0,1} in the blown-up projective space Pr in general points. The notion of (−1)-curves was analyzed in the early days of mirror symmetry by Kontsevich, with the motivation of counting curves on a Calabi–Yau threefold. In dimension two, Nagata studied planar (−1)-curves in order to construct a counterexample to Hilbert’s 14th problem. We introduce the notion of classes of (0)- and (1)-curves in Pr with s points blown up, and we prove that their number is finite if and only if the space is a Mori Dream Space. We further introduce a bilinear form on a space of curves and a unique symmetric Weyl-invariant class, F (which we will refer to as the anticanonical curve class). For Mori Dream Spaces, we prove that (−1)-curves can be defined arithmetically by the linear and quadratic invariants determined by the bilinear form. Moreover, (0)- and (1)-Weyl lines give the extremal rays for the cone of movable curves in Pr with r+3 points blown up. As an application, we use the technique of movable curves to reprove that if F2≤0 then Y is not a Mori Dream Space, and we propose to apply this technique to other spaces.

Second Main Theorems for Holomorphic Curves in the Projective Space with Slowly Moving Hypersurfaces

Second Main Theorems for Holomorphic Curves in the Projective Space with Slowly Moving Hypersurfaces

The real-oriented cohomology of infinite stunted projective spaces

The real-oriented cohomology of infinite stunted projective spaces

Quadratic vector fields in class I

In [Ye et al., Theory of Limit Cycles, 1986], quadratic systems are classified into three different normal forms (I, II and III) with increasing number of parameters. The simplest family is I and even several subfamilies of it have been studied, and some global attempts have been done, up to this paper, the full study was still undone. In this article, we make an interdisciplinary global study of Class I. Since the family has four parameters, we have studied it using the same technique that has already been used in several papers with similar systems which is based on the algebraic invariants of the Sibirskii's school. The bifurcation diagram for this class, done in the adequate parameter space which is the 3-dimensional real projective space, is quite rich in its complexity and yields 261 subsets with 49 different phase portraits for Class I (2 of them corresponding to linear systems), 7 of which have limit cycles. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by an analytic set of curves corresponding to phase portraits which have separatrix connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections.

A new convergence theorem for mean curvature flow of hypersurfaces in quaternionic projective spaces

A new convergence theorem for mean curvature flow of hypersurfaces in quaternionic projective spaces

Rational Weighted Projective Hypersurfaces

Abstract A very general hypersurface of dimension $n$ and degree $d$ in complex projective space is rational if $d \leq 2$, but is expected to be irrational for all $n, d \geq 3$. Hypersurfaces in weighted projective space with degree small relative to the weights are likewise rational. In this paper, we introduce rationality constructions for weighted hypersurfaces of higher degree that provide many new rational examples over any field. We answer in the affirmative a question of Okada about the existence of very general terminal Fano rational weighted hypersurfaces in all dimensions $n \geq 6$.

A Compact, Coherent Representation of Stellar Surface Variation in the Spectral Domain

Time-varying inhomogeneities on stellar surfaces constitute one of the largest sources of radial velocity (RV) error for planet detection and characterization. We show that stellar variations, because they manifest on coherent, rotating surfaces, give rise to changes that are complex but usably compact and coherent in the spectral domain. Methods for disentangling stellar signals in RV measurements benefit from modeling the full domain of spectral pixels. We simulate spectra of spotted stars using starry and construct a simple spectrum projection space that is sensitive to the orientation and size of stellar surface features. Regressing measured RVs in this projection space reduces RV scatter by 60%–80% while preserving planet shifts. We note that stellar surface variability signals do not manifest in spectral changes that are purely orthogonal to a Doppler shift or exclusively asymmetric in line profiles; enforcing orthogonality or focusing exclusively on asymmetric features will not make use of all the information present in the spectra. We conclude with a discussion of existing and possible implementations on real data based on the presented compact, coherent framework for stellar signal mitigation.

On the stability of the quaternion projective space

On the stability of the quaternion projective space

3D-PSSIM: Projective Structural Similarity for 3D Mesh Quality Assessment Robust to Topological Irregularities.

Despite acceleration in the use of 3D meshes, it is difficult to find effective mesh quality assessment algorithms that can produce predictions highly correlated with human subjective opinions. Defining mesh quality features is challenging due to the irregular topology of meshes, which are defined on vertices and triangles. To address this, we propose a novel 3D projective structural similarity index ( 3D- PSSIM) for meshes that is robust to differences in mesh topology. We address topological differences between meshes by introducing multi-view and multi-layer projections that can densely represent the mesh textures and geometrical shapes irrespective of mesh topology. It also addresses occlusion problems that occur during projection. We propose visual sensitivity weights that capture the perceptual sensitivity to the degree of mesh surface curvature. 3D- PSSIM computes perceptual quality predictions by aggregating quality-aware features that are computed in multiple projective spaces onto the mesh domain, rather than on 2D spaces. This allows 3D- PSSIM to determine which parts of a mesh surface are distorted by geometric or color impairments. Experimental results show that 3D- PSSIM can predict mesh quality with high correlation against human subjective judgments, across the presence of noise, even when there are large topological differences, outperforming existing mesh quality assessment models.

Algebraic hyperbolicity of very general hypersurfaces in products of projective spaces

Algebraic hyperbolicity of very general hypersurfaces in products of projective spaces

Co-Creation in Research: Further Reflections From the 'Co-Creating Safe Spaces' Project.

Applied research using co-creation methods is rarely described or evaluated in detail. Practical evidence of co-creation processes and collaboration effectiveness is needed to better understand its complex and dynamic nature. Using a case study design and survey method, we assessed processes of co-implementation and co-evaluation grounded in our own experiences from the Co-Creating Safe Spaces project. We examine these in the context of a published systematic framework designed to improve clarity about co-creation processes and report on how co-creation was experienced by collaborative partners. Our study showed the interconnectedness between co-implementation and co-evaluation processes and the importance of aligning research with program processes to ensure it is responsive to emergent local needs and problems. Given relatively low levels of researcher embeddedness across sites, service champions played a pivotal role in data collection. Survey findings indicated strong support for a healthy collaboration with some concerns expressed over individual partner's areas of responsibility and ability to deliver on commitments. Co-creation can be a very robust approach to translational research but is a complex endeavour. Ongoing reflexivity and attention to relational aspects support genuine collaboration and provide a foundation for addressing challenges. People with lived experience of emotional distress and/or suicidal crisis, including researchers from both academic and non-research backgrounds, service managers, peer workers, carers and advocates, were involved in this research and authored this paper.

Voxel Map to Occupancy Map Conversion Using Free Space Projection for Efficient Map Representation for Aerial and Ground Robots

Voxel Map to Occupancy Map Conversion Using Free Space Projection for Efficient Map Representation for Aerial and Ground Robots

Channel Estimation for RIS-Assisted Multiuser mmWave Systems With Direct Channels Based on a Novel Space Projection Approach

Channel Estimation for RIS-Assisted Multiuser mmWave Systems With Direct Channels Based on a Novel Space Projection Approach

Action of ℝ-Fuchsian groups on ℙℂn

Let [Formula: see text] be the group of orientation preserving isometries of the [Formula: see text]-dimensional real hyperbolic space [Formula: see text], and let [Formula: see text] be the group of holomorphic isometries of the [Formula: see text]-dimensional complex hyperbolic space [Formula: see text]. The group [Formula: see text] is naturally a subgroup of [Formula: see text], so it acts on [Formula: see text] preserving a ball that serves as a model for [Formula: see text]. We consider discrete subgroups of [Formula: see text] with limit set an [Formula: see text]-dimensional real sphere and we look at their action on [Formula: see text] for [Formula: see text] via the natural embedding of [Formula: see text] into [Formula: see text]. We describe the Kulkarni limit set of these set of these actions and show that this is a real semi-algebraic set in [Formula: see text]. We also show that the Kulkarni region of discontinuity can only have either one or three connected components. We use Sylvester’s law of inertia when [Formula: see text]. In the other cases, we use some suitable projections of the [Formula: see text]-dimensional complex projective space to the [Formula: see text]-dimensional complex projective space.