Projective Variety Research Articles

Defect relation for holomorphic maps from complex discs into projective varieties and hypersurfaces

Defect relation for holomorphic maps from complex discs into projective varieties and hypersurfaces

Kawamata–Miyaoka type inequality for ℚ-Fano varieties with canonical singularities

Abstract Let 𝑋 be an 𝑛-dimensional normal ℚ-factorial projective variety with canonical singularities and Picard number one such that 𝑋 is smooth in codimension two, − K X -K_{X} is ample and n ≥ 2 n\geq 2 . We prove that 𝑋 satisfies the following Kawamata–Miyaoka type inequality: c 1 ⁢ ( X ) n < 4 ⁢ c 2 ⁢ ( X ) ⋅ c 1 ⁢ ( X ) n − 2 . c_{1}(X)^{n}<4c_{2}(X)\cdot c_{1}(X)^{n-2}. If additionally 𝑋 is a threefold with terminal singularities, then a stronger inequality is also obtained.

The history of hydrological studies on the Mekong floodplains – from colonial experiments to computational models

ABSTRACT This paper investigates the interwoven history of hydrological studies and water infrastructure development on the Mekong floodplains. On the basis of an extensive literature review, archival work, and key informant interviews, we unravel the making of an expert-led understanding of the Mekong floodplains – from the detailed reports written by colonial engineers during the French protectorate in the late 19th century to the development of the first computational models, developed in the 1960s on punch-cards and in Fortran. We show how these studies not only reflect the objectives of successive powers and their relationships to local communities but also shaped the landscape and hydrology of the Mekong floodplains and continue to do so through a variety of water infrastructure development projects. This serves as a call to conceive of hydrological knowledge as contingent rather than as a neutral depiction of physical processes that exists independently from the conditions of its production.

A remark on the relative Lie algebroid connections and their moduli spaces

We investigate the relative lie algebroid connections on a holomorphic vector bundle over a family of compact complex manifolds (or smooth projective varieties over [Formula: see text]). We provide a sufficient condition for the existence of a relative Lie algebroid connection on a holomorphic vector bundle over a complex analytic family of compact complex manifolds. We show that the relative Lie algebroid Chern classes of a holomorphic vector bundle admitting relative Lie algebroid connection vanish, if each of the fibers of the complex analytic family is compact and Kähler. Moreover, we consider the moduli space of relative Lie algebroid connections and we show that there exists a natural relative compactification of this moduli space.

Endomorphisms of varieties and Bott vanishing

We show that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. This is a new way to analyze which varieties have nontrivial endomorphisms. In particular, we extend some classification results on varieties admitting endomorphisms (for Fano threefolds of Picard number one and several other cases) to any characteristic. The classification results in characteristic zero are due to Amerik–Rovinsky–Van de Ven, Hwang–Mok, Paranjape–Srinivas, Beauville, and Shao–Zhong. Our method also bounds the degree of morphisms into a given variety. Finally, we relate endomorphisms to global F F -regularity.

On the moment measure conjecture

AbstractThe moment measure conjecture of Białynicki-Birula and Sommese gives a combinatorial characterization of all open substacks of a global quotient stack for a torus action on a normal projective variety that admit a proper good moduli space, in other words it characterizes the invariant open subvarieties that admit a proper quotient. In this article we prove the conjecture for actions on smooth varieties. This gives an instance of stability conditions defined by cohomological invariants that are not given by Chern classes of line bundles.

On the spectral variety for rank two Higgs bundles

AbstractIn this article, we study the Hitchin morphism over a smooth projective variety. The Hitchin morphism is a map from the moduli space of Higgs bundles to the Hitchin base, which in general not surjective when the dimension of the variety is greater than one. Chen–Ngô introduced the spectral base, which is a closed subvariety of the Hitchin base. They conjectured that the Hitchin morphism is surjective to the spectral base and also proved that the surjectivity is equivalent to the existence of finite Cohen–Macaulayfications of the spectral varieties. For rank two Higgs bundles over a projective manifold, we explicitly construct a finite Cohen–Macaulayfication of the spectral variety as a double branched covering, thereby confirming Chen–Ngô's conjecture. Moreover, using this Cohen–Macaulayfication, we can construct the Hitchin section for rank two Higgs bundles, which allows us to study the rigidity problem of the character variety and also to explore a generalization of the Milnor–Wood‐type inequality.

The performance of ChatGPT-4.0o in medical imaging evaluation: a cross-sectional study

This study investigated the performance of ChatGPT-4.0o in evaluating the quality of positioning in radiographic images. Thirty radiographs depicting a variety of knee, elbow, ankle, hand, pelvis, and shoulder projections were produced using anthropomorphic phantoms and uploaded to ChatGPT-4.0o. The model was prompted to provide a solution to identify any positioning errors with justification and offer improvements. A panel of radiographers assessed the solutions for radiographic quality based on established positioning criteria, with a grading scale of 1–5. In only 20% of projections, ChatGPT-4.0o correctly recognized all errors with justifications and offered correct suggestions for improvement. The most commonly occurring score was 3 (9 cases, 30%), wherein the model recognized at least 1 specific error and provided a correct improvement. The mean score was 2.9. Overall, low accuracy was demonstrated, with most projections receiving only partially correct solutions. The findings reinforce the importance of robust radiography education and clinical experience.

Self-dual projective toric varieties and their ideals

Self-dual projective toric varieties and their ideals

Vanishing theorems for Mather–Jacobian multiplier ideals on a Gorenstein projective variety

Abstract In this paper, we establish several results related to vanishing theorems for Mather–Jacobian multiplier ideals on a Gorenstein projective variety, including an injectivity theorem, a Nadel-type vanishing theorem, a Griffiths-type vanishing theorem for vector bundles, and a vanishing theorem for the asymptotic Mather–Jacobian multiplier ideals.

Convex Fujita numbers are not determined by the fundamental group

Abstract We study effective global generation of adjoint line bundles on smooth projective varieties. To measure the effectivity we introduce the concept of the convex Fujita number of a smooth projective variety and compute its value for a class of varieties with prescribed dimension d ≥ 2 and an arbitrary projective group as fundamental group.

Biochemical Society support in action: experiences from our community

Throughout the year, the Biochemical Society offers a programme of grants for all career stages supporting research, attendance at scientific conferences and the sponsorship of events. So far this year, we have awarded over £200,000 towards conference participation, summer vacation studentships, international collaboration and the organization of independent seminars and postgraduate days. In this article, we are delighted to share a few insights from our community that highlight the variety of projects and opportunities made possible by our funding. For more information about our funding opportunities and the application process, please visit our website.

Biochemical Society support in action: experiences from our community

Throughout the year, the Biochemical Society offers a programme of grants for all career stages supporting research, attendance at scientific conferences, and the sponsorship of events. So far this year, we have awarded over £312,000 towards conference participation, international collaboration, diversity projects and PhD scholarships. In this article, we are delighted to share a few insights from our community that highlight the variety of projects and opportunities made possible by our funding. For more information about our funding opportunities and the application process, please visit our website.

Symplectic Grassmannians and cyclic quivers

AbstractThe goal of this paper is to extend the quiver Grassmannian description of certain degenerations of Grassmann varieties to the symplectic case. We introduce a symplectic version of quiver Grassmannians studied in our previous papers and prove a number of results on these projective algebraic varieties. First, we construct a cellular decomposition of the symplectic quiver Grassmannians in question and develop combinatorics needed to compute Euler characteristics and Poincaré polynomials. Second, we show that the number of irreducible components of our varieties coincides with the Euler characteristic of the classical symplectic Grassmannians. Third, we describe the automorphism groups of the underlying symplectic quiver representations and show that the cells are the orbits of this group. Lastly, we provide an embedding into the affine flag varieties for the affine symplectic group.

StarMalloc: Verifying a Modern, Hardened Memory Allocator

We present StarMalloc, a verified, efficient, security-oriented, and concurrent memory allocator. Using the Steel separation logic framework, we show how to specify and verify a multitude of low-level patterns and delicate security mechanisms, by relying on a combination of dependent types, SMT, and modular abstractions to enable efficient verification. We produce a verified artifact, in C, that implements the entire API surface of an allocator, and as such works as a drop-in replacement for real-world projects, notably the Firefox browser. As part of StarMalloc, we develop several generic datastructures and proof libraries directly reusable in future systems verification projects. We also extend the Steel toolchain to express several low-level idioms that were previously missing. Finally, we show that StarMalloc exhibits competitive performance by evaluating it against 10 state-of-the-art memory allocators, and against a variety of real-world projects, such as Redis, the Lean compiler, and the Z3 SMT solver.

Curves on Brill–Noether special K3 surfaces

AbstractMukai showed that projective models of Brill–Noether general polarized K3 surfaces of genus 6–10 and 12 are obtained as linear sections of projective homogeneous varieties, and that their hyperplane sections are Brill–Noether general curves. In general, the question, raised by Knutsen, and attributed to Mukai, of whether the Brill–Noether generality of any polarized K3 surface is equivalent to the Brill–Noether generality of smooth curves in the linear system , is still open. Using Lazarsfeld–Mukai bundle techniques, we answer this question in the affirmative for polarized K3 surfaces of genus , which provides a new and unified proof even in the genera where Mukai models exist.

Additive actions on projective hypersurfaces with a finite number of orbits

An induced additive action on a projective variety [Formula: see text] is a regular action of the group [Formula: see text] on [Formula: see text] with an open orbit, which can be extended to a regular action on the ambient projective space [Formula: see text]. In this work, we classify all projective hypersurfaces admitting an induced additive action with a finite number of orbits.

A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2

A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2

Gromov-Witten Invariants in Complex and Morava-Local K-Theories

Given a closed symplectic manifold X, we construct Gromov-Witten-type invariants valued both in (complex) K-theory and in any complex-oriented cohomology theory \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{K}$\\end{document} which is Kp(n)-local for some Morava K-theory Kp(n). We show that these invariants satisfy a version of the Kontsevich-Manin axioms, extending Givental and Lee’s work for the quantum K-theory of complex projective algebraic varieties. In particular, we prove a Gromov-Witten type splitting axiom, and hence define quantum K-theory and quantum \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{K}$\\end{document}-theory as commutative deformations of the corresponding (generalised) cohomology rings of X; the definition of the quantum product involves the formal group of the underlying cohomology theory. The key geometric input of these results is a construction of global Kuranishi charts for moduli spaces of stable maps of arbitrary genus to X. On the algebraic side, in order to establish a common framework covering both ordinary K-theory and Kp(n)-local theories, we introduce a formalism of ‘counting theories’ for enumerative invariants on a category of global Kuranishi charts.

Instagram Story Effects Usage Scale (ISEUS): A scale for user tendencies in social media

In today's technology, especially on Instagram, story effects have been used frequently by women. These effects make the face perfect by performing aesthetic procedures and make-up on the face. As a result of this perfection, the use of Instagram story effects can negatively affect women's perceptions of beauty. Additionally, the usage of beauty affects might make women have poor body image by diminishing their self-esteem and confidence and raising their social anxiety. This may lead to a rise in young girls seeking plastic surgery, particularly during adolescence. A scale related to the use of Instagram story effects and its effect on the perception of beauty has not previously been developed in order to measure this situation. Including this scale in the literature will open the door for further research that will support and raise awareness of the issue. As a result, an 8-item Instagram Story Effects Usage Scale (ISEUS) was developed in this study using data from 147 Instagram story effects users. In order to reveal the construct validity of ISEUS, exploratory and confirmatory factor analysis was applied. The results show that the developed scale's psychometric properties are satisfactory. ISEUS is intended to be used in a variety of research projects that bring together different disciplines.