Projective Variety Research Articles

Mutations of noncommutative crepant resolutions in geometric invariant theory

Let X be a generic quasi-symmetric representation of a connected reductive group G. The GIT quotient stack X=[Xss(ℓ)/G]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {X}=[X^\ ext {ss}(\\ell )/G]$$\\end{document} with respect to a generic ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} is a (stacky) crepant resolution of the affine quotient X/G, and it is derived equivalent to a noncommutative crepant resolution (=NCCR) of X/G. Halpern-Leistner and Sam showed that the derived category Db(cohX)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ extrm{D}}^{\ extrm{b}}}({\ ext {coh}}\\mathfrak {X})$$\\end{document} is equivalent to certain subcategories of Db(coh[X/G])\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ extrm{D}}^{\ extrm{b}}}({\ ext {coh}}[X/G])$$\\end{document}, which are called magic windows. This paper studies equivalences between magic windows that correspond to wall-crossings in a hyperplane arrangement in terms of NCCRs. We show that those equivalences coincide with derived equivalences between NCCRs induced by tilting modules, and that those tilting modules are obtained by certain operations of modules, which is called exchanges of modules. When G is a torus, it turns out that the exchanges are nothing but iterated Iyama–Wemyss mutations. Although we mainly discuss resolutions of affine varieties, our theorems also yield a result for projective Calabi-Yau varieties. Using techniques from the theory of noncommutative matrix factorizations, we show that Iyama–Wemyss mutations induce a group action of the fundamental group π1(P1\\{0,1,∞})\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi _1(\\mathbb {P}^1\\,\\backslash \\{0,1,\\infty \\})$$\\end{document} on the derived category of a Calabi-Yau complete intersection in a weighted projective space.

Mutations of numerically exceptional collections on surfaces

A conjecture of Bondal–Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We show that the braid group acts transitively on the set of maximal numerically exceptional collections on rational surfaces up to isometries of the Picard lattice and twists with line bundles. Considering the blow-up of the projective plane in up to 9 points in very general position, these results lift to the derived category. More precisely, we prove that, under these assumptions, a maximal numerically exceptional collection consisting of line bundles is a full exceptional collection and any two of them are related by a sequence of mutations and shifts. The former extends a result of Elagin–Lunts and the latter a result of Kuleshov–Orlov, both concerning del Pezzo surfaces. In contrast, we show in concomitant work (Krah in Invent Math 235(3):1009–1018, 2024) that the blow-up of the projective plane in 10 points in general position admits a non-full exceptional collection of maximal length consisting of line bundles.

On anti-ample vector bundles and nef and big vector bundles

We prove that the direct image of an anti-ample vector bundle is anti-ample under any finite flat morphism of non-singular projective varieties. In the second part we prove some properties of big and nef vector bundles. In particular it is shown that the tensor product of a nef vector bundle with a nef and big vector bundle is again nef and big. This generalizes a result of Schneider.

Some remarks on the Kp,1$\mathcal {K}_{p,1}$ theorem

AbstractLet be a non‐degenerate projective irreducible variety of dimension , degree , and codimension over an algebraically closed field of characteristic 0. Let be the th graded Betti number of . Green proved the celebrating ‐theorem about the vanishing of for high values for and potential examples of nonvanishing graded Betti numbers. Later, Nagel–Pitteloud and Brodmann–Schenzel classified varieties with nonvanishing . It is clear that when there is an ‐dimensional variety of minimal degree containing , however, this is not always the case as seen in the example of the triple Veronese surface in .In this paper, we completely classify varieties with nonvanishing such that does not lie on an ‐dimensional variety of minimal degree. They are exactly cones over smooth del Pezzo varieties, whose Picard number is .

Labour-Intensive Filmmaking: An interview with Su Friedrich

In this interview, filmmaker Su Friedrich comments on a wide variety of projects, as well as her labour-intensive work process, the crucial role of feedback from friends and fellow filmmakers, and the importance of being specific. She also describes her experience researching unusual archives, her views on the current media landscape, and the satisfaction of creating Edited By, a website dedicated to honouring the many women who revolutionised film editing, playing a key role in cinema history that has yet to be fully recognised.

Difference analogues of the second main theorem for holomorphic curves and arbitrary families of hypersurfaces in projective varieties

Difference analogues of the second main theorem for holomorphic curves and arbitrary families of hypersurfaces in projective varieties

Inference of neutron-star properties with unified crust-core equations of state for parameter estimation

Context. Relating different global neutron-star (NS) properties, such as tidal deformability and radius, or mass and radius, requires an equation of state (EoS). Determining the NS EoS is therefore not only the science goal of a variety of observational projects, but it also enters in the analysis process; for example, to predict a NS radius from a measured tidal deformability via gravitational waves (GW) during the inspiral of a binary NS merger. To this aim, it is important to estimate the theoretical uncertainties on the EoS, one of which is the possible bias coming from an inconsistent treatment of the low-density region; that is, the use of a so called non-unified NS crust. Aims. We propose a numerical tool allowing the user to consistently match a nuclear-physics informed crust to an arbitrary high-density EoS describing the core of the star. Methods. We introduce an inversion procedure of the EoS close to saturation density that allows users to extract nuclear-matter parameters and extend the EoS to lower densities in a consistent way. For the treatment of inhomogeneous matter in the crust, a standard approach based on the compressible liquid-drop (CLD) model approach was used in our work. A Bayesian analysis using a parametric agnostic EoS representation in the high-density region is also presented in order to quantify the uncertainties induced by an inconsistent treatment of the crust. Results. We show that the use of a fixed, realistic-but-inconsistent model for the crust causes small but avoidable errors in the estimation of global NS properties and leads to an underestimation of the uncertainties in the inference of NS properties. Conclusions. Our results highlight the importance of employing a consistent EoS in inference schemes. The numerical tool that we developed to reconstruct such a thermodynamically consistent EoS, CUTER, has been tested and validated for use by the astrophysical community.

EcoLiDAR: An economical LiDAR scanner for ecological research.

Despite recent popularization and widespread use in modern electronic devices, LiDAR technology remains expensive for research purposes, in part due to the very high performance offered by commercially available LiDAR scanners. However, such high performance is not always needed, and the expensive price ends up making LiDAR scanners inaccessible for research projects with reduced budget, such as those in developing countries. Here we designed and built a simple ground-based LiDAR scanner, with performance sufficient to fulfil the requirements for a variety of ecological research projects, while being cheap and easy to build. We managed to assemble a LiDAR scanner under 400 USD (as of 2021), and it is simple enough to be built by personnel with minimal engineering background. We also demonstrated the quality of the resulting point clouds by scanning a test site and producing some common LiDAR products. Although not adequate for mapping large area due to its limited range, our LiDAR design is open, customizable, and can produce adequate results while costing ~1% of "low-cost" scanners available in the market. As such, our LiDAR scanner opens a world of new opportunities, particularly for projects in developing countries.

Modified defect relation for Gauss maps of minimal surfaces with hypersurfaces of projective varieties in the subgeneral position

AbstractIn this paper, we establish some modified defect relations for the Gauss map of a complete minimal surface into a ‐dimension projective subvariety with hypersurfaces of in ‐subgeneral position with respect to . In particular, we give the upper bound for the number if the image intersects each hypersurface a finite number of times and is nondegenerate over , where , that is, the image of is not contained in any hypersurface of degree with . Our results extend and generalize the previous results for the case of the Gauss map and hyperplanes in a projective space. The results and the method of this paper have been applied by some authors to study the unicity problem of the Gauss maps sharing families of hypersurfaces.

Sasakian geometry on sphere bundles II: Constant scalar curvature

In a previous paper [18] the authors employed the fiber join construction of Yamazaki [38] together with the admissible construction of Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman [2] to construct new extremal Sasaki metrics on odd dimensional sphere bundles over smooth projective algebraic varieties. In the present paper we continue this study by applying a recent existence theorem [14] that shows that under certain conditions one can always obtain a constant scalar curvature Sasaki metric in the Sasaki cone. Moreover, we explicitly describe this construction for certain sphere bundles of dimension 5 and 7.

Dimension Growth for Affine Varieties

Abstract We prove uniform upper bounds on the number of integral points of bounded height on affine varieties. If $X$ is an irreducible affine variety of degree $d\geq 4$ in ${\mathbb{A}}^{n}$, which is not the preimage of a curve under a linear map ${\mathbb{A}}^{n}\to{\mathbb{A}}^{n-\dim X+1}$, then we prove that $X$ has at most $O_{d,n,\varepsilon }(B^{\dim X - 1 + \varepsilon })$ integral points up to height $B$. This is a strong analogue of dimension growth for projective varieties, and improves upon a theorem due to Pila, and a theorem due to Browning–Heath-Brown–Salberger. Our techniques follow the $p$-adic determinant method, in the spirit of Heath-Brown, but with improvements due to Salberger, Walsh, and Castryck–Cluckers–Dittmann–Nguyen. The main difficulty is to count integral points on lines on an affine surface in ${\mathbb{A}}^{3}$, for which we develop point-counting results for curves in ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. We also formulate and prove analogous results over global fields, following work by Paredes–Sasyk.

Impact of Project Management Practices on Cost and Time Overruns in Construction Projects in Nigeria

A major contributor to economic growth in Nigeria, the construction sector is frequently beset by large cost and schedule overruns that jeopardize project profitability and impede the development of infrastructure. This study looks into how project management techniques affect these overruns in building construction projects in Nigeria. The goal of the study is to pinpoint the major project management variables that lead to overspending and delays while also offering recommendations for best practices to address these problems. A mixed-methods strategy was used, integrating qualitative case studies with quantitative data analysis. Project engineers, contractors, Architects, and Technicians were given standardized questionnaires to complete about a variety of Nigerian building projects. In-depth interviews and focus groups were also held to acquire a deeper understanding of the real-world difficulties and experiences encountered in the field. The study pinpoints several crucial project management techniques that affect time and cost performance. The results show that Inadequate construction site management and supervision ranked 1st with an RII of 0.7877, Improper construction techniques and lack of experience of the consultant came 2nd and 3rd with RII of 0.7215 and 0.7213 respectively. Late payments during work progress and Use of improper equipment are contributors to project time overrun; while, factors affecting cost overrun are fluctuation in building material costs ranked 1st with RII of 0.7354, frequent design changes, (0.7138), omissions and errors in bills of quantities (0.6754), import of construction materials (0.6754). The report suggests enhancing project management through effective resource management techniques, ongoing training and certification programs, and implementing advanced technology to improve communication, resource management, and planning precision, contributing to sustainable growth in Nigeria's construction industry. Keywords: stakeholder management, construction sector, project management, major contributor

Problems and related results in algebraic vision and multiview geometry

This article is a survey of results in algebraic vision and multiview geometry. The starting point is the study of projective algebraic varieties which are critical for scene reconstruction. Initially studied for reconstructing static scenes in three-dimensional spaces, these critical loci are later investigated for dynamic and segmented scenes in higher-dimensional projective spaces. The formal analysis of the ideals of critical loci employs Grassmann tensors, introduced as crucial tools for determining these ideals and aiding the reconstruction process away from critical loci. A long-term goal of the authors with other co-authors involves two main aspects: firstly studying properties of Grassmann tensors, as rank, multi-rank and core, along with the varieties that parameterize these tensors; concurrently conducting an analysis of families of critical loci in various scenarios.

Better life stories make better lives: a reply to Berg

Is it good for us if the different parts of our lives are connected to each other like the parts of a good story? Some philosophers have thought so, while others have firmly rejected it. In this paper, I focus on the state-of-the-art anti-narrativist arguments Amy Berg has recently presented in this journal. I argue that while she makes a good case that the best kind of lives for us do not revolve around a single project or theme, the best kind of narrativist views actually encourage us to pursue a variety of different projects, as long as they are mutually supportive. I claim that when interpreted in the most plausible way, prudentially good-making narrative coherence arises precisely out of this kind of unity in diversity. Well-roundedness and narrative coherence are thus not inherently in tension, but are both good-making holistic features of our lives.

Algebraic fibre spaces with strictly nef relative anti‐log canonical divisor

AbstractLet be a projective klt pair, and a fibration to a smooth projective variety with strictly nef relative anti‐log canonical divisor . We prove that is a locally trivial fibration with rationally connected fibres, and the base is a canonically polarized hyperbolic manifold. In particular, when is a single point, we establish that is rationally connected. Moreover, when and is strictly nef, we prove that is ample, which confirms the singular version of a conjecture by Campana and Peternell for threefolds.

Toric Sylvester forms

In this paper, we investigate the structure of the saturation of ideals generated by sparse homogeneous polynomials over a projective toric variety X with respect to the irrelevant ideal of X. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on X. In particular, we prove that toric Sylvester forms yield bases of some graded components of Isat/I, where I denotes an ideal generated by n+1 generic forms, n is the dimension of X and Isat is the saturation of I with respect to the irrelevant ideal of the Cox ring of X. Then, to illustrate the relevance of toric Sylvester forms we provide three consequences in elimination theory over smooth toric varieties: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we explore the computation of the toric residue of the product of two forms.

Solving equations using Khovanskii bases

We develop a new eigenvalue method for solving structured polynomial equations over any field. The equations are defined on a projective algebraic variety which admits a rational parameterization by a Khovanskii basis, e.g., a Grassmannian in its Plücker embedding. This generalizes established algorithms for toric varieties, and introduces the effective use of Khovanskii bases in computer algebra. We investigate regularity questions and discuss several applications.

Projective homogeneous varieties of Picard rank one in small characteristic

Projective homogeneous varieties of Picard rank one in small characteristic

James reduced product schemes and double quasisymmetric functions

Symmetric function theory is a key ingredient in the Schubert calculus of Grassmannians. Quasisymmetric functions are analogues that are similarly central to algebraic combinatorics, but for which the associated geometry is poorly developed. Baker and Richter (2008) showed that QSym manifests topologically as the cohomology ring of the loop suspension of infinite projective space or equivalently of its combinatorial homotopy model, the James reduced product JCP∞. In recent work, we used this viewpoint to develop topologically-motivated bases of QSym and initiate a Schubert calculus for JCP∞ in both cohomology and K-theory.Here, we study the torus-equivariant cohomology of JCP∞. We identify a cellular basis and introduce double monomial quasisymmetric functions as combinatorial representatives, analogous to the factorial Schur functions and double Schubert polynomials of classical Schubert calculus. We also provide a combinatorial Littlewood–Richardson-type rule for the structure coefficients of this basis.Furthermore, we introduce an algebro-geometric analogue of the James reduced product construction. In particular, we prove that the James reduced product of a complex (quasi-)projective variety also carries the structure of a (quasi-)projective variety.

On the canonicity of the singularities of quotients of the Fulton-MacPherson compactification

We prove that quotients of the Fulton-MacPherson compactification of configuration spaces of smooth projective varieties of dimension > 1 >1 by permutation groups have canonical singularities.