For smooth projective varieties [Formula: see text] and [Formula: see text] over a perfect field [Formula: see text] of positive characteristic [Formula: see text] so that [Formula: see text] and [Formula: see text] are derived equivalent, and admitting a closed embedding into [Formula: see text], we assume that [Formula: see text]. We prove that if [Formula: see text] is ordinary (resp., Hodge-Witt), then [Formula: see text] is ordinary (resp., Hodge-Witt).

Normal functions \nu provide a method for studying algebraic cycles Z_{t}\subset X_{t} varying in a family of smooth projective varieties. Associated with \nu is an infinitesimal invariant \delta\nu that reflects the first-order variation of pairs (X_{t},Z_{t}) . Over the years, \delta\nu has been widely used in the study of various geometric questions. We note that whereas \nu is a transcendental invariant, like periods of algebraic integrals, \delta\nu has a natural filtration whose associated graded gives algebraic sections of coherent sheaves. In a number of interesting cases, these sections have had geometric interpretations. In this paper, we will discuss an identification between the singularities \nu and \delta\nu . The formal proof of this result will be given in a separate work.

Abstract The purpose of this note is twofold. First, we give a quick proof of Ballico–Chiantini’s theorem stating that a Fano or Calabi–Yau variety of dimension at least 4 in codimension 2 is a complete intersection. Second, we improve Barth–Van de Ven’s result asserting that if the degree of a smooth projective variety of dimension n is less than approximately $$0.63 \cdot n^{1/2},$$ 0.63 · n 1 / 2 , then it is a complete intersection. We show that the degree bound can be improved to approximately $$0.79 \cdot n^{2/3}.$$ 0.79 · n 2 / 3 .

In Algebraic Statistics, M.A. Cueto, J. Morton and B. Sturmfels introduced a statistical model, the Restricted Boltzmann Machine, which introduced the Hadamard product of two or more vectors of an affine or projective space, i.e., the componentwise product of their entries, forcing Algebraic Geometry to enter. The Hadamard product X⋆Y of two subvarieties X,Y⊂Pn is defined as the Zariski closure of the Hadamard product of its elements. Recently, D. Antolini and A. Oneto introduced and studied the definition of Hadamard rank, and we prove some results on it. Moreover, we prove some theorems on the dimension and shape of the Hadamard powers of X. The aim is to describe the images of the Hadamard products without taking the Zariski closure. We also discuss several scenarios describing the case in which some of the data, i.e., the variety X, is wrong or it is not possible to recover it.

We consider the problem of smoothing algebraic cycles with rational coefficients on smooth projective complex varieties up to homological equivalence. We show that a solution to this problem would be incompatible with the validity of the Hartshorne conjecture on complete intersections in projective space. We also solve unconditionally a symplectic variant of this problem.

We construct a new Weil cohomology for smooth projective varieties over a field, universal among Weil cohomologies with values in rigid additive tensor categories. A similar universal problem for Weil cohomologies with values in rigid abelian tensor categories also has a solution. We give a variant for Weil cohomologies satisfying more axioms, like weak and strong Lefschetz. As a consequence, we get a different construction of André’s category of motives for motivated correspondences and show that it has a universal property. This theory extends over suitable bases.

CONTEXTperts (Context and Topic Experts) bring real-world experience in the context of the study topic and setting to enhance research team expertise, capacity, and creativity. They complement but do not replace authentic patient and community engagement, professional consultants, or formal advisory boards. Individual CONTEXTpert consultations or group meetings can help improve research questions, study designs, implementation plans, dissemination, and application of findings. They have added value to a variety of projects with varied research methods and study designs, including research with and without patient or community involvement. CONTEXTperts can bring vision, challenge, and reality checks to a variety of research teams with a practical, affordable model.

Galois subspaces for projective varieties

Abstract We prove that, over a smooth quasi-projective curve, the set of non-isotrivial, smooth and projective families of polarized varieties with a fixed Hilbert polynomial and semi-ample canonical bundle is bounded. This extends the boundedness results of Arakelov, Parshin and Kovács–Lieblich beyond the canonically polarized case.

A startup or start-up may be a company or project undertaken by an entrepreneur to hunt, develop, and validate a scalable business model. While entrepreneurship refers to all or any new businesses, including self-employment and businesses that never shall become registered, startups ask new businesses that shall grow large beyond the solo founder. At the start, startups face high uncertainty and have high rates of failure, but a minority of them do continue to achieve success and influential. Entrepreneurship is one of the outstanding factors for the development of a nation. The Government of India rigorously encourages innovation, startups and entrepreneurship and allocates huge funds to encourage the growth of the entrepreneurial environment. To encourage students to explore entrepreneurship as a career option, a variety of organizational programs and projects provide a support system and connect industries, institutions and skills development programs. This Research Paper will highlight the prospects and challenges faced by the new start-ups through Digital platform. This paper will generate new ideas of forming start-ups and creating entrepreneurs.

The restoration of native species-dominated ecosystems is critical for improving ecosystem health and meeting policy goals in the Sacramento–San Joaquin Delta and Suisun Marsh (upper San Francisco Estuary, collectively), one of the largest estuarine systems in North America. To accomplish large-scale restoration in this heavily altered system, a variety of projects, programs, and motivations inform restoration planning and implementation. Chapter 4, “Ecosystem,” of the Delta Plan synthesizes restoration goals across these efforts to produce comprehensive ecosystem restoration targets of between 60,000 and 80,000 acres across seven ecosystem types by 2050, but a comprehensive review of restoration progress and planning to date is needed. To fill this gap, this paper analyzes the current state of ecosystem restoration in the upper San Francisco Estuary in the context of the Delta Plan targets. We review current scientific and management literature and implementation approaches, and synthesize acreage totals across completed, in-progress, and planned projects for four ecosystem types where substantial development of restoration in the system has occurred: tidal wetland, non-tidal wetland, riparian, and floodplain. We find that tidal wetland restoration has progressed more rapidly than other ecosystem types, motivated by mitigation requirements related to the federal Endangered Species Act. Across all ecosystem types, we identify both promising progress and clear needs for accelerated planning and implementation of restoration projects to meet Delta Plan 2050 targets, and discuss ongoing needs related to science, funding, and implementation.

In this paper, we study the basic structures of degree- g g topological recursion relations on the moduli space of curves M ¯ g , n \overline {\mathcal {M}}_{g,n} : (i) the coefficient of the bouquet class on M ¯ g , n \overline {\mathcal {M}}_{g,n} , which gives the answer to a conjecture of T. Kimura and X. Liu [Comm. Math. Phys. 262 (2006), pp. 645–661]; (ii) linear relations among the coefficients of certain rational tails locus of M ¯ g , n \overline {\mathcal {M}}_{g,n} . Three applications of topological recursion relations will be discussed: (i) coefficients of universal equations for Gromov–Witten invariants for any smooth projective variety; (ii) the coefficient of the bouquet class in the double ramification formula of the top Hodge class λ g \lambda _g ; (iii) a new recursive formula for computing the intersection numbers on the moduli space of stable curves.

This paper presents some nontrivial computational results on derived category and Fourier–Mukai technique in algebraic geometry. In particular, it aims at presenting calculations involving spherical twists as a certain class of Fourier–Mukai functors and its cohomological descent on the singular rational cohomology of smooth projective variety. The purpose of this investigation is to present a new perspective, based upon Fourier–Mukai technique, on solving classical problems involving characteristic classes: in particular, the Chern and the Euler characteristics.

Abstract In this paper, we establish a structure theorem for a minimal projective klt variety satisfying Miyaoka's equality . Specifically, we prove that the canonical divisor is semi‐ample and that the Kodaira dimension is equal to 0, 1, or 2. Furthermore, based on this abundance result, we show that a maximally quasi‐étale cover of is smooth, and we explicitly describe the structure of the Iitaka fibration. In addition, we prove an analogous result for projective klt varieties with numerically effective anti‐canonical divisor.

Corvaja and Zannier asked whether a smooth projective integral variety with a dense set of rational points over a number field satisfies the weak Hilbert property. We introduce an extension of the weak Hilbert property for schemes over arithmetic base rings by considering near-integral points, extending Corvaja–Zannier’s question beyond the projective case. Building on work of Bary-Soroker–Fehm–Petersen and Corvaja–Demeio–Javanpeykar–Lombardo–Zannier, we prove several properties of this more general notion, in particular its persistence under products. We also answer positively Corvaja–Zannier’s question for all algebraic groups over finitely generated fields of characteristic zero.

This work introduces the first in-depth study of h -free and h -full elements in abelian monoids, providing a unified approach for understanding their role in various mathematical structures. Let 𝔪 be an element of an abelian monoid, with ω ( 𝔪 ) denoting the number of distinct prime elements generating 𝔪 . We study the moments of ω ( 𝔪 ) over subsets of h -free and h -full elements, establishing the normal order of ω ( 𝔪 ) within these subsets. Our findings are then applied to number fields, global function fields, and geometrically irreducible projective varieties, demonstrating the broad relevance of this approach.

Abstract Working in the category of smooth projective varieties over an algebraically closed field of characteristic 0, we review notions of ampleness and numerical nefness for Higgs bundles which “feel” the Higgs field and formulate criteria of the Barton-Kleiman type for these notions. We give an application to minimal surfaces of general type that saturate the Miyaoka-Yau inequality, showing that their cotangent bundle is ample.

Given a vector bundle \mathcal{E} on a smooth projective variety B , the flag bundle \mathcal{F}l(1,2,\mathcal{E}) admits two projective bundle structures over the Grassmann bundles \mathcal{G}r(1,\mathcal{E}) and \mathcal{G}r (2,\mathcal{E}) . The data of a general section of a suitably defined line bundle on \mathcal{F}l(1,2,\mathcal{E}) defines two varieties: a cover X_{1} of B , and a fibration X_{2} on B with general fiber isomorphic to a smooth Fano variety. We construct a semiorthogonal decomposition of the derived category of X_{2} which consists of a list of exceptional objects and a subcategory equivalent to the derived category of X_{1} . As a by-product, we obtain a new full exceptional collection for the Fano fourfold of degree 12 and genus 7 . Any birational map of smooth projective varieties which is resolved by blowups with exceptional divisor \mathcal{F}l(1, 2, \mathcal{E}) is an instance of a so-called Grassmann flip: we prove that the DK conjecture of Bondal–Orlov and Kawamata holds for such flips. This generalizes a previous result of Leung and Xie to a relative setting.

In the search of a projective analog of Kunz’s theorem and a Frobenius-theoretic analog of Mori–Hartshorne’s theorem, we investigate the positivity of the kernel of the Frobenius trace (equivalently, the negativity of the cokernel of the Frobenius endomorphism) on a smooth projective variety over an algebraically closed field of positive characteristic. For instance, such a kernel is ample for projective spaces. Conversely, we show that for curves, surfaces, and threefolds the Frobenius trace kernel is ample only for Fano varieties of Picard rank 1.

ABSTRACT Adaptive reuse has gained global recognition as an effective approach to achieve the Sustainable Development Goals. Yet, despite worldwide acclaim, the revitalisation strategy has only recently been gaining traction in Indonesia. Nevertheless, a gradual increase in projects that qualify as “adaptive reuse” can be detected in the country. In Jakarta, the strategy has been adopted in a variety of revitalisation projects including architecture dating back to the colonial era, as well as young industrial heritage. This article scrutinises the factors that have hindered the adoption of adaptive reuse in Jakarta in the past, and the factors that have, on the contrary, contributed to its recent success and incremental application. It is argued that, despite (historical) policy flaws, the bottom-up implementation of adaptive reuse in restoration projects is impacting the Indonesian heritage practice on all levels. This is substantiated by the case study of Pos Bloc, a former colonial post office that is being repurposed as a cultural and commercial hub in Central Jakarta.