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We give a geometric characterization of plus-one generated projective line arrangements that are next-to-free. We present new succinct proofs, via associated line bundles, for some properties of plus-one generated projective line arrangements. [ABSTRACT FROM AUTHOR]

A tangent category is a categorical abstraction of the tangent bundle construction for smooth manifolds. In that context, Cockett and Cruttwell develop the notion of differential bundle which, by work of MacAdam, generalizes the notion of smooth vector bundle to the abstract setting. Here we provide a new characterization of differential bundles and show that, up to isomorphism, a differential bundle is determined by its projection map and zero section. We show how these results can be used to quickly identify differential bundles in various tangent categories. [ABSTRACT FROM AUTHOR]

We calculate the structure sets in the sense of surgery theory of total spaces of bundles over eight-dimensional sphere with fibre a seven-dimensional sphere, in which manifolds homotopy equivalent to the total spaces are organized, and we investigate the question, which of the elements in these structure sets can be realized as such bundles. [ABSTRACT FROM AUTHOR]

A new approach to finite elasto-plasticity of crystalline materials, with a differential geometry point of view toward the material description, is proposed. In order to define the plastic and elastic distortions, traditionally it is accepted the existence of an intermediate configuration, which is endowed with a differential manifold structure. A more restrictive but physically and mathematically well-motivated structure is assigned to the intermediate configuration, namely, a vector bundle structure. Two approaches to multiplicative decomposition of the deformation gradient can be rebuilt if on the same total space (intermediate configuration) the vector bundle structures are considered, but either the reference configuration, B (i.e., plastic assumption) or the deformed configuration, B t (i.e., elastic assumption) stands for the base space. As the base spaces are homeomorphic, a pullback vector bundle over one base space is provided by the other one through the motion diffeomorphism. Thus, the total space is endowed with different differential manifold structures. The plastic distortion and the inverse elastic distortion, respectively, are defined for any material points as linear, invertible, and non-integrable maps from the tangent vector space at their material points, with the value on the attached vector fiber. The multiplicative decompositions into their non-integrable components are derived, based on the associate inverse elastic distortion and plastic distortion, respectively. When the plastic and elastic assumptions are simultaneously accepted, then a three-term multiplicative decomposition of the deformation gradient is achieved. The transition functions, which characterize the compatibility of the overlapping charts which belong to these different structures, namely of the vector bundle and of the pullback vector bundle over the same configuration, say B define the non-uniqueness of the two-term multiplicative decomposition. The material symmetry transformation is defined for elastic-type materials. The compatibility of the models and examples of the bundle vector structures are also discussed. [ABSTRACT FROM AUTHOR]

We study Seshadri constants of certain ample vector bundles on projective varieties. Our main motivation is the following question: Under what conditions are the Seshadri constants of ample vector bundles at least 1 at all points of the variety? We exhibit some conditions under which this question has an affirmative answer. We primarily consider ample bundles on projective spaces and Hirzebruch surfaces. We also show that Seshadri constants of ample vector bundles can be arbitrarily small. [ABSTRACT FROM AUTHOR]

We define vector bundles for tropical schemes, and explore their properties. The paper largely consists of three parts (1) we study free modules over zero-sum free semirings, which provide the necessary algebraic background for the theory, (2) we relate vector bundles on tropical schemes to topological vector bundles and vector bundles on monoid schemes, and finally (3) we show that all line bundles on a tropical scheme can be lifted to line bundles on a usual scheme in the affine case. [ABSTRACT FROM AUTHOR]

In this paper, we consider spaces whose geometry is generated by a homogeneous function of degree m ≥ 2, which is invariant under the action of some subgroup of the linear group of the given space. A general method is proposed and examples of realization of such spaces on linear algebras are given. [ABSTRACT FROM AUTHOR]

This paper classifies rank two vector bundles on a del Pezzo threefold X of degree five whose projectivizations are weak Fano. This classification is then used to determine properties of the moduli spaces of such vector bundles on X , and we determine precisely when the moduli spaces are smooth, irreducible, and fine. We also prove that such a bundle on a del Pezzo threefold of degree one or two splits, and as result gives a classification of weak Fano bundles of rank two on a del Pezzo threefold of Picard rank one. [ABSTRACT FROM AUTHOR]

We present geometric realizations of horospherical two-orbit varieties, by showing that their blow-up along the unique closed invariant orbit is the zero locus of a general section of a homogeneous vector bundle over some auxiliary variety. As an application, we compute the cohomology ring of the G_2-variety, including its quantum version. We also consider the Spin_7-variety, which deserves a different treatment. [ABSTRACT FROM AUTHOR]

We use a correspondence between Ulrich bundles on a projective variety and quiver representations to prove that certain del Pezzo surfaces satisfy the Ulrich trichotomy, for any given polarization. [ABSTRACT FROM AUTHOR]

We complete the classification of globally generated vector bundles with c1≤5$c_1 \le 5$ on projective spaces by treating the case c1=5$c_1 = 5$ on Pn$\mathbb {P}^n$, n≥4$n \ge 4$. It turns out that there are very few indecomposable bundles of this kind: besides some obvious examples there are, roughly speaking, only the (first twist of the) rank 5 vector bundle which is the middle term of the monad defining the Horrocks bundle of rank 3 on P5$\mathbb {P}^5$, and its restriction to P4$\mathbb {P}^4$. We recall, in an appendix, from one of our previous papers, the main results allowing the classification of globally generated vector bundles with c1=5$c_1 = 5$ on P3$\mathbb {P}^3$. Since there are many such bundles, a large part of the main body of the paper is occupied with the proof of the fact that, except for the simplest ones, they do not extend to P4$\mathbb {P}^4$ as globally generated vector bundles. [ABSTRACT FROM AUTHOR]

In this paper, we construct a correspondence between the complex vector bundles of finite rank on smooth manifolds and the representation of the generated vector space by a basis of the vector bundles and derive its non-triviality via the representation of the generated space. We also provide some applications of this correspondence. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the geometry of vector bundle manifolds equipped with spherically symmetric metrics. We will compute the curvature tensor, the sectional curvature, the Ricci curvature tensor and the scalar curvature. We will then characterize spherically symmetric metrics of constant sectional curvature. Moreover, we will establish some rigidity results regarding the scalar curvature and the Ricci curvature. Finally, we investigate geodesics using the fundamental tensors of a submersion. [ABSTRACT FROM AUTHOR]

We give an algorithm to determine whether a kernel sheaf over a smooth projective curve over an algebraically closed field is semistable. The algorithm uses symmetric powers to make destabilizing subbundles visible as global sections. [ABSTRACT FROM AUTHOR]

We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter k R -spaces and locally convex spaces E such that E × E is a k R -space. [ABSTRACT FROM AUTHOR]

This paper treats the strict semi-stability of the symmetric powers S k E of a stable vector bundle E of rank 2 with even degree on a smooth projective curve C of genus g ≥ 2. The strict semi-stability of S 2 E is equivalent to the orthogonality of E or the existence of a bisection on the ruled surface ℙ C (E) whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of E with strictly semi-stable S 3 E. Moreover, it is shown that when S 2 E is stable, every symmetric power S k E is stable for all but a finite number of E in the moduli of stable vector bundles of rank  2 with fixed determinant of even degree on C. [ABSTRACT FROM AUTHOR]

We equip an arbitrary vector bundle over a Riemannian manifold, endowed with a fiber metric and a compatible connection, with a spherically symmetric metric (cf. [4]), and westudy harmonicity of its sections firstly as smooth maps and then as critical points of the energy functional with variations through smooth sections.We also characterize vertically harmonic sections. Finally, we give some examples of special vector bundles, recovering in some situations some classical harmonicity results. [ABSTRACT FROM AUTHOR]

We describe completely the so called jet like functors of a vector bundle E over an m-dimensional manifold M, i.e. bundles FE over M canonically depending on E such that F(E1 ×M E2) = FE1 ×M FE2 for any vector bundles E1 and E2 over M. Then we study how a linear vector field on E can induce canonically a vector field on FE. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective complex curve of genus g ≥ 2 and let MX(2,Λ) be the moduli space of semi-stable rank-2 vector bundles over X with fixed determinant Λ. We show that the wobbly locus, i.e. the locus of semi-stable vector bundles admitting a non-zero nilpotent Higgs field, is a union of divisors 𝓦k ⊂ MX(2,Λ). We show that on one wobbly divisor the set of maximal subbundles is degenerate. We also compute the class of the divisors 𝓦k in the Picard group of MX(2, Λ). [ABSTRACT FROM AUTHOR]

We consider the problem of constructing matrices of linear forms of constant rank by focusing on the associated vector bundles on projective spaces. Important examples are given by the classical Steiner bundles, as well as some special (duals of) syzygy bundles that we call Drézet bundles. Using the classification of globally generated vector bundles with small first Chern class on projective spaces, we are able to describe completely the indecomposable matrices of constant rank up to six; some of them come from rigid homogeneous vector bundles, some other from Drézet bundles related either to plane quartics or to instanton bundles on P 3. [ABSTRACT FROM AUTHOR]

We construct wave operators and a scattering operator for the scattering of a charged particle on the Dirac magnetic monopole. The analysis features a two-Hilbert-space approach in which the identification operator matches states of the same angular momentum. [ABSTRACT FROM AUTHOR]

Let R = K [ x , y , z ] be a standard graded polynomial ring where K is an algebraically closed field of characteristic zero. Let M = ⊕ j M j be a finite length graded R -module. We say that M has the Weak Lefschetz Property if there is a homogeneous element L of degree one in R such that the multiplication map × L : M j → M j + 1 has maximal rank for every j. The main result of this paper is to show that if E is a locally free sheaf of rank 2 on P 2 then the first cohomology module of E , H ⁎ 1 (P 2 , E) , has the Weak Lefschetz Property. [ABSTRACT FROM AUTHOR]

We give examples of surfaces which are Ulrich-wild, i.e., that support families of dimension p of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrarily large p. [ABSTRACT FROM AUTHOR]

We deal with instanton bundles on the product ℙ 1 × ℙ 2 {\mathbb{P}^{1}\times\mathbb{P}^{2}} and the blow up of ℙ 3 {\mathbb{P}^{3}} along a line. We give an explicit construction leading to instanton bundles. Moreover, we also show that they correspond to smooth points of a unique irreducible component of their moduli space. [ABSTRACT FROM AUTHOR]

We ask when the CM (Castelnuovo–Mumford) regularity of a vector bundle on a projective variety X is numerical, and address the case when X is an abelian variety. We show that the continuous CM-regularity of a semihomogeneous vector bundle on an abelian variety X is a piecewise-constant function of Chern data, and we also use generic vanishing theory to obtain a sharp upper bound for the continuous CM-regularity of any vector bundle on X. From these results we conclude that the continuous CM-regularity of many semihomogeneous bundles — including many Verlinde bundles when X is a Jacobian — is both numerical and extremal. [ABSTRACT FROM AUTHOR]

A closed subscheme of codimension two T ⊂ P 2 is a quasi complete intersection (q.c.i.) of type (a , b , c) if there exists a surjective morphism O (− a) ⊕ O (− b) ⊕ O (− c) → I T. We give bounds on deg ⁡ (T) in function of a , b , c and r , the least degree of a syzygy between the three polynomials defining the q.c.i. (see Theorem 6). As a by-product we recover a theorem of du Plessis-Wall on the global Tjurina number of plane curves (see Theorem 20) and some other related results. [ABSTRACT FROM AUTHOR]

We give the complete classification of Mukai pairs of dimension 4 and rank 2 with Picard number one, that is, pairs (X, E) where X is a Fano 4-fold with Picard number one, and E is an ample vector bundle of rank 2 on X with c1(X) = c1(E). Equivalently, the present paper completes the classification of ruled Fano 5-folds with index two, which was partially done by C. Novelli and G. Occhetta in 2007. [ABSTRACT FROM AUTHOR]

Commutative K-theory, a cohomology theory built from spaces of commuting matrices, has been explored in recent work of Adem, Gómez, Gritschacher, Lind and Tillman. In this article, we use unstable methods to construct explicit representatives for the real commutative K-theory classes on surfaces. These classes arise from commutative O ⁢ (2) {O(2)} -valued cocycles and are analyzed via the point-wise inversion operation on commutative cocycles. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective curve of genus g = 2 over an algebraically closed field k of characteristic p ≥ 0, and let FX : X → X be the absolute Frobenius morphism. Let MsX(r, d) be the moduli space of stable vector bundles of rank r and degree d on X. We study the Frobenius stratification of MsX(3, 0) in terms of Harder-Narasimhan polygons of Frobenius pull backs of stable vector bundles and obtain the irreducibility and dimension of each nonempty Frobenius stratum in the case in which (p, g) = (3, 2). [ABSTRACT FROM AUTHOR]

Let (X , L) be a complex polarized n -fold with the structure of a classical scroll over a smooth projective threefold Y. The Hilbert curve of such a pair (X , L) is a complex affine plane curve of degree n , consisting of n − 3 evenly spaced parallel lines plus a cubic. This paper is devoted to a detailed study of this cubic. In particular, existence of triple points, behavior with respect to the line at infinity, and non-reducedness, are analyzed in connection with the structure of (X , L). Special attention is reserved to the case n = 4 , where various examples are presented and the possibility that the cubic is itself the Hilbert curve of the base threefold Y for a suitable polarization is discussed. [ABSTRACT FROM AUTHOR]

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$. We show that for any integers $r$ and $d$ with $0 , there exists a maximally Frobenius destabilised stable vector bundle of rank $r$ and degree $d$ on $X$ if and only if $r\mid d$. [ABSTRACT FROM AUTHOR]

We construct equivariant vector bundles over quantum projective spaces using parabolic Verma modules over the quantum general linear group. Using an alternative realization of the quantized coordinate ring of the projective space as a subalgebra in the algebra of functions on the quantum group, we reformulate quantum vector bundles in terms of quantum symmetric pairs. We thus prove the complete reducibility of modules over the corresponding coideal stabilizer subalgebras, via the quantum Frobenius reciprocity. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions to specify vector bundles of conformal blocks for 𝔰𝔩2m with rectangular weights which have ranks 0, 1, and larger than 1. We show that the first Chern classes of such rank one bundles determine a finitely generated subcone of the nef cone. [ABSTRACT FROM AUTHOR]

In this paper, we study invariants of linear differential operators with respect to algebraic Lie pseudogroups. Then we use these invariants and the principle of n-invariants to get normal forms (or models) of the differential operators and solve the equivalence problem for actions of algebraic Lie pseudogroups. As a running example of application of the methods, we use the pseudogroup of local symplectomorphisms. [ABSTRACT FROM AUTHOR]

Tangent categories provide an axiomatic approach to key structural aspects of differential geometry that exist not only in the classical category of smooth manifolds but also in algebraic geometry, homological algebra, computer science, and combinatorics. Generalizing the notion of (linear) connection on a smooth vector bundle, Cockett and Cruttwell have defined a notion of connection on a differential bundle in an arbitrary tangent category. Herein, we establish equivalent formulations of this notion of connection that reduce the amount of specified structure required. Further, one of our equivalent formulations substantially reduces the number of axioms imposed, and others provide useful abstract conceptualizations of connections. In particular, we show that a connection on a differential bundle E over M is equivalently given by a single morphism K that induces a suitable decomposition of T E as a biproduct. We also show that a connection is equivalently given by a vertical connection K for which a certain associated diagram is a limit diagram. [ABSTRACT FROM AUTHOR]

Almost paracontact and normal almost paracontact Finsler structures on a vector bundle are defined. Finding some conditions, integrability of these structures is studied. Moreover, we define paracontact metric, para- Sasakian and K-paracontact Finsler structures and study some properties of these structures. For a K-paracontact Finsler structure, we find vertical and horizontal ag curvatures. Then, defining the vertical φ-ag curvature, we prove that every locally symmetric para-Sasakian Finsler structure has a negative vertical φ-ag curvature. Finally, we define the horizontal and vertical Ricci tensors of a para-Sasakian Finsler structure and study some curvature properties of them. [ABSTRACT FROM AUTHOR]

The problems of analysis and modeling of spherical trajectories, that is, continuous longitudinal data on , are important in several disciplines. These problems are challenging for two reasons: (1) nonlinear geometry of and (2) the presence of phase variability in given data. This article develops a geometric framework for separating phase variability from given trajectories, leaving only the shape or the amplitude variability. The key idea is to represent each trajectory with a pair of variables, a starting point, and a transported square-root velocity curve (TSRVC), a curve in the tangent (vector) space at the starting point. The space of all such curves forms a vector bundle and the norm, along with the standard Riemannian metric on , provides a natural, warping-invariant metric on this vector bundle. This leads to an efficient algorithm for registration of trajectories, that is, phase-amplitude separation, and computational tools, such as clustering, sample means, and principal component analysis (PCA) of the two components separately. It also helps derive simple statistical models of phase-amplitude components of spherical trajectories. This comprehensive framework is demonstrated using two datasets: a set of bird-migration trajectories and a set of hurricane paths in the Atlantic ocean. Supplementary material for this article is available online. [ABSTRACT FROM AUTHOR]

We prove that, given the isomorphism class of the parabolic Deligne–Hitchin moduli space over a smooth projective curve, we can recover the isomorphism class of the curve and the parabolic points. [ABSTRACT FROM AUTHOR]

this paper we study a class of real vector bundles from the partial trivial viewpoint. We define a partial trivial vector bundle as a couple (ξ, ρp) where ξ = (E, л,M) is a vector bundle and ρp is a global partial frame of ξ. We give an incomplete answer to the question: when and how is it possible to obtain a partial trivial structure in a real vector bundle? The construction of some partial trivial structures is reduced to a problem of "adjunction and lifting" for Gauss map in a certain commutative diagram homotopy. Pontryagin classes of a partial trivial vector bundle are considered. The structure of induced partial trivial tangent bundle of a foliated manifold is presented. [ABSTRACT FROM AUTHOR]

Let be a surface with and endowed with a very ample line bundle such that . We show that supports special (often stable) Ulrich bundles of rank , extending a recent result by A. Beauville. Moreover, we show that such an supports families of dimension of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large except for very few cases. We also show that the same is true for each linearly normal non-special surface with in of degree at least , Enriques surface and anticanonical rational surface. [ABSTRACT FROM AUTHOR]

Given a closed manifold M and a vector bundle ξ of rank n over M , by gluing two copies of the disc bundle of ξ , we can obtain a closed manifold D ( ξ , M ) , the so-called double manifold. In this paper, we firstly prove that each sphere bundle S r ( ξ ) of radius r > 0 is an isoparametric hypersurface in the total space of ξ equipped with a connection metric, and for r > 0 small enough, the induced metric of S r ( ξ ) has positive Ricci curvature under the additional assumptions that M has a metric with positive Ricci curvature and n ≥ 3 . As an application, if M admits a metric with positive Ricci curvature and n ≥ 2 , then we construct a metric with positive Ricci curvature on D ( ξ , M ) . Moreover, under the same metric, D ( ξ , M ) admits a natural isoparametric foliation. For a compact minimal isoparametric hypersurface Y n in S n + 1 ( 1 ) , which separates S n + 1 ( 1 ) into S + n + 1 and S − n + 1 , one can get double manifolds D ( S + n + 1 ) and D ( S − n + 1 ) . Inspired by Tang, Xie and Yan's work on scalar curvature of such manifolds with isoparametric foliations (cf. [25] ), we study Ricci curvature of them with isoparametric foliations in the last part. [ABSTRACT FROM AUTHOR]

Let $S\subseteq \mathbb{P}^{d}$ be an anticanonically embedded surface of degree $d\geq 3$. In this note, we classify stable Ulrich bundles on $S$ of rank two. We also study their moduli spaces. [ABSTRACT FROM PUBLISHER]

In this article we suggest a vector bundle description for multi-plier modules of vector bundles over noncompact spaces. We prove that the isomorphism classes of multiplier modules are dependent on the isomorphism classes of their underlying modules. This gives a way to evaluate the set of extensions of Hilbert modules in topological terms in simple cases. [ABSTRACT FROM AUTHOR]

We compute the Hilbert–Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen–Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals. [ABSTRACT FROM AUTHOR]

We show that the singularities of the Fresnel surface for Maxwell's equation on an anisotrpic material can be accounted from purely topological considerations. The importance of these singularities is that they explain the phenomenon of conical refraction predicted by Hamilton. We show how to desingularise the Fresnel surface, which will allow us to use Morse theory to find lower bounds for the number of critical wave velocities inside the material under consideration. Finally, we propose a program to generalise the results obtained to the general case of hyperbolic differential operators on differentiable bundles. [ABSTRACT FROM AUTHOR]

We prove explicit formulas for Chern classes of tensor products of virtual vector bundles, whose coefficients are given by certain universal polynomials in the ranks of the two bundles. [ABSTRACT FROM AUTHOR]

In this paper we obtain an explicit formula for the number of curves in P 2 , of degree d , passing through ( d ( d + 3 ) / 2 − k ) generic points and having a singularity X , where X is of type A k ≤ 7 , D k ≤ 7 or E k ≤ 7 . Our method comprises of expressing the enumerative problem as the Euler class of an appropriate bundle and using a purely topological method to compute the degenerate contribution to the Euler class. These numbers have also been computed by M. Kazarian using the existence of universal formulas for Thom polynomials. [ABSTRACT FROM AUTHOR]

The aim of this note is to construct sequences of vector bundles with unbounded rank and discriminant on an arbitrary algebraic surface. This problem, on principally polarized abelian varieties with cyclic Neron–Severi group generated by the polarization, was considered by Nakashima in connection with the Douglas–Reinbacher–Yau conjecture on the Strong Bogomolov Inequality. In particular we show that on any surface, the Strong Bogomolov Inequality S B I l is false for all l > 4 . [ABSTRACT FROM AUTHOR]

In this paper, we show that the secant variety to a smooth projective variety embedded by a sufficiently positive line bundle is normal. As an application, we deduce that the secant variety to a general canonical curve of genus at least 7 is normal. [ABSTRACT FROM AUTHOR]

An irreducible algebraic stack is called unirational if there exists a surjective morphism, representable by algebraic spaces, from a rational variety to an open substack. We prove unirationality of the stack of prioritary omalous bundles on Hirzebruch surfaces, which implies also the unirationality of the moduli space of omalous H -stable bundles for any ample line bundle H on a Hirzebruch surface (compare with Costa and Miro-Ŕoig, 2002). To this end, we find an explicit description of the duals of omalous rank-two bundles with a vanishing condition in terms of monads. Since these bundles are prioritary, we conclude that the stack of prioritary omalous bundles on a Hirzebruch surface different from P 1 × P 1 is dominated by an irreducible section of a Segre variety, and this linear section is rational (Ionescu, 2015). In the case of the space quadric, the stack has been explicitly described by N. Buchdahl. As a main tool we use Buchdahl’s Beilinson-type spectral sequence. Monad descriptions of omalous bundles on hypersurfaces in P 4 , Calabi–Yau complete intersection, blowups of the projective plane and Segre varieties have been recently obtained by A.A. Henni and M. Jardim (Henni and Jardim, 2013), and monads on Hirzebruch surfaces have been applied in a different context in Bartocci et al. (2015). [ABSTRACT FROM AUTHOR]