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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]

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]

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]

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]

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]

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]

Abstract: Let be either the real number field or the complex number field and the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given -vector bundle over to be stably extendible to for every . In this paper, we simplify the theorems and apply them to the tangent bundle of , its complexification, the normal bundle associated to an immersion of in , and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5]. [Copyright &y& Elsevier]

Abstract: We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation, i.e. one whose kernel consists only of ineffective arrows, on a smooth vector bundle. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as an extension, by some bundle of compact Lie groups, of some action groupoid with G compact. [Copyright &y& Elsevier]

Abstract: Methods from stochastic analysis are combined with functional analytic methods in order to prove a Feynman–Kac formula för Schrödinger type operators with nonnegative locally square integrable potentials on vector bundles over complete Riemannian manifolds. In particular, we obtain a Feynman–Kac–Itô formula on manifolds for Schrödinger operators with magnetic fields. [ABSTRACT FROM AUTHOR]

Abstract: A theory of double affine and special double affine bundles, i.e. differential manifolds with two compatible (special) affine bundle structures, is developed as an affine counterpart of the theory of double vector bundles. The motivation and basic examples come from Analytical Mechanics, where double affine bundles have been recognized as a proper geometrical tool in a frame independent description of many important systems. Different approaches to the (special) double affine bundles are compared and carefully studied together with the problems of double vector bundle models and hulls, duality, and relations to associated phase spaces, contact structures, and other canonical constructions. [Copyright &y& Elsevier]

Abstract: Let denote the -dimensional standard lens space mod 3. In this paper, we study the conditions for a given real vector bundle over to be stably extendible to for every , and establish the formula on the power (k-fold) of a real vector bundle ζ over . Moreover, we answer the stable splitting problem for real vector bundles over by means of arithmetic conditions. [Copyright &y& Elsevier]

Abstract: Let F be the real number field R or the complex number field C, and let denote the real projective n-space. In this paper, we study the conditions for a given F-vector bundle over to be stably extendible to for every , and establish the formulas on the power (r-fold) of an F-vector bundle ζ over . Our results are improvements of the previous papers [T. Kobayashi, H. Yamasaki, T. Yoshida, The power of the tangent bundle of the real projective space, its complexification and extendibility, Proc. Amer. Math. Soc. 134 (2005) 303–310] and [Y. Hemmi, T. Kobayashi, Min Lwin Oo, The power of the normal bundle associated to an immersion of , its complexification and extendibility, Hiroshima Math. J. 37 (2007) 101–109]. Furthermore, we answer the stable splitting problem for F-vector bundles over by means of arithmetic conditions. [Copyright &y& Elsevier]

Abstract: It is a classical theorem of Milnor that for every vector bundle over , all the Stiefel–Whitney classes vanish if and only if . We describe a space B as W-trivial (except for one dimension) if for every vector bundle over B, all the Stiefel–Whitney classes vanish (except for a single fixed dimension). We establish theorems which state that certain high-connectivities of B imply these trivialities as well as a theorem which states that there are infinitely many “W-trivial except for one dimension” spaces. [Copyright &y& Elsevier]

Abstract: Let G be a simply connected linear algebraic group, defined over the field of complex numbers, whose Lie algebra is simple. Let P be a proper parabolic subgroup of G. Let E be a holomorphic vector bundle over such that E admits a homogeneous structure. Assume that E is not stable. Then E admits a homogeneous structure with the following property: There is a nonzero subbundle left invariant by the action of G such that . [Copyright &y& Elsevier]

Abstract: We develop a method, initially due to Salamon, for computing the space of “invariant” forms on an associated bundle , with a suitable notion of invariance. We determine sufficient conditions for this space to be -closed. We apply our method to the construction of Calabi–Yau metrics on and . [Copyright &y& Elsevier]

Active Contour modeling (ACM) has been shown to be a powerful method in object boundary extraction. In this paper, a new ACM based on vector bundle constraint for particle swarm optimization (VBCPSO-ACM) is proposed. Different from the traditional particle swarm optimization (PSO), in the process of velocity update, a vector bundle is predefined for each particle and velocity update of the particle is restricted to its bundle. Applying this idea to ACM, control points on the contour are treated as particles in PSO and the evolution of the contour is driven by the particles. Meanwhile, global searching is shifted to local searching in ACM by decreasing the number of neighbors and inertia. In addition, the addition and deletion of particles on the active contour make this new model possible for representing the real boundaries more precisely. The proposed VBCPSO-ACM can avoid self-intersection during contour evolving and also extract inhomogeneous boundaries. The simulation results proved its great performance in performing contour extraction. [ABSTRACT FROM AUTHOR]

Abstract: We show that every bad orbifold vector bundle can be realized as the restriction of a good orbifold vector bundle to a suborbifold of the base space. We give an explicit construction of this result in which the Chen–Ruan orbifold cohomologies of the two base spaces are isomorphic (as additive groups). This construction is used to indicate an extension of the Chern–Weil construction of characteristic classes to bad orbifold vector bundles. In particular, we apply this construction to the orbifold Euler class and demonstrate that it acts as an obstruction to the existence of nonvanishing sections. [Copyright &y& Elsevier]

Abstract: Let p be a prime number. It is proven that for the product of free actions of on modp homology spheres , , where s are assumed to be odd if p is odd, for any continuous map the set has dimension at least , provided for all i (). [Copyright &y& Elsevier]

Abstract: Let X be a smooth projective curve of genus and let be the moduli space of semistable bundles over X of rank 2 with trivial determinant. Three different desingularizations of have been constructed by Seshadri (Proceedings of the International Symposium on Algebraic Geometry, 1978, 155), Narasimhan–Ramanan (C. P. Ramanujam—A Tribute, 1978, 231), and Kirwan (Proc. London Math. Soc. 65(3) (1992) 474). In this paper, we construct a birational morphism from Kirwan''s desingularization to Narasimhan–Ramanan''s, and prove that the Narasimhan–Ramanan''s desingularization (called the moduli space of Hecke cycles) is the intermediate variety between Kirwan''s and Seshadri''s as was conjectured recently in (Math. Ann. 330 (2004) 491). As a by-product, we compute the cohomology of the moduli space of Hecke cycles. [Copyright &y& Elsevier]

The equivalence problem for linear differential operators of the second order, acting in vector bundles, is discussed. The field of rational invariants of symbols is described and connections, naturally associated with differential operators, are found. These geometrical structures are used to solve the problems of local as well as global equivalence of differential operators. [ABSTRACT FROM AUTHOR]

Abstract: Let be a complete connected smooth (compact) Riemannian manifold of dimension . Let be a smooth vector bundle over . Let be a second order differential operator on , where is a Laplace-Type operator on the sections of the vector bundle and a smooth vector field on . Let be the heat kernel of relative to . In this paper we will derive an exact and an asymptotic expansion for where is the center of normal coordinates defined on , is a point in the normal neighborhood centered at . The leading coefficients of the expansion are then computed at in terms of the linear and quadratic Riemannian curvature invariants of the Riemannian manifold , of the vector bundle , and of the vector bundle section and its derivatives. We end by comparing our results with those of previous authors (I. Avramidi, P. Gilkey, and McKean–Singer). [Copyright &y& Elsevier]