The strength for line bundles

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Edoardo Ballico ◽  
Emanuele Ventura
Keyword(s):  
Line Bundle ◽  
Algebraic Variety ◽  
Commutative Algebra ◽  
Upper Bounds ◽  
Line Bundles ◽  
Homogeneous Polynomials

We introduce the strength for sections of a line bundle on an algebraic variety. This generalizes the strength of homogeneous polynomials that has been recently introduced to resolve Stillman's conjecture, an important problem in commutative algebra. We establish the first properties of this notion and give some tool to obtain upper bounds on the strength in this framework. Moreover, we show some results on the usual strength such as the reducibility of the set of strength two homogeneous polynomials.

scholarly journals Upper bounds for some Brill–Noether loci over a finite field

2018 ◽  
Vol 14 (03) ◽  
pp. 739-749 ◽  
Author(s):  
Kamal Khuri-Makdisi
Keyword(s):  
Line Bundle ◽  
Finite Field ◽  
Classical Result ◽  
Base Point ◽  
Upper Bounds ◽  
Line Bundles ◽  
Effective Version ◽  
General Line ◽  
Genus Formula ◽  
Explicit Constant

Let [Formula: see text] be a smooth projective algebraic curve of genus [Formula: see text], over the finite field [Formula: see text]. A classical result of H. Martens states that the Brill–Noether locus of line bundles [Formula: see text] in [Formula: see text] with [Formula: see text] and [Formula: see text] is of dimension at most [Formula: see text], under conditions that hold when such an [Formula: see text] is both effective and special. We show that the number of such [Formula: see text] that are rational over [Formula: see text] is bounded above by [Formula: see text], with an explicit constant [Formula: see text] that grows exponentially with [Formula: see text]. Our proof uses the Weil estimates for function fields, and is independent of Martens’ theorem. We apply this bound to give a precise lower bound of the form [Formula: see text] for the probability that a line bundle in [Formula: see text] is base point free. This gives an effective version over finite fields of the usual statement that a general line bundle of degree [Formula: see text] is base point free. This is applicable to the author’s work on fast Jacobian group arithmetic for typical divisors on curves.

scholarly journals Projective surfaces with K-very ample line bundles of degree ≤ 4K + 4

1994 ◽  
Vol 136 ◽  
pp. 57-79 ◽  
Author(s):  
Edoardo Ballico ◽  
Andrew J. Sommese
Keyword(s):  
Line Bundle ◽  
Projective Space ◽  
Smooth Surface ◽  
Negative Integer ◽  
Line Bundles ◽  
Restriction Map ◽  
Projective Surface ◽  
Projective Surfaces

A line bundle, L, on a smooth, connected projective surface, S, is defined [7] to be k-very ample for a non-negative integer, k, if given any 0-dimensional sub-scheme with length , it follows that the restriction map is onto. L is 1-very ample (respectively 0-very ample) if and only if L is very ample (respectively spanned at all points by global sections). For a smooth surface, S, embedded in projective space by | L | where L is very ample, L being k-very ample is equivalent to there being no k-secant Pk−1 to S containing ≥ k + 1 points of S.

scholarly journals Determinants, Pfaffians and Quasi-Free Representations of the CAR Algebra

1998 ◽  
Vol 10 (05) ◽  
pp. 705-721 ◽  
Author(s):  
Mauro Spera ◽  
Tilmann Wurzbacher
Keyword(s):  
Hilbert Space ◽  
Line Bundle ◽  
Line Bundles ◽  
Complex Structures ◽  
Square Root ◽  
Purification Procedure ◽  
Algebraic Construction ◽  
Infinite Dimensional ◽  
Weil Type ◽  
Determinant Line Bundle

In this paper we apply the theory of quasi-free states of CAR algebras and Bogolubov automorphisms to give an alternative C*-algebraic construction of the Determinant and Pfaffian line bundles discussed by Pressley and Segal and by Borthwick. The basic property of the Pfaffian of being the holomorphic square root of the Determinant line bundle (after restriction from the Hilbert space Grassmannian to the Siegel manifold, or isotropic Grassmannian, consisting of all complex structures on an associated Hilbert space) is derived from a Fock–anti-Fock correspondence and an application of the Powers–Størmer purification procedure. A Borel–Weil type description of the infinite dimensional Spin c- representation is obtained, via a Shale–Stinespring implementation of Bogolubov transformations.

scholarly journals Numerical dimension and a Kawamata–Viehweg–Nadel-type vanishing theorem on compact Kähler manifolds

2014 ◽  
Vol 150 (11) ◽  
pp. 1869-1902 ◽  
Author(s):  
Junyan Cao
Keyword(s):  
Line Bundle ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Line Bundles ◽  
Kahler Manifolds ◽  
Vanishing Theorem ◽  
Kahler Manifold ◽  
Singular Metrics ◽  
Compact Kähler Manifold

AbstractLet $X$ be a compact Kähler manifold and let $(L,{\it\varphi})$ be a pseudo-effective line bundle on $X$. We first define a notion of numerical dimension for pseudo-effective line bundles with singular metrics, and then discuss the properties of this numerical dimension. Finally, we prove a very general Kawamata–Viehweg–Nadel-type vanishing theorem on an arbitrary compact Kähler manifold.

SINGULAR CURVES WITH LINE BUNDLES L DEFINED OVER ${\mathbb F}_q$ AND WITH H0(L) = H1(L) = 0

2013 ◽  
Vol 06 (02) ◽  
pp. 1350023
Author(s):  
Edoardo Ballico
Keyword(s):  
Line Bundle ◽  
Line Bundles ◽  
Projective Curve ◽  
Similar Line

Here we prove the existence of several pairs (X, L), where X is a geometrically integral projective curve defined over 𝔽q and L is a line bundle on X defined over 𝔽q and with H0(X, L) = H1(X, L) = 0. These examples are obtained using the existence of similar line bundles on the normalization of X, i.e. a case studied by C. Ballet, C. Ritzenthaler and R. Roland.

scholarly journals The Morse inequalities for line bundles

1970 ◽  
Vol 11 (3) ◽  
pp. 260-264
Author(s):  
Samir Khabbaz
Keyword(s):  
Line Bundle ◽  
Critical Point ◽  
Critical Points ◽  
Differentiable Manifold ◽  
Line Bundles ◽  
Structural Group ◽  
Morse Inequalities ◽  
Degenerate Critical Point ◽  
Two Indices

In place of a real valued differentiable (C2) function on a closedn-dimensional differentiable manifoldM, we may more generally consider a differentiable section s in any line bundleLonM, assumed to have structural groupZ2, the group of integers modulo two. Since the usual definitions of a critical point and of a non-degenerate critical point are local in nature, and since composing a differentiable real valued function with the functiont→—t does not change its set of critical points or its set of non-degenerate critical point, it is clear that we may speak of critical points and nondegenerate critical points of the section s. Unless the bundle has a fixed trivialization however, the index of a non-degenerate critical point must be thought of as a set of two numbers {k, n—k), corresponding to the two indices arising from the two trivializations possible forLrestricted to a small enough neighborhood of the point, i.e. corresponding to the two possible ways of reading the index. With this understanding we extend the usual definitions, and call a differentiable (C2) section s of L a Morse section if each of its critical points is non-degenerate.

scholarly journals Positivity properties of metrics and delta-forms

2019 ◽  
Vol 2019 (752) ◽  
pp. 141-177 ◽  
Author(s):  
Walter Gubler ◽  
Klaus Künnemann
Keyword(s):  
Algebraic Variety ◽  
Abelian Varieties ◽  
Toric Varieties ◽  
Convex Geometry ◽  
Piecewise Smooth ◽  
Line Bundles ◽  
Canonical Metrics ◽  
On Line

Abstract In previous work, we have introduced δ-forms on the Berkovich analytification of an algebraic variety in order to study smooth or formal metrics via their associated Chern δ-forms. In this paper, we investigate positivity properties of δ-forms and δ-currents. This leads to various plurisubharmonicity notions for continuous metrics on line bundles. In the case of a formal metric, we show that many of these positivity notions are equivalent to Zhang’s semipositivity. For piecewise smooth metrics, we prove that plurisubharmonicity can be tested on tropical charts in terms of convex geometry. We apply this to smooth metrics, to canonical metrics on abelian varieties and to toric metrics on toric varieties.

scholarly journals Remarks on Determinant Line Bundles, Chern-Simons Forms and Invariants

2002 ◽  
Vol 91 (1) ◽  
pp. 5 ◽  
Author(s):  
Johan L. Dupont ◽  
Flemming Lindblad Johansen
Keyword(s):  
Line Bundle ◽  
Line Bundles ◽  
Manifolds With Boundary ◽  
Principal Bundles ◽  
Determinant Line Bundles ◽  
Chern Simons ◽  
Determinant Line Bundle

We study generalized determinant line bundles for families of principal bundles and connections. We explore the connections of this line bundle and give conditions for the uniqueness of such. Furthermore we construct for families of bundles and connections over manifolds with boundary, a generalized Chern-Simons invariant as a section of a determinant line bundle.

scholarly journals On the growth of von Neumann dimension of harmonic spaces of semipositive line bundles over covering manifolds

2016 ◽  
Vol 27 (11) ◽  
pp. 1650093 ◽  
Author(s):  
Huan Wang
Keyword(s):  
Line Bundle ◽  
Main Tool ◽  
Line Bundles ◽  
Harmonic Space ◽  
Dolbeault Cohomology ◽  
Harmonic Forms ◽  
Von Neumann ◽  
Hermitian Manifolds ◽  
Von Neumann Dimension ◽  
Tensor Powers

We study the harmonic space of line bundle valued forms over a covering manifold with a discrete group action, and obtain an asymptotic estimate for the von Neumann dimension of the space of harmonic [Formula: see text]-forms with values in high tensor powers of a semipositive line bundle. In particular, we estimate the von Neumann dimension of the corresponding reduced [Formula: see text]-Dolbeault cohomology group. The main tool is a local estimate of the pointwise norm of harmonic forms with values in semipositive line bundles over Hermitian manifolds.

scholarly journals CHEN–RUAN COHOMOLOGY OF SOME MODULI SPACES, II

2010 ◽  
Vol 21 (04) ◽  
pp. 497-522 ◽  
Author(s):  
INDRANIL BISWAS ◽  
MAINAK PODDAR
Keyword(s):  
Riemann Surface ◽  
Line Bundle ◽  
Prime Number ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Line Bundles ◽  
Holomorphic Line Bundle ◽  
Stable Vector Bundles

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξ → X a holomorphic line bundle such that r is not a divisor of degree ξ. Let [Formula: see text] denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γ we will denote the group of line bundles L over X such that L⊗r is trivial. This group Γ acts on [Formula: see text] by the rule (E, L) ↦ E ⊗ L. We compute the Chen–Ruan cohomology of the corresponding orbifold.

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