scholarly journals Effective Faithful Tropicalizations Associated to Adjoint Linear Systems

2018 ◽  
Vol 2019 (19) ◽  
pp. 6089-6112
Author(s):  
Shu Kawaguchi ◽  
Kazuhiko Yamaki
Keyword(s):  
Linear System ◽  
Line Bundle ◽  
Projective Variety ◽  
Ample Line Bundle ◽  
Smooth Projective Variety ◽  
Discrete Valuation Ring ◽  
Complete Discrete Valuation Ring ◽  
System L ◽  
Semistable Model ◽  
Adjoint Bundle

Abstract Let R be a complete discrete valuation ring of equi-characteristic zero with fraction field K. Let X be a connected smooth projective variety of dimension d over K, and let L be an ample line bundle over X. We assume that there exist a regular strictly semistable model ${\mathscr {X}}$ of X over R and a relatively ample line bundle ${\mathscr {L}}$ over ${\mathscr {X}}$ with $\left .{{\mathscr {L}}}\right \vert_{{X}} \cong L$. Let $S({\mathscr {X}})$ be the skeleton associated to ${\mathscr {X}}$ in the Berkovich analytification Xan of X. In this article, we study when $S({\mathscr {X}})$ is faithfully tropicalized into tropical projective space by the adjoint linear system |L⊗m ⊗ ωX|. Roughly speaking, our results show that if m is an integer such that the adjoint bundle is basepoint free, then the adjoint linear system admits a faithful tropicalization of $S({\mathscr {X}})$.

SEMISTABILITY AND NUMERICALLY EFFECTIVENESS IN POSITIVE CHARACTERISTIC

2011 ◽  
Vol 22 (01) ◽  
pp. 25-46 ◽  
Author(s):  
INDRANIL BISWAS ◽  
YOGISH I. HOLLA
Keyword(s):  
Line Bundle ◽  
Parabolic Subgroup ◽  
Projective Variety ◽  
Positive Characteristic ◽  
Ample Line Bundle ◽  
Smooth Projective Variety ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Smooth Projective Curve ◽  
Trivial Center

Let G be a connected reductive linear algebraic group defined over an algebraically closed field k of positive characteristic. Let Z(G) ⊂ G be the center, and [Formula: see text], where each Gi is simple with trivial center. For i ∈ [1, m], let ρi : G → Gi be the natural projection. Fix a proper parabolic subgroup P of G such that for each i ∈ [1, m], the image ρi(G) ⊂ Gi is a proper parabolic subgroup. Fix a strictly anti-dominant character χ of P such that χ is trivial on Z(G). Let M be a smooth projective variety, defined over k, equipped with a very ample line bundle ξ. Let EG → M be a principal G-bundle. We prove that the following six statements are equivalent: (1) The line bundle EG(χ) → EG/P associated to the principal P-bundle EG → EG/P for the character χ is numerically effective. (2) The sequence of principal G-bundles [Formula: see text] is bounded, where FM is the absolute Frobenius morphism of M. (3) The principal G-bundle EG is strongly semistable with respect to ξ, and c2( ad (EG)) is numerically equivalent to zero. (4) The principal G-bundle EG is strongly semistable with respect to ξ, and [c2( ad (EG))c1(ξ)d-2] = 0. (5) The adjoint vector bundle ad (EG) is numerically effective. (6) For every pair of the form (Y,ψ), where Y is an irreducible smooth projective curve and ψ : Y → M is a morphism, the principal G-bundle ψ*EG → Y is semistable.

scholarly journals A lower bound for $$\chi ({\mathcal {O}}_S)$$

2021 ◽  
Author(s):  
Vincenzo Di Gennaro
Keyword(s):  
Lower Bound ◽  
Linear System ◽  
Line Bundle ◽  
Hyperplane Section ◽  
Ample Line Bundle ◽  
Complex Surface ◽  
Veronese Surface ◽  
Very Ample Line Bundle ◽  
Rational Normal Scroll

AbstractLet $$(S,{\mathcal {L}})$$ ( S , L ) be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $${\mathcal {L}}$$ L of degree $$d > 25$$ d > 25 . In this paper we prove that $$\chi (\mathcal O_S)\ge -\frac{1}{8}d(d-6)$$ χ ( O S ) ≥ - 1 8 d ( d - 6 ) . The bound is sharp, and $$\chi ({\mathcal {O}}_S)=-\frac{1}{8}d(d-6)$$ χ ( O S ) = - 1 8 d ( d - 6 ) if and only if d is even, the linear system $$|H^0(S,{\mathcal {L}})|$$ | H 0 ( S , L ) | embeds S in a smooth rational normal scroll $$T\subset {\mathbb {P}}^5$$ T ⊂ P 5 of dimension 3, and here, as a divisor, S is linearly equivalent to $$\frac{d}{2}Q$$ d 2 Q , where Q is a quadric on T. Moreover, this is equivalent to the fact that a general hyperplane section $$H\in |H^0(S,{\mathcal {L}})|$$ H ∈ | H 0 ( S , L ) | of S is the projection of a curve C contained in the Veronese surface $$V\subseteq {\mathbb {P}}^5$$ V ⊆ P 5 , from a point $$x\in V\backslash C$$ x ∈ V \ C .

ON THE DEGREE AND THE BIRATIONALITY OF THE SECOND ADJUNCTION MAPPING

1999 ◽  
Vol 10 (06) ◽  
pp. 707-719 ◽  
Author(s):  
MAURO C. BELTRAMETTI ◽  
ANDREW J. SOMMESE
Keyword(s):  
Linear System ◽  
Line Bundle ◽  
Mild Condition ◽  
Three Dimensional ◽  
Ample Line Bundle ◽  
Kodaira Dimension ◽  
Projective Manifold ◽  
Very Ample Line Bundle ◽  
System 2 ◽  
Complete Linear System

Let ℒ be a very ample line bundle on ℳ, a projective manifold of dimension n ≥3. Under the assumption that Kℳ + (n-2) ℒ has Kodaira dimension n, we study the degree of the map ϕ associated to the complete linear system |2(KM + (n-2) L)|, where (M, L) is the first reduction of (ℳ, ℒ). In particular we show that under a number of conditions, e.g. n ≥ 5 or Kℳ + (n-3)ℒ having nonnegative Kodaira dimension, the degree of ϕ is one, i.e. ϕ is birational. We also show that under a mild condition on the linear system |KM + (n-2) L| satisfied for all known examples, ϕ is birational unless (ℳ, ℒ) is a three dimensional variety with very restricted invariants. Moreover there is an example with these invariants such that deg ϕ= 2.

scholarly journals On relative base point freeness of adjoint bundle

1997 ◽  
Vol 146 ◽  
pp. 185-197
Author(s):  
Shigeharu Takayama
Keyword(s):  
Line Bundle ◽  
Ample Line Bundle ◽  
Base Point ◽  
Adjoint Bundle ◽  
Projective Morphism ◽  
Effective Result

Abstract.We give an effective result on the relative base point freeness of an adjoint bundle for a pair of a projective morphism and a relatively ample line bundle.

scholarly journals On principal bundles over a projective variety defined over a finite field

2009 ◽  
Vol 4 (2) ◽  
pp. 209-221 ◽  
Author(s):  
Indranil Biswas
Keyword(s):  
Line Bundle ◽  
Finite Field ◽  
Fundamental Group ◽  
Projective Variety ◽  
Rational Point ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Linear Algebraic Group ◽  
The Third ◽  
Extra Assumption

AbstractLet M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x0. Let (M,x0/ denote the corresponding fundamental group-scheme introduced by Nori. Let EG be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξ on M. We prove that the following three statements are equivalent:1. The principal G-bundle EG over M is given by a homomorphism (M,x0)→G.2. There are integers b > a ≥ 1, such that the principal G-bundle (FbM)* EG is isomorphic to (FaM) * EG where FM is the absolute Frobenius morphism of M.3. The principal G-bundle EG is strongly semistable, the degree(c2(ad(EG))c1 (ξ)d−2 = 0, where d = dimM, and the degree(c1(EG(χ))c1(ξ)d−1) = 0 for every character χ of G, where EG(χ) is the line bundle over M associated to EG for χ.In [16], the equivalence between the first statement and the third statement was proved under the extra assumption that dimM = 1 and G is semisimple.

On a criterion for hyperplane sections

1988 ◽  
Vol 103 (1) ◽  
pp. 59-67 ◽  
Author(s):  
Lucian Bǎdescu
Keyword(s):  
Line Bundle ◽  
Normal Bundle ◽  
Projective Variety ◽  
Ample Line Bundle ◽  
Cartier Divisor ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Hyperplane Sections

Throughout this paper we shall fix an algebraically closed field k. Consider the following:Problem. Let (Y, L) be a normal polarized variety over k, i.e. a normal projective variety Y over k together with an ample line bundle L on Y. Then one may ask under which conditions the following statement holds:(*) Every normal projective variety X containing Y as an ample Cartier divisor such that the normal bundle of Y in X is L, is isomorphic to the projective cone over Y.

scholarly journals Manifolds Covered by Lines and Extremal Rays

2012 ◽  
Vol 55 (4) ◽  
pp. 799-814 ◽  
Author(s):  
Carla Novelli ◽  
Gianluca Occhetta
Keyword(s):  
Line Bundle ◽  
Projective Variety ◽  
Ample Line Bundle ◽  
Rational Curves ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Cone Of Curves ◽  
Complex Projective ◽  
Extremal Ray

AbstractLet X be a smooth complex projective variety, and let H ∈ Pic(X) be an ample line bundle. Assume that X is covered by rational curves with degree one with respect to H and with anticanonical degree greater than or equal to (dimX – 1)/2. We prove that there is a covering family of such curves whose numerical class spans an extremal ray in the cone of curves NE(X).

scholarly journals Homotopy abelianity of the DG-Lie algebra controlling deformations of pairs (variety with trivial canonical bundle, line bundle)

2021 ◽  
pp. 2150086
Author(s):  
Donatella Iacono ◽  
Marco Manetti
Keyword(s):  
Line Bundle ◽  
Lie Algebra ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Canonical Bundle ◽  
Algebraically Closed Field

We investigate the deformations of pairs [Formula: see text], where [Formula: see text] is a line bundle on a smooth projective variety [Formula: see text], defined over an algebraically closed field [Formula: see text] of characteristic 0. In particular, we prove that the DG-Lie algebra controlling the deformations of the pair [Formula: see text] is homotopy abelian whenever [Formula: see text] has trivial canonical bundle, and so these deformations are unobstructed.

scholarly journals Log Picard Algebroids and Meromorphic Line Bundles

2019 ◽  
Author(s):  
Marco Gualtieri ◽  
Kevin Luk
Keyword(s):  
Line Bundle ◽  
Projective Variety ◽  
Hodge Structure ◽  
Smooth Projective Variety ◽  
Line Bundles ◽  
Mixed Hodge Structure ◽  
Natural Class ◽  
De Rham ◽  
Integrality Condition ◽  
Holomorphic Line Bundles

Abstract We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure. We then solve the prequantization problem, showing that under the appropriate integrality condition, a log Picard algebroid is the Lie algebroid of symmetries of what is called a meromorphic line bundle, a generalization of the usual notion of line bundle in which the fibers degenerate along the divisor. We give a geometric description of such bundles and establish a classification theorem for them, showing that they correspond to a subgroup of the holomorphic line bundles on the divisor complement. Importantly, these holomorphic line bundles need not be algebraic. Finally, we provide concrete methods for explicitly constructing examples of meromorphic line bundles, such as for a smooth cubic divisor in the projective plane.

scholarly journals Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type

2021 ◽  
Author(s):  
Lie Fu ◽  
Robert Laterveer ◽  
Charles Vial
Keyword(s):  
Projective Variety ◽  
Arbitrary Dimension ◽  
Smooth Projective Variety ◽  
The Other ◽  
Fano Variety ◽  
Fano Varieties ◽  
Intersection Product ◽  
Canonical Class ◽  
Main Input

AbstractGiven a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.

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