scholarly journals Global generation of adjoint bundles

1996 ◽  
Vol 142 ◽  
pp. 5-16 ◽  
Author(s):  
Hajime Tsuji
Keyword(s):  
Line Bundle ◽  
Ample Line Bundle ◽  
Base Point ◽  
Projective Surface ◽  
Smooth Projective Surface

In 1988, I. Reider proved that for a smooth projective surface X and an ample line bundle L on X, Kx + 3L is globally generated and Kx + 4L is very ample ([12]). In fact his theorem is much stronger than this (see [12] for detail). Recently a lot of results have been obtained about effective base point freeness (cf. [1, 3, 8, 13, 14, 15]). In particular J. P. Demailly proved that 2KX + 12nnL is very ample for a smooth projective n-fold X and an ample line bundle L on X. [2] will give a good overview for these recent results. The motivation of these works is the following conjecture posed by T. Fujita.

scholarly journals Towards the weighted Bounded Negativity Conjecture for blow-ups of algebraic surfaces

2019 ◽  
Vol 163 (3-4) ◽  
pp. 361-373
Author(s):  
Roberto Laface ◽  
Piotr Pokora
Keyword(s):  
Line Bundle ◽  
Characteristic Zero ◽  
Algebraic Surface ◽  
Intersection Number ◽  
Algebraic Surfaces ◽  
The Self ◽  
Projective Surface ◽  
Intersection Numbers ◽  
Weighted Version ◽  
Smooth Projective Surface

AbstractIn the present paper we focus on a weighted version of the Bounded Negativity Conjecture, which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a function depending on the intesection of curve with an arbitrary big and nef line bundle that is positive on the curve. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.

The sheaf of relative differentials of a fibred surface

1993 ◽  
Vol 114 (3) ◽  
pp. 461-470
Author(s):  
Fernando Serrano
Keyword(s):  
Linear Systems ◽  
Smooth Curve ◽  
Base Point ◽  
Projective Surface ◽  
Positive Part ◽  
Rational Singularities ◽  
Smooth Projective Surface ◽  
Genus 2

AbstractLet Φ: S → C denote a fibration from a smooth projective surface onto a smooth curve, with fibres of genus ≥2. The double dual of the sheaf of relative differentials has been studied by F. Serrano [14]. There, it was proved that dim grows asymptotically as the square of n in case Φ is not isotrivial (i.e. fibres vary in modulus), and the converse holds true in most cases, in a way that can be made precise. In the non-isotrivial case, the present paper provides further information about by analysing the linear systems for large n. If P denotes the positive part of in its Zariski decomposition, then it is shown that |rP| is eventually base-point free for some r > 0. Furthermore, Proj is a normal projective surface, fibred over C, birational to S, and with only rational singularities.

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 Universal polynomials for singular curves on surfaces

2014 ◽  
Vol 150 (7) ◽  
pp. 1169-1182 ◽  
Author(s):  
Jun Li ◽  
Yu-jong Tzeng
Keyword(s):  
Vector Bundle ◽  
Linear System ◽  
Line Bundle ◽  
Vector Bundles ◽  
Projective Surface ◽  
Analytic Singularity ◽  
Smooth Projective Surface ◽  
Chern Numbers ◽  
Cutting Out ◽  
Curves On Surfaces

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ be a complex smooth projective surface and $L$ be a line bundle on $S$. For any given collection of isolated topological or analytic singularity types, we show the number of curves in the linear system $|L|$ with prescribed singularities is a universal polynomial of Chern numbers of $L$ and $S$, assuming $L$ is sufficiently ample. More generally, we show for vector bundles of any rank and smooth varieties of any dimension, similar universal polynomials also exist and equal the number of singular subvarieties cutting out by sections of the vector bundle. This work is a generalization of Göttsche’s conjecture.

scholarly journals On the set of divisors with zero geometric defect

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinh Tuan Huynh ◽  
Duc-Viet Vu
Keyword(s):  
Line Bundle ◽  
Zero Divisor ◽  
Ample Line Bundle ◽  
Holomorphic Section ◽  
Projective Manifold ◽  
Holomorphic Curve ◽  
Very Ample Line Bundle ◽  
Complex Projective ◽  
Geometric Defect

AbstractLet {f:\mathbb{C}\to X} be a transcendental holomorphic curve into a complex projective manifold X. Let L be a very ample line bundle on {X.} Let s be a very generic holomorphic section of L and D the zero divisor given by {s.} We prove that the geometric defect of D (defect of truncation 1) with respect to f is zero. We also prove that f almost misses general enough analytic subsets on X of codimension 2.

scholarly journals Some remarks on the Gromov width of homogeneous Hodge manifolds

2014 ◽  
Vol 11 (09) ◽  
pp. 1460029 ◽  
Author(s):  
Andrea Loi ◽  
Roberto Mossa ◽  
Fabio Zuddas
Keyword(s):  
Line Bundle ◽  
Upper Bound ◽  
Ample Line Bundle ◽  
Seshadri Constant ◽  
Gromov Width

We provide an upper bound for the Gromov width of compact homogeneous Hodge manifolds (M, ω) with b2(M) = 1. As an application we obtain an upper bound on the Seshadri constant ϵ(L) where L is the ample line bundle on M such that [Formula: see text].

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 .

scholarly journals Symplectic Capacities of Toric Manifolds and Related Results

2006 ◽  
Vol 181 ◽  
pp. 149-184 ◽  
Author(s):  
Guangcun Lu
Keyword(s):  
Line Bundle ◽  
Blow Up ◽  
Ample Line Bundle ◽  
Symplectic Manifolds ◽  
Toric Manifolds ◽  
Seshadri Constant ◽  
Combinatorial Data

AbstractIn this paper we give concrete estimations for the pseudo symplectic capacities of toric manifolds in combinatorial data. Some examples are given to show that our estimates can compute their pseudo symplectic capacities. As applications we also estimate the symplectic capacities of the polygon spaces. Other related results are impacts of symplectic blow-up on symplectic capacities, symplectic packings in symplectic toric manifolds, the Seshadri constant of an ample line bundle on toric manifolds, and symplectic capacities of symplectic manifolds withS1-action.

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

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