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 ADJUNCTION FOR VARIETIES WITH A ℂ* ACTION

2020 ◽  
Author(s):  
ELEONORA A. ROMANO ◽  
JAROSŁAW A. WIŚNIEWSKI
Keyword(s):  
Fixed Point ◽  
Line Bundle ◽  
Normal Bundle ◽  
Ample Line Bundle ◽  
Projective Manifold ◽  
Fano Manifolds ◽  
Adjunction Theory ◽  
Rank 2 ◽  
General Orbit ◽  
Complex Projective

Abstract Let X be a complex projective manifold, L an ample line bundle on X, and assume that we have a ℂ* action on (X;L). We classify such triples (X; L;ℂ*) for which the closure of a general orbit of the ℂ* action is of degree ≤ 3 with respect to L and, in addition, the source and the sink of the action are isolated fixed points, and the ℂ* action on the normal bundle of every fixed point component has weights ±1. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension 11 and 13 are homogeneous if their group of automorphisms is reductive of rank ≥ 2.

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 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 Explicit Noether–Lefschetz for arbitrary threefolds

2007 ◽  
Vol 143 (2) ◽  
pp. 323-342 ◽  
Author(s):  
ANGELO FELICE LOPEZ ◽  
CATRIONA MACLEAN
Keyword(s):  
Line Bundle ◽  
Ample Line Bundle ◽  
Very Ample Line Bundle ◽  
Regularity Properties ◽  
Explicit Bounds

AbstractWe study the Noether–Lefschetz locus of a very ample line bundle L on an arbitrary smooth threefold Y. Building on results of Green, Voisin and Otwinowska, we give explicit bounds, depending only on the Castelnuovo–Mumford regularity properties of L, on the codimension of the components of the Noether–Lefschetz locus of |L|.

scholarly journals EXPECTED TOPOLOGY OF RANDOM REAL ALGEBRAIC SUBMANIFOLDS

2014 ◽  
Vol 14 (4) ◽  
pp. 673-702 ◽  
Author(s):  
Damien Gayet ◽  
Jean-Yves Welschinger
Keyword(s):  
Holomorphic Vector Bundle ◽  
Betti Numbers ◽  
Ample Line Bundle ◽  
Connected Components ◽  
Projective Manifold ◽  
Positive Probability ◽  
Random Real ◽  
Smooth Complex ◽  
Real Locus ◽  
Complex Projective

Let$X$be a smooth complex projective manifold of dimension$n$equipped with an ample line bundle$L$and a rank$k$holomorphic vector bundle$E$. We assume that$1\leqslant k\leqslant n$, that$X$,$E$and$L$are defined over the reals and denote by$\mathbb{R}X$the real locus of$X$. Then, we estimate from above and below the expected Betti numbers of the vanishing loci in$\mathbb{R}X$of holomorphic real sections of$E\otimes L^{d}$, where$d$is a large enough integer. Moreover, given any closed connected codimension$k$submanifold${\it\Sigma}$of$\mathbb{R}^{n}$with trivial normal bundle, we prove that a real section of$E\otimes L^{d}$has a positive probability, independent of$d$, of containing around$\sqrt{d}^{n}$connected components diffeomorphic to${\it\Sigma}$in its vanishing locus.

scholarly journals Classification of algebraic non-ruled surfaces with sectional genus less than or equal to six

1985 ◽  
Vol 100 ◽  
pp. 1-9 ◽  
Author(s):  
Elvira Laura Livorni
Keyword(s):  
Line Bundle ◽  
Hyperplane Section ◽  
Ample Line Bundle ◽  
Ruled Surfaces ◽  
Sectional Genus ◽  
Rational Surfaces ◽  
Very Ample Line Bundle ◽  
Similar Classification

In this paper we have given a biholomorphic classification of smooth, connected, protective, non-ruled surfaces X with a smooth, connected, hyperplane section C relative to L, where L is a very ample line bundle on X, such that g = g(C) = g(L) is less than or equal to six. For a similar classification of rational surfaces with the same conditions see [Li].

On Topological Properties of Some Coverings. An Addendum to a Paper of Lanteri and Struppa

1992 ◽  
Vol 44 (1) ◽  
pp. 206-214
Author(s):  
Jarosław A. Wiśniewski
Keyword(s):  
Line Bundle ◽  
Projective Space ◽  
Betti Numbers ◽  
Ample Line Bundle ◽  
Double Cover ◽  
Topological Properties ◽  
Surjective Morphism ◽  
Projective Manifolds ◽  
Dimensional Projective Space ◽  
Complex Projective

AbstractLet π: X′ → X be a finite surjective morphism of complex projective manifolds which can be factored by an embedding of X′ into the total space of an ample line bundle 𝓛 over X. A theorem of Lazarsfeld asserts that Betti numbers of X and X′ are equal except, possibly, the middle ones. In the present paper it is proved that the middle numbers are actually non-equal if either 𝓛 is spanned and deg π ≥ dim X, or if X is either a hyperquadric or a projective space and π is not a double cover of an odd-dimensional projective space by a hyperquadric.

scholarly journals Formality conjecture for K3 surfaces

2019 ◽  
Vol 155 (5) ◽  
pp. 902-911 ◽  
Author(s):  
Nero Budur ◽  
Ziyu Zhang
Keyword(s):  
Line Bundle ◽  
Stability Condition ◽  
Ample Line Bundle ◽  
Main Tool ◽  
K3 Surfaces ◽  
Derived Category ◽  
K3 Surface ◽  
Coherent Sheaves ◽  
Dg Algebra ◽  
Complex Projective

We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the differential graded (DG) algebra$\operatorname{RHom}^{\bullet }(F,F)$is formal for any sheaf$F$polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.

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 Special Ulrich bundles on non-special surfaces with pg = q = 0

2017 ◽  
Vol 28 (08) ◽  
pp. 1750061 ◽  
Author(s):  
Gianfranco Casnati
Keyword(s):  
Line Bundle ◽  
Recent Result ◽  
Ample Line Bundle ◽  
Rational Surface ◽  
Enriques Surface ◽  
Very Ample Line Bundle ◽  
Dimension Formula ◽  
Special Surface

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

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