Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles

2008 ◽  
Vol 144 (1) ◽  
pp. 109-118 ◽  
Author(s):  
ANTONIO LANTERI ◽  
HIDETOSHI MAEDA
Keyword(s):  
Vector Bundles ◽  
Projective Variety ◽  
Ample Line Bundle ◽  
Global Section ◽  
Sectional Genus ◽  
Ample Vector Bundle ◽  
Complex Projective Variety ◽  
Projective Manifolds ◽  
Complex Projective ◽  
Zero Locus

AbstractLet ϵ be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of ϵ whose zero locus Z is a smooth subvariety of dimension n-r ≥ 2 of X. Let H be an ample line bundle on X such that the restriction HZ of H to Z is very ample. Triplets (X, ϵ, H) with g(Z, HZ) = 3 are classified, where g(Z, HZ) is the sectional genus of (Z, HZ).

Ample Vector Bundles of Curve Genus One

1999 ◽  
Vol 42 (2) ◽  
pp. 209-213 ◽  
Author(s):  
Antonio Lanteri ◽  
Hidetoshi Maeda
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Projective Variety ◽  
Ample Vector Bundle ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Curve Genus ◽  
Genus One ◽  
Complex Projective ◽  
Zero Locus

AbstractWe investigate the pairs (X, ε) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle ε of rank n − 1 on X such that ε has a section whose zero locus is a smooth elliptic curve.

scholarly journals Some Adjunction Properties of Ample Vector Bundles

2001 ◽  
Vol 44 (4) ◽  
pp. 452-458
Author(s):  
Hironobu Ishihara
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Projective Variety ◽  
Sectional Genus ◽  
Genus Zero ◽  
Ample Vector Bundle

AbstractLet ε be an ample vector bundle of rank r on a projective variety X with only log-terminal singularities. We consider the nefness of adjoint divisors KX +(t−r) det ε when t ≥ dim X and t > r. As an application, we classify pairs (X, ε) with cr-sectional genus zero.

Special varieties in adjunction theory and ample vector bundles

2001 ◽  
Vol 130 (1) ◽  
pp. 61-75 ◽  
Author(s):  
A. LANTERI ◽  
H. MAEDA
Keyword(s):  
Vector Bundles ◽  
Projective Variety ◽  
Global Section ◽  
Smooth Projective Variety ◽  
Ample Vector Bundle ◽  
Projective Varieties ◽  
Adjunction Theory ◽  
Set Up ◽  
Important Position ◽  
Smooth Projective Varieties

In this paper varieties are always assumed to be defined over the field [Copf ] of complex numbers.Given a smooth projective variety Z, the classification of smooth projective varieties X containing Z as an ample divisor occupies an extremely important position in the theory of polarized varieties and it is well-known that the structure of Z imposes severe restrictions on that of X. Inspired by this philosophy, we set up the following condition ([midast ]) in [LM1] in order to generalize several results on ample divisors to ample vector bundles:([midast ]) [Escr ] is an ample vector bundle of rank r [ges ] 2 on a smooth projective variety X of dimension n such that there exists a global section s ∈ Γ([Escr ]) whose zero locus Z = (s)0 is a smooth subvariety of X of dimension n − r [ges ] 1.

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 AMPLE VECTOR BUNDLES WITH SECTIONS VANISHING ON SPECIAL VARIETIES

1999 ◽  
Vol 10 (06) ◽  
pp. 677-696 ◽  
Author(s):  
MARCO ANDREATTA ◽  
GIANLUCA OCCHETTA
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Projective Manifold ◽  
Ample Vector Bundle ◽  
Smooth Submanifold ◽  
Del Pezzo ◽  
Complex Projective ◽  
Zero Locus

Let ε be an ample vector bundle of rank r on a complex projective manifold X such that there exists a section s ∈ Γ(ε) whose zero locus Z = (s = 0) is a smooth submanifold of the expected dimension dim X-r:= n - r. Assume that Z is not minimal; we investigate the hypothesis under which the extremal contractions of Z can be lifted to X. Finally we study in detail the cases in which Z is a scroll, a quadric bundle or a del Pezzo fibration.

scholarly journals LINEAR SYSTEMS ON IRREGULAR VARIETIES

2019 ◽  
Vol 19 (6) ◽  
pp. 2087-2125 ◽  
Author(s):  
Miguel Ángel Barja ◽  
Rita Pardini ◽  
Lidia Stoppino
Keyword(s):  
Continuous Function ◽  
Line Bundle ◽  
Linear Systems ◽  
Projective Variety ◽  
Geometrical Properties ◽  
Complex Projective Variety ◽  
Wide Range ◽  
Image Position ◽  
Irregular Varieties ◽  
Complex Projective

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety of dimension $m$ (possibly $T=X$) and $a:X\rightarrow A$ a morphism to an abelian variety such that $\text{Pic}^{0}(A)$ injects into $\text{Pic}^{0}(T)$; let $L$ be a line bundle on $X$ and $\unicode[STIX]{x1D6FC}\in \text{Pic}^{0}(A)$ a general element.We introduce two new ingredients for the study of linear systems on $X$. First of all, we show the existence of a factorization of the map $a$, called the eventual map of $L$ on $T$, which controls the behavior of the linear systems $|L\otimes \unicode[STIX]{x1D6FC}|_{|T}$, asymptotically with respect to the pullbacks to the connected étale covers $X^{(d)}\rightarrow X$ induced by the $d$-th multiplication map of $A$.Second, we define the so-called continuous rank function$x\mapsto h_{a}^{0}(X_{|T},L+xM)$, where $M$ is the pullback of an ample divisor of $A$. This function extends to a continuous function of $x\in \mathbb{R}$, which is differentiable except possibly at countably many points; when $X=T$ we compute the left derivative explicitly.As an application, we give quick short proofs of a wide range of new Clifford–Severi inequalities, i.e., geographical bounds of the form $$\begin{eqnarray}\displaystyle \text{vol}_{X|T}(L)\geqslant C(m)h_{a}^{0}(X_{|T},L), & & \displaystyle \nonumber\end{eqnarray}$$ where $C(m)={\mathcal{O}}(m!)$ depends on several geometrical properties of $X$, $L$ or $a$.

scholarly journals On the geometry of Poincaré's problem for one-dimensional projective foliations

2001 ◽  
Vol 73 (4) ◽  
pp. 475-482 ◽  
Author(s):  
MARCIO G. SOARES
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Projective Variety ◽  
Holomorphic Foliation ◽  
One Dimensional ◽  
Complex Projective Variety ◽  
Geometric Objects ◽  
Complex Projective

We consider the question of relating extrinsic geometric characters of a smooth irreducible complex projective variety, which is invariant by a one-dimensional holomorphic foliation on a complex projective space, to geometric objects associated to the foliation.

Vector Fields and the Cohomology Ring of Toric Varieties

2005 ◽  
Vol 48 (3) ◽  
pp. 414-427 ◽  
Author(s):  
Kiumars Kaveh
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Projective Variety ◽  
Cohomology Ring ◽  
Holomorphic Vector Field ◽  
Zero Set ◽  
Graded Algebras ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective

AbstractLetXbe a smooth complex projective variety with a holomorphic vector field with isolated zero setZ. From the results of Carrell and Lieberman there exists a filtrationF0⊂F1⊂ · · · ofA(Z), the ring of ℂ-valued functions onZ, such thatas graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a 1-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra ofX.

scholarly journals Hyperbolicity related problems for complete intersection varieties

2013 ◽  
Vol 150 (3) ◽  
pp. 369-395 ◽  
Author(s):  
Damian Brotbek
Keyword(s):  
Projective Space ◽  
Existence Theorem ◽  
Complete Intersection ◽  
Cotangent Bundle ◽  
Projective Variety ◽  
Complete Intersections ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective ◽  
Intersection Surface

AbstractIn this paper we examine different problems regarding complete intersection varieties of high multidegree in a smooth complex projective variety. First we prove an existence theorem for jet differential equations that generalizes a theorem of Diverio. Then we show how one can deduce hyperbolicity for generic complete intersections of high multidegree and high codimension from the known results on hypersurfaces. Finally, motivated by a conjecture of Debarre, we focus on the positivity of the cotangent bundle of complete intersections, and prove some results towards this conjecture; among other things, we prove that a generic complete intersection surface of high multidegree in a projective space of dimension at least four has an ample cotangent bundle.

scholarly journals Nef anti-canonical divisors and rationally connected fibrations

2019 ◽  
Vol 155 (7) ◽  
pp. 1444-1456
Author(s):  
Sho Ejiri ◽  
Yoshinori Gongyo
Keyword(s):  
Projective Variety ◽  
Kodaira Dimension ◽  
Canonical Divisor ◽  
Complex Projective Variety ◽  
Rationally Connected ◽  
Complex Projective

We study the Iitaka–Kodaira dimension of nef relative anti-canonical divisors. As a consequence, we prove that given a complex projective variety with klt singularities, if the anti-canonical divisor is nef, then the dimension of a general fibre of the maximal rationally connected fibration is at least the Iitaka–Kodaira dimension of the anti-canonical divisor.

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