Rational curves on fibered varieties

Author(s):

Fabrizio Anella

Keyword(s):

Projective Variety ◽

Rational Curves ◽

Canonical Bundle ◽

Minimal Genus ◽

Codimension One ◽

Genus One ◽

Abstract Let X be a complex projective variety with log terminal singularities and vanishing augmented irregularity. In this paper, we prove that if X admits a relatively minimal genus-one fibration, then it contains a subvariety of codimension one covered by rational curves contracted by the fibration. We then focus on the case of varieties with numerically trivial canonical bundle and we discuss several consequences of this result.

Ample Vector Bundles of Curve Genus One

Canadian Mathematical Bulletin ◽

10.4153/cmb-1999-025-9 ◽

1999 ◽

Vol 42 (2) ◽

pp. 209-213 ◽

Cited By ~ 3

Author(s):

Antonio Lanteri ◽

Hidetoshi Maeda

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Projective Variety ◽

Ample Vector Bundle ◽

Smooth Complex ◽

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.

Manifolds Covered by Lines and Extremal Rays

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-119-7 ◽

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 ◽

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).

Rational curves on genus-one fibrations

manuscripta mathematica ◽

10.1007/s00229-020-01259-2 ◽

2020 ◽

Author(s):

Fabrizio Anella

Keyword(s):

Projective Variety ◽

Sufficient Conditions ◽

Rational Curves ◽

Necessary And Sufficient Conditions ◽

Minimal Genus ◽

Necessary And Sufficient ◽

Genus One

AbstractIn this paper we look for necessary and sufficient conditions for a genus-one fibration to have rational curves. We show that a projective variety with log terminal singularities that admits a relatively minimal genus-one fibration $$X\rightarrow B$$ X → B does contain vertical rational curves if and only if it not isomorphic to a finite étale quotient of a product $$\tilde{B}\times E$$ B ~ × E over B. Many sufficient conditions for the existence of rational curves in a variety that admits a genus-one fibration are proved in this paper.

LINEAR SYSTEMS ON IRREGULAR VARIETIES

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748019000069 ◽

2019 ◽

Vol 19 (6) ◽

pp. 2087-2125 ◽

Cited By ~ 3

Author(s):

Miguel Ángel Barja ◽

Rita Pardini ◽

Lidia Stoppino

Keyword(s):

Continuous Function ◽

Line Bundle ◽

Linear Systems ◽

Projective Variety ◽

Geometrical Properties ◽

Wide Range ◽

Image Position ◽

Irregular Varieties ◽

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 $a$.

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

Anais da Academia Brasileira de Ciências ◽

10.1590/s0001-37652001000400001 ◽

2001 ◽

Vol 73 (4) ◽

pp. 475-482 ◽

Cited By ~ 6

Author(s):

MARCIO G. SOARES

Keyword(s):

Projective Space ◽

Complex Projective Space ◽

Projective Variety ◽

Holomorphic Foliation ◽

One Dimensional ◽

Geometric Objects ◽

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

Canadian Mathematical Bulletin ◽

10.4153/cmb-2005-039-1 ◽

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 ◽

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.

Hyperbolicity related problems for complete intersection varieties

Compositio Mathematica ◽

10.1112/s0010437x13007458 ◽

2013 ◽

Vol 150 (3) ◽

pp. 369-395 ◽

Cited By ~ 10

Author(s):

Damian Brotbek

Keyword(s):

Projective Space ◽

Existence Theorem ◽

Complete Intersection ◽

Cotangent Bundle ◽

Projective Variety ◽

Complete Intersections ◽

Smooth Complex ◽

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.

Nef anti-canonical divisors and rationally connected fibrations

Compositio Mathematica ◽

10.1112/s0010437x19007383 ◽

2019 ◽

Vol 155 (7) ◽

pp. 1444-1456

Author(s):

Sho Ejiri ◽

Yoshinori Gongyo

Keyword(s):

Projective Variety ◽

Kodaira Dimension ◽

Canonical Divisor ◽

Rationally Connected ◽

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.

Rank 3 rigid representations of projective fundamental groups

Compositio Mathematica ◽

10.1112/s0010437x18007182 ◽

2018 ◽

Vol 154 (7) ◽

pp. 1534-1570 ◽

Author(s):

Adrian Langer ◽

Carlos Simpson

Keyword(s):

Irreducible Representation ◽

Projective Variety ◽

Fundamental Groups ◽

Projective Varieties ◽

Smooth Complex ◽

Geometric Origin ◽

Rank 2 ◽

Complex Projective ◽

Smooth Projective Varieties

Let$X$be a smooth complex projective variety with basepoint$x$. We prove that every rigid integral irreducible representation$\unicode[STIX]{x1D70B}_{1}(X\!,x)\rightarrow \operatorname{SL}(3,\mathbb{C})$is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by Corlette and the second author in the rank 2 case and answers one of their questions.

MINIMAL MODEL PROGRAM WITH SCALING AND ADJUNCTION THEORY

International Journal of Mathematics ◽

10.1142/s0129167x13500079 ◽

2013 ◽

Vol 24 (02) ◽

pp. 1350007 ◽

Cited By ~ 5

Author(s):

MARCO ANDREATTA

Keyword(s):

Minimal Model ◽

Projective Variety ◽

Model Program ◽

Minimal Model Program ◽

Projective Varieties ◽

Adjunction Theory ◽

Birational Equivalence ◽

Big Line Bundle ◽

Let (X, L) be a quasi-polarized pair, i.e. X is a normal complex projective variety and L is a nef and big line bundle on it. We study, up to birational equivalence, the positivity (nefness) of the adjoint bundles KX + rL for high rational numbers r. For this we run a Minimal Model Program with scaling relative to the divisor KX + rL. We give then some applications, namely the classification up to birational equivalence of quasi-polarized pairs with sectional genus 0, 1 and of embedded projective varieties X ⊂ ℙN with degree smaller than 2 codim ℙN(X) + 2.