Manifolds Covered by Lines and Extremal Rays

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).

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Higher-dimensional Clifford–Severi equalities

Communications in Contemporary Mathematics ◽

10.1142/s0219199719500792 ◽

2019 ◽

Vol 22 (08) ◽

pp. 1950079 ◽

Cited By ~ 1

Author(s):

Miguel Ángel Barja ◽

Rita Pardini ◽

Lidia Stoppino

Keyword(s):

Lower Bound ◽

Line Bundle ◽

Abelian Variety ◽

Projective Variety ◽

Line Bundles ◽

Smooth Complex ◽

Higher Dimensional ◽

Complex Projective Variety ◽

Irregular Varieties ◽

Complex Projective

Let [Formula: see text] be a smooth complex projective variety, [Formula: see text] a morphism to an abelian variety such that [Formula: see text] injects into [Formula: see text] and let [Formula: see text] be a line bundle on [Formula: see text]; denote by [Formula: see text] the minimum of [Formula: see text] for [Formula: see text]. The so-called Clifford–Severi inequalities have been proven in [M. A. Barja, Generalized Clifford–Severi inequality and the volume of irregular varieties, Duke Math. J. 164(3) (2015) 541–568; M. A. Barja, R. Pardini and L. Stoppino, Linear systems on irregular varieties, J. Inst. Math. Jussieu (2019) 1–39; doi:10.1017/S1474748019000069]; in particular, for any [Formula: see text] there is a lower bound for the volume given by: [Formula: see text] and, if [Formula: see text] is pseudoeffective, [Formula: see text] In this paper, we characterize varieties and line bundles for which the above Clifford–Severi inequalities are equalities.

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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 ◽

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 $a$.

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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 ◽

Cited By ~ 1

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.

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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 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.

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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 ◽

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.

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Rank 3 rigid representations of projective fundamental groups

Compositio Mathematica ◽

10.1112/s0010437x18007182 ◽

2018 ◽

Vol 154 (7) ◽

pp. 1534-1570 ◽

Cited By ~ 1

Author(s):

Adrian Langer ◽

Carlos Simpson

Keyword(s):

Irreducible Representation ◽

Projective Variety ◽

Fundamental Groups ◽

Projective Varieties ◽

Smooth Complex ◽

Complex Projective Variety ◽

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.

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

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004107000813 ◽

2008 ◽

Vol 144 (1) ◽

pp. 109-118 ◽

Cited By ~ 5

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).

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Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2013-0037 ◽

2015 ◽

Vol 2015 (702) ◽

pp. 1-40 ◽

Cited By ~ 10

Author(s):

Timo Schürg ◽

Bertrand Toën ◽

Gabriele Vezzosi

Keyword(s):

Algebraic Geometry ◽

Projective Variety ◽

Obstruction Theory ◽

Important Ingredient ◽

Smooth Complex ◽

Perfect Complexes ◽

Derived Algebraic Geometry ◽

Complex Projective Variety ◽

Complex Projective

AbstractA quasi-smooth derived enhancement of a Deligne–Mumford stack 𝒳 naturally endows 𝒳 with a functorial perfect obstruction theory in the sense of Behrend–Fantechi. We apply this result to moduli of maps and perfect complexes on a smooth complex projective variety.ForWe give two further applications toAn important ingredient of our construction is a

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Unramified Brauer group of the moduli spaces of PGLr(ℂ)-bundles over curves

Open Mathematics ◽

10.2478/s11533-014-0403-4 ◽

2014 ◽

Vol 12 (8) ◽

Author(s):

Indranil Biswas ◽

Amit Hogadi ◽

Yogish Holla

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Projective Variety ◽

Brauer Group ◽

Projective Curve ◽

Connected Component ◽

Smooth Complex ◽

Complex Projective Variety ◽

Complex Projective ◽

Complex Projective Curve

AbstractLet X be an irreducible smooth complex projective curve of genus g, with g ≥ 2. Let N be a connected component of the moduli space of semistable principal PGLr (ℂ)-bundles over X; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of N is trivial.

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