Positive curves in polarized manifolds
type: Item
instance of: scholarly article
title: Positive curves in polarized manifolds
zbMATH Open document ID: 0899.14009
DOI: 10.1007/BF02678200
author: Lucian Bădescu
author: Mauro C. Beltrametti
author: Paolo Francia
published in: Manuscripta Mathematica
publication date: 1997-06-01T00:00:00Z
full work available at URL: https://eudml.org/doc/156271
review text: Curves \(Y\) on a smooth complex polarized variety \((X,A)\) of dimension \(k \geq 3\) are studied in order to generalize a paper by \textit{R. Paoletti} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 6, No. 4, 259-274 (1995; Zbl 0874.14018)] in case \(k=3\) about the Seshadri constant \(\varepsilon (Y,A)\) and positivity conditions on \(Y\). Let \(\pi : X_Y \rightarrow X\) be the blow up of \(X\) at the smooth curve \(Y\), \(E\) the exceptional divisor of the blow up, then the Seshadri constant is defined by: \(\varepsilon := \varepsilon (Y,A) := \sup\{\eta \in {\mathbb{Q}}\mid \pi ^* A-\eta E\) is ample\}. Then for all \(\eta \in (0,\varepsilon)\cap {\mathbb{Q}}\), \(\eta = {n\over m}\), we can assume \(m(\pi^* A)- nE\) to be very ample, so, if \(H_1,\dots,H_{k-2}\) are generic elements in \(m(\pi^* A)- nE\), \({\mathcal H} = H_1\cap \dots \cap H_{k-2}\) and \(Y' = {\mathcal H}\cap Y\) will be smooth and connected (of dimension 2 and 1 respectively). The curve \(Y\) is said to be \(A\)-ample if it exists \(\eta \in (0,\varepsilon)\) such that \(Y'\) is ample in \({\mathcal H}\), and is said to be A-big if \({Y'}^2 > 0\). The main results in the paper are: -- any \(A\)-ample curve \(Y\) meets every hypersurface of \(X\), and any \(A\)-big curve meets every hypersurface on \(X\) which is smooth and with an ample normal bundle; -- an \(A\)-big curve \(Y\) is not \(A\)-ample if and only if there is an irreducible hypersurface in \(X\) which does not meet \(Y\); -- when \(Y\) is \(A\)-ample or \(A\)-big, criteria to compute the cohomological dimension of \(X-Y\) are given; -- \(A\)-ampleness and \(A\)-bigness are open properties in flat families.
reviewed by: Alessandro Gimigliano
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 1002960
zbMATH Keywords: polarized variety
zbMATH Keywords: Seshadri constant
zbMATH Keywords: hypersurfaces
zbMATH Keywords: ampleness
zbMATH Keywords: bigness
zbMATH Keywords: ample curve
zbMATH Keywords: big curve
MaRDI profile type: Publication
cites work: Introduction to Grothendieck duality theory
cites work: Q3347989
cites work: Notes on embeddings of blowups
cites work: The positivity of the Chern classes of an ample vector bundle
cites work: Q4288628
cites work: Q3328651
cites work: On the connectedness of degeneracy loci and special divisors
cites work: Q2551490
cites work: Cohomological dimension of algebraic varieties
cites work: Ample subvarieties of algebraic varieties. Notes written in collaboration with C. Musili
cites work: Formal functions and formal embeddings
cites work: Ample Vector Bundles on Algebraic Surfaces
cites work: Seshadri constants, gonality of space curves, and restriction of stable bundles
cites work: Seshadri positive curves in a smooth projective 3-fold
cites work: Some remarks on set-theoretic intersection curves in \(\mathbb{P}^ 3\)
cites work: Submanifolds of Abelian varieties
cites work: Cohomological Dimension and Abelian Varieties
description: scientific article; zbMATH DE number 1002960