Classification of bi-polarized 3-folds \((X, L_{1}, L_{2})\) with \(h^{0}(K_{X}+L_{1}+L_{2})=1\)
type: Item
instance of: scholarly article
title: Classification of bi-polarized 3-folds \((X, L_{1}, L_{2})\) with \(h^{0}(K_{X}+L_{1}+L_{2})=1\)
zbMATH Open document ID: 1400.14027
author: Yoshiaki Fukuma
published in: Hiroshima Mathematical Journal
publication date: 2018-10-25T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.hmj/1533088829
review text: Consider \(X\) a smooth complex projective variety of dimension \(n\) and \(L\) an ample line bundle on \(X\). Several questions on the adjoint bundles \(K_X+tL\), \(t\) a positive integer, are of interest, in particular those on the interplay between their effectivity and nefness (see the Introduction of this paper and references therein). More generally, one can consider multipolarized manifolds \((X,L_1, \dots, L_{n-1})\) (\(L_i\) ample), and it is conjectured (see Conj. 3) that for \(n \geq 3\), if \(K_X+L_1+\dots+L_{n-1}\) is nef then it is effective, i.e., \(h^0(K_X+L_1+\dots+L_{n-1})>0\). This conjecture has been proved when \(n=3\) leading to a classification of \((X,L_1,L_2)\) such that \(K_X+L_1+L_2\) is not effective (\(X\) is either a linear space, or a quadric, or a \(\mathbb{P}^2\)-bundle over a smooth projective curve, the \(L_i\)'s are precisely described, proper references in Conj. 3). As a natural continuation of this research, the paper under review presents a complete classification of multipolarized threefolds \((X,L_1,L_2)\) such that \(h^0(K_X+L_1+L_2)=1\). The complete list of possibilities is collected in Section 3, Main result. There are seven cases: a Del Pezzo variety, a linear space, a quadric, a product of a line and a plane, a scroll or a quadric fibration over an elliptic curve, and a scroll over a smooth projective surface; the possible \(L_i\)'s are also presented.
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J30
zbMATH DE Number: 6965539
zbMATH Keywords: polarized manifold
zbMATH Keywords: adjoint bundle
zbMATH Keywords: sectional geometric genus
zbMATH Keywords: effectivity and nefness
reviewed by: Roberto Muñoz
MaRDI profile type: Publication
OpenAlex ID: W2902523012
DOI: 10.32917/HMJ/1533088829
description: scientific article; zbMATH DE number 6965539