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2011•Journal of Algebraic GeometryOpen access

On a conjecture of Beltrametti and Sommese

Andreas Höring

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Abstract

Let X X be a projective manifold of dimension n n . Beltrametti and Sommese conjectured that if A A is an ample divisor such that K X + ( n − 1 ) A K_X+ (n-1)A is nef, then K X + ( n − 1 ) A K_X+(n-1)A has non-zero global sections. We prove a weak version of this conjecture in arbitrary dimension. In dimension three, we prove the stronger non-vanishing conjecture of Ambro, Ionescu and Kawamata and give an application to Seshadri constants.

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What this paper is about

Let X X be a projective manifold of dimension n n . Beltrametti and Sommese conjectured that if A A is an ample divisor such that K X + ( n − 1 ) A K_X+ (n-1)A is nef, then K X + ( n − 1 ) A K_X+(n-1)A has non-zero global sections. We prove a weak version of this conjecture in arbitrary dimension. In dimension three, we prove the stronger non-vanishing conjecture of Ambro, Ionescu and Kawamata and give an application to Seshadri constants.

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Available abstract

Let X X be a projective manifold of dimension n n . Beltrametti and Sommese conjectured that if A A is an ample divisor such that K X + ( n − 1 ) A K_X+ (n-1)A is nef, then K X + ( n − 1 ) A K_X+(n-1)A has non-zero global sections. We prove a weak version of this conjecture in arbitrary dimension. In dimension three, we prove the stronger non-vanishing conjecture of Ambro, Ionescu and Kawamata and give an application to Seshadri constants.

Key concepts: Conjecture, Mathematics, Dimension (graph theory), Divisor (algebraic geometry), Zero (linguistics), Pure mathematics, Manifold (fluid mechanics), Combinatorics

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