On the numerically fixed parts of line bundles (Q1075386)

Let L be a line bundle on a nonsingular projective variety V defined over an algebraically closed field. The author introduces the notions of numerical base locus and numerical fixed part of L and studies its basic properties. The stable base locus SBs(L) of L is by definition the intersection of the base loci of all powers of L. If A is an ample line bundle on V then \(NBs(L)=\cup^{\infty}_{n-1}SBs(A+nL)\)\ is independent of A and called the numerical base locus of L. For an integer \(m\geq 1\) let F(m,L) denote the fixed part of the linear system \(| mL|\). In Div(V)\(\otimes {\mathbb{R}}\) the lower bound of the sequence \((m^{-1}F(m,L))_{m\in {\mathbb{N}}}\) makes sense and is denoted by F(L). If L is pseudoeffective and A an ample line bundle, and if \(F(A+nL)=\sum a_{\Gamma}(n;A,L)\Gamma \quad is\) an irreducible decomposition with \(a_{\Gamma}(n;A,L)\in {\mathbb{R}}\) and \(\Gamma\) a prime divisor, then \(a_{\Gamma}(A,L)=\lim_{n\to \infty}n^{-1}a_{\Gamma}(n;A,L)<\infty \quad and\) \(\sum a_{\Gamma}(A,L)<\infty.\quad Hence\) \(\sum a_{\Gamma}(A,L)\Gamma \quad is\) an \({\mathbb{R}}\)-divisor with possibly infinitely many components, which turns out to depend only on the numerical equivalence class of L. It is called the numerical fixed part of L. Details for the proofs of the given results shall appear elsewhere.