Numerically positive divisors on algebraic surfaces (Q1345113)

Let \(S\) be a smooth projective surface over \(\mathbb{C}\) and \(L\) a numerically positive line bundle, i.e. \(L \cdot C > 0\) for all curves \(C\). A result of Kleiman implies \(L^2 \geq 0\). If \(L^2 > 0\) then \(L\) is ample by the Nakai-Moishezon criterion. The authors of the paper under review are interested in numerically positive line bundles which are not ample and therefore satisfy \(L^2 = 0\). Denote by \(g(L) = 1 + {1 \over 2} (L^2 + K_S \cdot L)\) the arithmetic genus of \(L\). It is shown that for numerically positive line bundles \(g(L) \geq 0\) holds. The main result is that numerically positive non-ample line bundles satisfy \(g(L) \geq 2\). Mumford produced examples where \(S\) is a certain \(\mathbb{P}^1\)-bundle over a curve \(B\) of genus \(q \geq 2\) with a non-ample numerically positive line bundle \(L\) of genus \(g(L) = q\). The authors construct a surface \(S\) by blowing up \(\mathbb{P}^2\) in 13 points and give examples for non-ample numerically positive line bundles on \(S\) for every genus \(\geq 2\).