On sectional genus of \(k\)-very ample line bundles on smooth surfaces with non-negative Kodaira dimension (Q1282077)

Let \(C\) be a smooth complex projective surface polarized by an ample line bundle \(L\). In the paper under review the relations between the sectional genus \(g(L)\) and the irregularity \(q(X)\) are investigated under the assumption that \(L\) is \(k\)-very ample, i.e., for any 0-dimensional subscheme \((Z,{\mathcal O}_Z)\) with length \(({\mathcal O}_Z)\leq k+1\), the restriction homomorphism \(\Gamma(L)\to\Gamma(L\otimes{\mathcal O}_Z)\) is surjective. The author conjectures that if \(X\) has Kodaira dimension \(\kappa(X)\geq 0\) then \(g(L)\geq(k+2)q(X)\). To include the classical conjecture for polarized surfaces one needs to put \(k=-1\) if \(L\) is ample but not spanned by global sections. The conjectured inequality is proved for \(k\geq 0\) in the following cases: (i) \(\kappa(X)=0\), (ii) \(\kappa(X)=1\), and (iii) \(\kappa(X)=2\) and \(X\) fibres over a smooth curve \(C\) with \(q(X)\leq g(C)+1\). Moreover in cases (i) and (ii) the pairs \((X,L)\) for which \(g(L)=(k+2)q(X)\) are characterized.