Algebraic surfaces containing an ample divisor of arithmetic genus two (Q1103014)

A polarized smooth surface \((S,L)\) is a pair of a smooth projective surface S and an ample line bundle \(L\) on \(S\). The sectional genus g of such a pair is definedby the formula \(2g-2=(K_ S+L)L\). The classification of polarized smooth surfaces with \(g=0\) and \(g=1\) is known. This article classifies the case \(g=2\). The possible \(S\) with the numerical class of \(L\) are listed. A notable example is a pair \((J(C),{\mathcal O}(C))\) where \(J(C)\) is the Jacobian of a smooth curve C of genus 2. The authors admit that the existence problem for the case \(\kappa (S)=1\) is still open. A similar result together with the classification of the higher dimensional case is independently obtained by \textit{T. Fujita} [''Classification of polarized manifolds of sectional genus two'', in Algebraic Geometry and Commutative Algebra, dedicated to M. Nagata, (1988)].