Polarized surfaces of $\Delta$-genus $3$

Abstract

Let $X$ be a smooth, complex, algebraic, projective surface and let $L$ be an ample line bundle on it. Let $\Delta = \Delta (X,L)= {c_1}{(L)^2} + 2 - {h^0}(L)$ denote the $\Delta$-genus of the pair $(X,L)$. The purpose of this paper is to classify such pairs under the assumption that $\Delta = 3$ and the complete linear system $| L |$ contains a smooth curve. If $d \geq 7$ and $g \geq \Delta$, Fujita has shown that $L$ is very ample and $g= \Delta$. If $d \geq 7$ and $g < \Delta = 3$, then $g= 2$ and those pairs have been studied by Fujita and Beltrametti, Lanteri, and Palleschi. To study the remaining cases we have examined the two possibilities of $L + tK$ being nef or not, for $t= 1,2$. In the cases in which $L + 2K$ is nef it turned out to be very useful to iterate the adjunction mapping for ample line bundles as it was done by Biancofiore and Livorni in the very ample case. If $g > \Delta$ there are still open cases to solve in which completely different methods are needed.