Elliptic surfaces with an ample divisor of genus two

Abstract

Beltrametti, Lanteri and Palleschi have recently started the classification of smooth algebraic surfaces having an ample divisor of arithmetic genus two (Arkiv für Mat. 25 (1987), 189-210). Their results for the class of elliptic surf aces can be considerably improved. The present paper focuses on elliptic surfaces S with Kodaira dimension one, xOs = O, and such that the (unique) elliptic fibration has a rational base. The result is the following : if S contains a genus two ample divisor then S is of the form S = (D x E)/G where G is a group acting on two curves D and E, E is elliptic, G is either Z2 x Z2 , Z2 x Z6 or Z4 x Z4 and D has genus 2,2 and 3 respectively. Moreover, the existence of such polarized surfaces is shown by a concrete example.