Elliptic surfaces with an ample divisor of genus two

Fernando Serrano

Vol. 152 (1992), No. 1, 187–199

DOI: 10.2140/pjm.1992.152.187

Abstract

Beltrametti, Lanteri and Palleschi have recently started the classification of smooth algebraic surfaces having an ample divisor of arithmetic genus two (Arkiv för Mat. 25 (1987), 189-210). Their results for the class of elliptic surfaces can be considerably improved. The present paper focuses on elliptic surfaces S with Kodaira dimension one, χ𝒪_S = 0, 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 ×E)∕G where G is a group acting on two curves D and E, E is elliptic, G is either ℤ₂ × ℤ₂, ℤ₂ × ℤ₆ or ℤ₄ × ℤ₄ and D has genus 2, 2 and 3 respectively. Moreover, the existence of such polarized surfaces is shown by a concrete example.

Mathematical Subject Classification 2000

Primary: 14J27

Secondary: 14J10

Milestones

Received: 12 July 1990

Revised: 10 August 1990

Published: 1 January 1992