False Hyperelliptic Surfaces with Section

✍ Scribed by Yoshifumi Takeda Nara

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 847 KB
• Volume: 167
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19941670113

✦ Synopsis

Abstract

Let X be a nonsingular relatively minimal projective surface over an algebraically closed field of characteristic p > 0. We call X a false hyperelliptic surface if X satisfies the following conditions: (1) c~2~(X) = 0, c~1~(X)^2^ = 0, dim Alb (X) = 1, and (2) All fibres of the Albanese mapping of X are rational curves with only one cusp of type x____p^v^ + y^n^ = 0. In this article, we consider a false hyperelliptic surface whose Albanese mapping has a cross‐section. We prove that every false hyperellyptic surface with section arises from an elliptic ruled surface and that every false hyperelliptic surface has an elliptic fibration with multiple fibre. Moreover, we construct an example of false hyperelliptic surface with section, whose elliptic fibration has a multiple fibre of supersingular elliptic curve of multiplicity p^v^ (v > 1).