Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-18T19:32:07.637Z Has data issue: false hasContentIssue false

SOLUBILITY OF CERTAIN PENCILS OF CURVES OF GENUS 1, AND OF THE INTERSECTION OF TWO QUADRICS IN ℙ4

Published online by Cambridge University Press:  23 August 2001

A. O. BENDER
Affiliation:
Robinson College, Cambridge CB3 9ET, a.o.bender@dpmms.cam.ac.uk
PETER SWINNERTON-DYER
Affiliation:
Trinity College, Cambridge CB2 1TQ, hpfs100@newton.cam.ac.uk
Get access

Abstract

The main part of the paper finds necessary conditions for solubility of a pencil of curves of genus 1, each of which is a 2-covering of an elliptic curve with at least one 2-division point. As in previous work, these are proved subject to Schinzel's Hypothesis and to the finiteness of the Tate-\u{S}afarevi\u{c} group of elliptic curves defined over a number field. It thus generalizes earlier work of Colliot-Thélène, Skorobogatov and the second author.

The final section gives necessary conditions (though of a rather ugly nature) for the solubility of a Del Pezzo surface of degree 4.

2000 Mathematical Subject Classification: 11D25.

Type
Research Article
Copyright
2001 London Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)