Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-24hb2 Total loading time: 0 Render date: 2024-03-28T10:27:05.054Z Has data issue: false hasContentIssue false

On the Minimality of Certain Hilbert Modular Surfaces

Published online by Cambridge University Press:  03 May 2010

Get access

Summary

Introduction

For some time now Hirzebruch and others have studied certain fields of Hilbert modular functions from a geometric point of view (see [2], [3], [5]). This leads to the introduction of a non-singular algebraic surface Y0(p) for all square-free positive integers p. In [5] the question is settled how the surfaces Y0(p) fit into the rough classification of algebraic surfaces, at least for those values of p which are prime and congruent 1 mod 4. It turns out that for p=5, 13 and 17 the surface Y0(p) is rational, that for p = 29, 37 and 41 this surface is an elliptic K3-surface, that for p = 53, 61 and 73 it is a minimal honestly elliptic surface, and that for p ≥ 89 the surface Y0(p) is of general type. As already follows from this description, the surfaces Y0(p) are minimal (i.e. without exceptional curves of the first kind) for 29 ≤ p ≤ 73. Now it is stated as a conjecture in [5] (p. 21) that this remains true for all p ≥ 89. Of course, it would be very interesting if this conjecture could be proved. In fact, if you know that a certain (simply connected) surface is of general type, and even if you know in addition its arithmetical genus and its Euler characteristic, but you don't know whether it is minimal, your knowledge does not amount to very much.

Type
Chapter
Information
Complex Analysis and Algebraic Geometry
A Collection of Papers Dedicated to K. Kodaira
, pp. 137 - 150
Publisher: Cambridge University Press
Print publication year: 1977

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.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×