We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 04 May 2010
Short Courses
Fields of Definition of Covers; Embedding Problems over Large Fields. Pierre Dèbes.
I. The first talk deals with joint work with Jean–Claude Douai. Let f : X → B be a finite cover defined over the separable closure Ks of a field K, withB an algebraic variety defined over K. Assume that f is isomorphic to each of its conjugates under G(Ks/K). The field K is called the field of moduli of the cover. Does it follow that the given cover can be defined over K? The answer is “No” in general: there is an obstruction to the field of moduli being a field of definition. Still, how can the obstruction be measured? We present a general approach for this problem. The obstruction is entirely of a cohomological nature. This was known only in the case of G–covers,i.e., Galois covers given together with their automorphisms. This special case happens to be the simplest one. In the situation of mere covers, the problem is shown to be controlled not by one, as for G–covers, but by several characteristic classes in H2(K, Z(G), L) (for a certain action L of G(Ks/K) on the center Z(G) the group of the cover). Furthermore our approach reveals a more hidden obstruction coming on top of the main one, called the first obstruction and which does not exist for G–covers.
Our Main Theorem yields quite concrete criteria for the field of moduli to be a field of definition. Such criteria were not available in the general situation of mere covers.
To save this book to your Kindle, first ensure no-reply@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.
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.
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.
