Rational points on the special fibre of a scheme over a valuation ring
by
Antoine Ducros
Université de Rennes 1 (France)

Let K be a field, let v be a valuation on K, let A be the associated ring and suppose the characteristic of the residue field k of A is zero. Let L be a finitely generated field extension of K. Call X a model of L over A if X is an integral A-scheme whose function field is L. Then we show that the following are equivalent:

- v extends to a valuation of L with residue field k.

- every model X of L proper over A has a k-point on his special fibre.

The proof uses Zariski's description of the valuation spectrum of L over A as the inverse limit of the proper models of A, and a result of Kuhlman and Prestel.

Additionnally, we can show that models of L (and more generally flat schemes of finite type over valuation rings) are finitely presented.

Date received: March 5, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacv-15.