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Arithmetic on singular cubic surfaces

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Arithmetic on singular cubic surfaces COMPOSITIO MATHEMATICA DANIEL F. CORAY Arithmetic on singular cubic surfaces Compositio Mathematica, tome 33, no 1 (1976), p. 55-67. <> © Foundation Compositio Mathematica, 1976, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// implique l’accord avec les conditions générales d’utilisation ( Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 55 ARITHMETIC ON SINGULAR CUBIC SURFACES Daniel F. Coray* COMPOSITIO MATHEMATICA, Vol. 33, Fasc. 1, 1976, pag. 55-67 Noordhoff International Publishing Printed in the Netherlands In this note we shall give a proof of the following: PROPOSITION 1: (a) Let V C p3 be a cubic surface, defined over the infinite perfect field k, and having exactly 3 singular points Q,, Q2, Q3. Then V is k-birationally equivalent to a non-singular cubic surface W containing a k-rational set of 3 skew lines. (b) Conversely, every non-singular cubic surface W CP3 containing a k-rational set of 3 skew lines is k-birationally equivalent to a cubic surface V with exactly 3 double points. A proof of this result was already outlined in a little known paper of B. Segre [10], but the crucial fact that W is non-singular receives no justification there. Moreover, Segre states the converse with an additional assumption, which is actually not needed, as we shall see (lemma 3). This proposition has a number of applications to arithmetical ques- tions ; we begin by discussing a few of them: (i) When k is a number field, it was shown by Skolem [12] that singular cubic surfaces satisfy the Hasse principle (i.e. if V ha

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