Arithmétique des Surfaces Cubiques Diagonales

Finite descent obstructions and rational points on curves

“…For example, a smooth cubic surface X in ‫ސ‬ 3 has X ‫ށ(‬ k ) f-cov • = X ‫ށ(‬ k ) • (since it has trivial geometric fundamental group), but may well have X ‫ށ(‬ k ) Br 1 • = ∅, even though there are points everywhere locally. See [Colliot-Thélène et al 1987a], where the algebraic Brauer-Manin obstruction is computed for all smooth diagonal cubic surfaces X : a 1 x 3 1 + a 2 x 3 2 + a 3 x 3 3 + a 4 x 3 4 = 0 with integral coefficients 0 < a i < 100, thereby verifying that it is the only obstruction against rational points on X (and thus providing convincing experimental evidence that this may be true for smooth cubic surfaces in general). This computation produces a list of 245 such surfaces with points everywhere locally, but no rational points, since X ‫ށ(‬ ‫ޑ‬ ) Br 1 • = ∅.…”

On the equations for universal torsors over del Pezzo surfaces

“…Thus the existence of s follows from Proposition 3.1. We prove the uniqueness by induction in r. For r = 4 the statement is clear, since the only elements of S that leave Gr (2,5) invariant are the elements of T (see the proof of Lemma 2.2). Assume r ≥ 5.…”

“…Hence Ver ω 1 maps (G/P ) a surjectively onto p ⊥ ω 1 . QED Arguing by induction as in this proof, it is easy to show starting with the case of the Plücker coordinates on Gr (2,5), that p µ (x) is the sum of all the monomials of weight µ with non-zero coefficients.…”

Evaluating Azumaya algebras on cubic surfaces

“…However, for curves the picture is much clearer: if C is a curve over a p-adic field k, then the local pairing extends to a pairing Pic C × Br C → Br k, which Lichtenbaum [12] showed can be identified with that arising from the Tate pairing, and is therefore non-degenerate. In [6], Colliot-Thélène, Kanevsky and Sansuc gave a thorough description of both Br X and the local evaluation maps in the case when X is a diagonal cubic surface. In particular, they showed that the evaluation map is constant for places v where X has good reduction or where X is rational over L v ; and, in contrast (Proposition 2), at finite places v where the reduction of X at v is a cone, the local evaluation map takes all possible values on X(L v ).…”