On large Picard groups and the Hasse Principle for curves and K3 surfaces
but what is known exactly when X has no k-rational point? In the present paper we begin by collecting several properties of BP. Most of them are well known, but they are difficult to find all at one place in the literature. In spite of its simplicity, this study already raises several questions. In Section 2 we investigate rational curves and smooth quadrics in Pk. We shall see that BP behaves in a rather unexpected fashion (Example 2.9). Over a number field, this is intimately connected with the Hasse Principle.