I'm currently taking a course called Advanced Algebraic Geometry with the following description:
This course will introduce the study of rational points on higher-dimensional varieties, concentrating on surfaces. On the geometric side, we will cover the basic geometry of surfaces including divisors, the Picard group, intersection theory and the Riemann-Roch theorem; on the algebraic side, we will introduce p-adic numbers, the Hasse principle and Brauer groups of fields. The two branches come together with the Brauer-Manin obstruction.
The notes they provide are mostly draft versions of some chapters of a book that isn't finished yet (see here). Note that these notes don't use the language of schemes.
I wonder if there are books (that are complete and published) out there that treat similar topics, maybe several books that would combine to cover most of the material?
The lecturer says that Hartshorne's algebraic geometry book of Silverman's elliptic curves book touches some parts of the material, but the problem is that we are not working over algebraically clsoed fields or perfect fields...
Currently I'm looking into Poonen's book Rational Points on Varieties, hoping that that book could serve as an anchor, and ideally I would have some additional books on the side (apart from the draft book the course provides).