Should set up a database for varieties, to absorb examples, and things that are known. For example all the exercises in Hartshorne (at least the first chapter). Other sources: Dieudonne
Brauer-Severi varieties, see Jahnel in AG/Various
Kollar: Rational curves on algebraic varieties. Gives some general introduction to higher-dim varieties I think. In folder AG/Various
Albanese variety
arXiv:1008.3825 Algebraic surfaces and hyperbolic geometry from arXiv Front: math.AG by Burt Totaro This is a survey of the Kawamata-Morrison cone conjecture on the structure of Calabi-Yau varieties and more generally Calabi-Yau pairs. We discuss the proof of the cone conjecture for algebraic surfaces, with plenty of examples
We show that the automorphism group of a K3 surface need not be commensurable with an arithmetic group, which answers a question by Mazur.
arXiv:0908.4241 Lectures on curves on varieties – Lisbon, 2009 from arXiv Front: math.AG by János Kollár These lectures give a short introduction to the study of curves on algebraic varieties. After an elementary proof of the dimension formula for the space of curves, we summarize the basic properties of uniruled and of rationally connected varieties. At the end, some connections with symplectic geometry are considered.
arXiv:1103.3156 The Expanding Zoo of Calabi-Yau Threefolds from arXiv Front: math.AG by Rhys Davies This is a short review of recent constructions of new Calabi-Yau threefolds with small Hodge numbers and/or non-trivial fundamental group, which are of particular interest for model-building in the context of heterotic string theory. The two main tools are topological transitions and taking quotients by actions of discrete groups. Both of these techniques can produce new manifolds from existing ones, and they have been used to bring many new specimens to the previously sparse corner of the Calabi-Yau zoo where both Hodge numbers are small. Two new manifolds are also obtained here from hyperconifold transitions, including the first example with fundamental group S3, the smallest non-Abelian group.
nLab page on Varieties examples