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Quid of Hirzebruch surfaces?
Many have probably already heard about (holomorphic -- algebraic) vector bundles over a (complex -- algebraic) manifold. Let's make things a little more projective by considering projective varieties given as projective bundles over projective spaces. We will discuss the classification of P^1-bundles over P^1 and the construction of homogeneous coordinates on these bundles if time allows it. As we will see, there is always a weighted projective space hiding nearby.
