Can someone recommend a few books on complex (algebraic) geometry, containing many examples of blow-ups and blow-downs explicitly worked out? I am not so much interested in abstract proj constructions, and various scheme-theoretic subtleties for now (of course these have their virtues!), but more interested in very concrete classical examples worked out (classical but ranging in complexity from very simple constructions to more complicated ones).
[Math] References with many examples of blow-ups and blow-downs in complex geometry
algebraic-geometryblowup
Best Answer
I'm currently reading these notes and they are quite good, many exercises inside for becoming familiar with blow-ups.
Interesting example of blow-up often comes as a linear system. Consider a family of curves of degree $d$ in the plane : $\lambda F + \mu G = 0$ with $F,G$ polynomials of degree $d$. To each point of $\mathbb P^2$ we can associate a unique $(\lambda : \mu) \in \mathbb P^1$ such that $p \in Z(\lambda F + \mu G)$ except when $p \in Z(F) \cap Z(G)$ (this locus is called the base-point of the linear system). For obtaining a well-defined map, you need to blow up the plane at $d^2$ points : you obtain then a well-defined map $Bl_Y(\mathbb P^2) \to \mathbb P^1$ and you can study your linear system which is a fibration (possibly with singular fibers).
The book " 4-manifolds and Kirby calculus" has also a nice section about blow-up.
Finally, any notes online about algebraic geometry also have a chapter about blow-up and classical examples.
(This is rather a comment that an answer, but I don't have enough space in comments).