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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).

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    Blowing up is a straight forward construction. To get a good feeling about blowing down, may be you should carefully study the theorem of Castelnuovo? – 2017-01-09
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    @Mohan thank you. I shall study it, time permitting. – 2017-01-09

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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).

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    Thank you. I really liked the example you have presented, which is very nice. I will have a look at the online notes and also the book (if I can get a hold of a copy). – 2017-01-08
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    You're welcome ! Another nice example is the Nagata transform on Hirzebruch surface, which is a combinaison of a blow-up and a blow-down on/outside the exceptional divisor. – 2017-01-08
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    Where can I find more details of the Nagata transform? This is precisely the kind of things I want to learn about, when you have a blowup followed by a blowdown. I guess I will start by using a search engine! – 2017-01-08
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    For be honest, I couldn't myself find any details about it. I only need it for the case of $\Sigma_1 \to \Sigma_2$ which is not too complicated. If you can find anything about it by the way, let me know ! (I think I read an article when it was explicitly computed but notation were really confusing for me. I can try to find it) – 2017-01-08
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    http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B38/pdf/B38-07.pdf https://arxiv.org/pdf/math/0207131.pdf Here Nagata transform are mentioned, and I guess it's the best you can obtain ... – 2017-01-08
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    For instance, I would like to understand better the diagrams used by some mathematicians such as in Nobuhiro Honda's twistor theory papers, and how to think about such birational transformations using diagrams. – 2017-01-09
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    I took a look at one of the diagram. I would say this document could be useful : https://www.renyi.hu/~nemethi/JEGYZET.ps (but usually, people take opposite convention of Honda for the graph of resolution : edge correspond to divisors, and there is an edge between two divisors if they intersect.) (and by the way, papers by Honda looks super interesting !) – 2017-01-09
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    Thank you for this other useful reference! – 2017-01-09
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    Sure ! Good luck ! And by the way, could you post any useful reference if you find other one ? I would be quite interested as well ! – 2017-01-09
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    I shall post useful notes if I find. – 2017-01-09
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    I found the chapter on surfaces in Griffiths and Harris to contain several examples of blowups and blowdowns, particularly in some proofs. In particular, there is a nice section on quadratic transformations (which generate all Cremona transformations), which are special birational transformations of the complex projective plane associated to some given $3$ non-collinear points. I find quadratic transformations quite interesting, geometrically. – 2017-01-10
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    Great ! I tried to read it, and it is very well written, it is quite big though. – 2017-01-12