For special varieties (curves, various types of surfaces, special three- and four-folds, etc.) people have accumulated a large amount of more ad-hoc methods. At the basic level, contemplating the geometry of simplest such varieties provides an excellent playground to truly absorb the power of abstract machinery from Hartshorne. Further on, while doing research you will probably need a very good knowledge of the geometry of your particular variety at hand, so you cannot go wrong with investing some time to learn this material. I will only mention one bookhere only the basics:
Curves:
- Chapter 4 of Hartshorne or
- Chapter 19 of Ravi Vakil's notes
Provide the absolute minimum here.
Surfaces:
- Chapter 5 of Hartshorne,
- Complex algebraic surfaces, Arnaud Beauville
- Algebraic surfaces and holomorphic vector bundles, Robert Friedman
The first two items are standard sources, again just becausewhile R. Friedman's book is an interesting synthesis of how smoothly it reads:classical geometry of surfaces and analysis of coherent sheaves on them, beautifully written.
K3 surfaces is truly an amazing class of varieties whose geometry is studied through a multitude of different techniques, soand this book does an excellent job showing those multiple facets of research in the area.
These notes are still work in progress, but as the previous item these notes talk about an amazing variety of techniques: moduli spaces, Hodge theoretic methods, etc. There is also a good testing ground for how well one has mastered themdiscussion of higher dimensional cases of three- and four-folds.
The rest is more idiosyncratic since it is the field I am workinginterested in -
