As others have said, after Hartshorne you could branch into too many different things, whose relative importance is subjective. Here is my take on what could constitute a good post-Hartshorne curriculum:
**Intersection theory**
- *Intersection theory*, William Fulton.
Classical foundational book, beautifully written.
- *3264 and all that*, David Eisenbud and Joe Harris.
This one actually tells you how to compute Chow rings in a multitude of situations. But it also contains a wealth of useful information apart from how to compute Chow rings, such as geometry of Grassmanians, Chern classes, Hilbert schemes and a toolbox for Riemann-Roch.
**Complex Algebraic Geometry**
- *Complex Geometry, An Introduction*, Daniel Huybrechts
This concise book covers nicely the foundational material of complex-analytic approach (which people used to learn from chapters 0 and 1 of Griffiths-Harris).
- *Hodge theory and complex complex algebraic geometry, I and II*, Claire Voisin.
It took me time to develop love for this book. Probably because it strives to reach modern techniques as fast as possible, making it hard for someone just entering the world of Hodge theory.
**Derived categories of coherent sheaves**
- *Fourier-Mukai Transforms in Algebraic Geometry*, Daniel Huybrechts
Just as all other of Huybrechts' books, this one is a true gem. The book teaches you how not to be scared of $D^b (Coh (X))$.
**Geometry of special varieties.**
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 mention here 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, while R. Friedman's book is an interesting synthesis of classical geometry of surfaces and analysis of coherent sheaves on them, beautifully written.
- *Lectures on K3 surfaces*, Daniel Huybrechts
K3 surfaces is truly an amazing class of varieties whose geometry is studied through a multitude of different techniques, and this book does an excellent job showing those multiple facets of research in the area.
- *[The geometry of cubic hypersurfaces][1]*, Daniel Huybrechts
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 discussion of higher dimensional cases of three- and four-folds.
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The rest is more idiosyncratic since it is the field I am interested in -
**Moduli spaces of sheaves.**
- *The Geometry of Moduli Spaces of Sheaves*, D. Huybrechts and M. Lehn
A systematic treatment of foundations of moduli of semistable sheaves. The book discusses an incredible amount of general techniques used in this area.
- *Lectures on Vector Bundles*, J.Le Potier.
Not the best read with respect to foundations (for that see the previous item in the list), but fully explains the best-studied case of moduli of sheaves on $\mathbb{P}^2$, which still serves as a beacon for research in this area.
- *Fundamental Algebraic Geometry. Grothendieck's FGA explained*, B. Fantechi et al.
An introduction into some more advanced fundamental techniques useful for moduli problems: descent theory, Hilbert and Quot schemes, elementary deformation theory and Picard scheme.
- *Deformation theory*, R. Hartshorne
A nice concise account of algebraic approach to deformation theory, with a lot of examples.
[1]: http://www.math.uni-bonn.de/people/huybrech/Notes.pdf