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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
7
votes
Books for (Complex) Algebraic Curves
When I was a student I liked a lot the book by Egbert Brieskorn and Horst Knörrer: Plane Algebraic Curves. I can recommend it.
4
votes
Good book on Riemann surfaces and Galois theory?
Try Askold Khovanskii: Galois Theory, Coverings and Riemann Surfaces.
2
votes
Algebraic Geometry - Blowup
An elementary and extremely detailed account of this is given in the book Brieskorn, Knörrer: Plane Algebraic Curves.
2
votes
Intuitive explanation of concentration of the measure for spheres
You find a discussion of it in Yuri Manin: Mathematics and Physics, Birkhäuser.
2
votes
Differential Algebra Book
Besides the book by van der Put and Singer, which is mentioned above, I recommend that you have a look at Werner Seiler's Involution. This comprehensively covers many topics in the areas you list.
1
vote
What is the exact statement of "there are 27 lines on a cubic"?
You can see a photo of a model of the cubic in the case when they are all real in the book by Gerd Fischer: Mathematische Modelle/Mathematical Models. The 27 lines are drawn on the surface.