Reference request: algebraic methods in geometry

Question

I am a (soon to be) third year undergraduate who has just finished courses in linear and abstract algebra. While I enjoyed the study of algebraic structures in their own right, my favorite part of the courses were the applications of the algebraic machinery developed to geometric problems (i.e. the connection between Galois theory and compass-straightedge constructions.) As such, I was hoping for a reference that developed this general approach of using algebraic techniques to better understand problems in geometry. I realize this is a really broad request, but unfortunately as I know almost nothing about this "algebraic geometry" I don't know what to specifically ask for. Anything that falls under this category is fine with me.

My understanding of algebra is still fairly elementary, so I would appreciate references that don't require too much background (read: I have nowhere near enough knowledge of commutative algebra to embark on the study of algebraic geometry, at least in its modern form.) If it helps, I specifically have covered almost all of Axler's Linear Algebra Done Right, and all of the material on groups and rings, as well as a little bit on fields, from Dummit and Foote. Thanks!

Answer 1

I would recommend that you get a copy of Hartshorne's Foundations of Projective Geometry. It gives a wonderful sense of the relationship between algebraic properties and geometric ones, and assumes nothing more than some group theory, familiarity with Euclidean geometry, and a bit of linear algebra, together with some mathematical sophistication, which you seem to have.

Answer 2

Perhaps try these books:

• Conics and Cubics: A Concrete Introduction to Algebraic Curves by Bix.

• Introduction to plane algebraic curves by Kunz and Belshoff.

• Plane algebraic curves by Brieskorn and Knörrer.

• Algebraic curves, an introduction to algebraic geometry by Fulton.

Answer 3

There are many possibilities here. A particular example is the study of geometric structures on Lie groups. This involves geometry, Euclidean, affine, hyperbolic structures, discrete groups, crystallographic groups, Lie groups, Lie algebras, representations, cohomology and many other tools. A very nice survey article is the following article by Bill Goldman:

Locally homogeneous geometric manifolds.

Goldman explaines how many questions about geometric structures can be studied by algebraic methods. This goes back to E. Cartan and W. Thurston.