Algebraic geometry where to begin?

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In summary, the speaker is seeking recommendations for learning algebraic geometry relevant to mathematical physics. They have completed first year math coursework and taken additional courses in various math subjects. Others have suggested Basic Algebraic Geometry by Shafarevich, Algebraic Curves by Fulton, and Algebraic Geometry by Hartshorne. The speaker has found these texts useful so far, but is still seeking recommendations for further reading. They are also considering Rings Varieties and Algorithms by Cox, Little, and O'Shea as a companion text.
  • #1
styler
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Howdy folks,
I was wondering if anybody could recommend a good place to start learning algebraic geometry as might be relevant for current/recent mathematical physics.
I have opened Griffiths and Harris and found it sprawling ad frustruating.
For the sake of judging what might be appropriate, i am mainly interested in physics but i have completed stadarnd first year math course work in topology, algebra and analysis and additionally taken courses in functional analysis, differnetial topology, differetial geometry and alebraic topology...so i guess i "know" about what a very modest beginning third year grad student in math has learned.
Recomedations for several complex variables would be appreciated as well.
Any ideas?
Thanks in advance.
 
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  • #2
Others suggested these texts for me:

Basic Algebraic Geometry - Shafarevich
Algebraic Curves - Fulton
Algebraic Geometry - Hartshorne

And I've found them useful so far. If you just wanted to look at one, I'd probably suggest Shafarevich.

Keep in mind that I've just started teaching myself the subject, so I'm by no means an expert on the available choices. :smile:
 
  • #3
cool. thanks. I have heard about the first one...how elemetary/advanced is it?
 
  • #4
It starts off, IMHO, fairly elementary. (admittedly, "elementary" is a relative term!) The later chapters go into schemes and are fairly more advanced, but Shafarevich is still nowhere near as difficult as Hartshorne. (I haven't peeked at the Complex Manifold chapters in Shafarevich, though)
 
  • #5
hartshorne has as a precursor commutative algebra by eisenbud.

an intro alg. geometry book is rings varieties and algorithms by cox, little, and o'shea. it's prolly a good thing to have as a companion when looking at real alg. geometry.
 
  • #6
thanks guys
maybe i'll post some questiogs as i go alog.
 

1. What is algebraic geometry?

Algebraic geometry is a branch of mathematics that studies the relationship between algebraic equations and geometric shapes. It uses tools from both algebra and geometry to understand and solve problems related to polynomial equations.

2. What are some real-world applications of algebraic geometry?

Algebraic geometry has many applications in fields such as computer graphics, robotics, cryptography, and physics. It is also used in the study of curves and surfaces in engineering and architecture.

3. How do I get started with algebraic geometry?

The best way to begin learning algebraic geometry is to have a strong foundation in algebra, geometry, and calculus. It is also helpful to have a good understanding of abstract algebra and topology. There are many textbooks, online resources, and courses available to help you get started.

4. What are the main branches of algebraic geometry?

The main branches of algebraic geometry include classical or synthetic algebraic geometry, which deals with the study of algebraic varieties and their properties, and scheme theory, which provides a more abstract and general framework for studying algebraic geometry. Other important branches include intersection theory, birational geometry, and moduli theory.

5. What are some common techniques used in algebraic geometry?

Some common techniques used in algebraic geometry include solving systems of polynomial equations using elimination and substitution, studying the behavior of solutions under deformations, and using sheaf theory to study the local properties of algebraic varieties. Other techniques include the use of projective and affine spaces, homological algebra, and the theory of algebraic curves and surfaces.

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