5 ms·

What are some good resources for grasping or learning more about algebraic geometry?

ginnungagap 8y ago
You'll need a good background in commutative algebra to learn algebraic geometry, if you have that and want to see the modern approach, schemes and everything else that Grothendieck did, I suggest Vakil's notes, they're freely available from his homepage. (Disclaimer: I only studied the first 15 chapters and the one on Kähler differentials which should be the 21st, but I suppose the second half is as good as the first, I'll find out for sure next term)
- MaysonL 8y ago
  Vakil's notes have an interesting epigram from Grothendieck: I can illustrate the … approach with the … image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!… A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration … the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it … yet finally it surrounds the resistant substance. — A. Grothendieck
  - ginnungagap 8y ago
    Vakil seems very fond of that quote, the full title of his notes is "The Rising Sea - Foundations of Algebraic Geometry"!
edflsafoiewq 8y ago
Try some projective geometry to get your feet wet.
kevinventullo 8y ago
Atiyah-Macdonald is a small but dense book with most of the commutative algebra you'll need to start learning algebraic geometry. You'll need to know some abstract algebra as a prerequisite (groups, rings, fields; covered in undergraduate algebra courses).
- ginnungagap 8y ago
  A less dense book for commutative algebra is Miles Reid's Undergraduate Commutative Algebra, but it doesn't cover tensor product of modules which will definitely be needed for algebraic geometry. Also for everyone interested in Atiyah-Macdonald you MUST do the exercises, half of the book is in the exercises!
throwawaymath 8y ago
This isn't a fun answer, but you basically can't without substantial background in undergrad mathematics. The subject has some of the highest prerequisites of any upper undergrad/lower grad level math course, as it touches on (and uses) just about everything you'd learn through an undergrad math degree. I don't want to discourage you, I'm just being realistic. If you want to work towards algebraic geometry, you can certainly do that. You'll need to first master linear algebra and abstract algebra. You should have a strong understanding of fields, groups, rings, vector spaces and modules. Someone else mentioned commutative algebra - that is more of a circular dependency with algebraic geometry than a hard one. It's good to have walking in, but realistically you can't master the subject without knowing algebraic geometry. You'll also need analysis, in particular complex analysis for curves. Real analysis and topology should also be covered but I suppose with tenacity you could get by without them. To translate these into concrete suggestions, in your position I'd try to work through the following, in order: 1. Linear Algebra Done Right (Axler) 2. Abstract Algebra (Dummit & Foot) 3. Complex Analysis (Ahlfors) 4. Algebraic Curves (Fulton) The last one is a standard upper undergraduate introduction to the subject. If possible you should organize a study group or take a class though, because trying to learn math on your own from a textbook is rough.
- NiftyWalrus 8y ago
  Could you describe what Grothendieck was doing in algerbiac geometry? Would studying the above get you up to the point of his work? If not, what are concrete suggestions that would get you there?
  - obastani 8y ago
    The "Background and history" section of the following article gives a very high-level idea of Grothendieck's contribution: https://en.wikipedia.org/wiki/Weil_conjectures https://en.wikipedia.org/wiki/Weil_conjectures
    - NiftyWalrus 8y ago
      What is the modern path to study this area now? I’m sure it’s better understood now and one wouldn’t have to follow the historical approach to study the same concepts.
mcslearner 8y ago
Depends how much math you know, but Chapter 15 (Commutative Rings and Algebraic Geometry) of Dummit and Foote may be a good starting place.
mtzet 8y ago
I highly recommend the book "An Invitation to Algebraic Geometry" by Karen E. Smith et. al., as an introduction. It'll give you a taste of the field, without delving too deeply into technicalities.
vitriol83 8y ago
The difficulty with learning 'modern' algebraic geometry is not only is it very dense and general, but that means the original motivation can become lost. So I think understanding Weil conjectures are key for modern algebraic geometry. And it's always easier to understand algebraic curves (algebraic geometry with dimension 1) and their connection to Riemann surfaces (algebraic curves over the complex numbers with analytic rather then algebraic structure), as they provide motivation for many of the results and constructions. A good introduction to Algebraic Curves and the Weil conjectures I've found is following https://math.mit.edu/~poonen/papers/curves.pdf https://math.mit.edu/~poonen/papers/curves.pdf For general algebraic geometry, JS Milne's notes are rather good https://www.jmilne.org/math/CourseNotes/ag.html https://www.jmilne.org/math/CourseNotes/ag.html and for an introduction to commutative algebra Atiyah-Macdonald's book is great.
kaitai 8y ago
If you do have some math background, "An Invitation to Algebraic Geometry" by Karen Smith et al is a lovely, slim volume that gets to important ideas right away. "Ideals, Varieties, and Algorithms" is an "invitation to computational geometry" by Cox, Little, and O'Shea. I think parts of it might really appeal to HN readers and it's supposed to be for an undergrad math major audience.
johncarlosbaez 8y ago
My favorite intros to algebraic geometry are these: Igor R. Shafarevich, Basic Algebraic Geometry, two volumes, third edition, Springer, 2013. Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry, 1994. (Especially nice if you like complex analysis, differential geometry and de Rham theory.) David Eisenbud and Joseph Harris, The Geometry of Schemes, Springer, 2010. Currently at https://www.maths.ed.ac.uk/~v1ranick/papers/eisenbudharris.pdf https://www.maths.ed.ac.uk/~v1ranick/papers/eisenbudharris.p... But whenever I want to learn some new field of math I start with Wikipedia. This is a pretty good overview: https://en.wikipedia.org/wiki/Algebraic_geometry https://en.wikipedia.org/wiki/Algebraic_geometry

You'll need a good background in commutative algebra to learn algebraic geometry, if you have that and want to see the modern approach, schemes and everything else that Grothendieck did, I suggest Vakil's notes, they're freely available from his homepage. (Disclaimer: I only studied the first 15 chapters and the one on Kähler differentials which should be the 21st, but I suppose the second half is as good as the first, I'll find out for sure next term)

MaysonL 8y ago
Vakil's notes have an interesting epigram from Grothendieck: I can illustrate the … approach with the … image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!… A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration … the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it … yet finally it surrounds the resistant substance. — A. Grothendieck
- ginnungagap 8y ago
  Vakil seems very fond of that quote, the full title of his notes is "The Rising Sea - Foundations of Algebraic Geometry"!

edflsafoiewq 8y ago

Try some projective geometry to get your feet wet.

kevinventullo 8y ago

Atiyah-Macdonald is a small but dense book with most of the commutative algebra you'll need to start learning algebraic geometry. You'll need to know some abstract algebra as a prerequisite (groups, rings, fields; covered in undergraduate algebra courses).

ginnungagap 8y ago
A less dense book for commutative algebra is Miles Reid's Undergraduate Commutative Algebra, but it doesn't cover tensor product of modules which will definitely be needed for algebraic geometry. Also for everyone interested in Atiyah-Macdonald you MUST do the exercises, half of the book is in the exercises!

throwawaymath 8y ago

This isn't a fun answer, but you basically can't without substantial background in undergrad mathematics. The subject has some of the highest prerequisites of any upper undergrad/lower grad level math course, as it touches on (and uses) just about everything you'd learn through an undergrad math degree. I don't want to discourage you, I'm just being realistic. If you want to work towards algebraic geometry, you can certainly do that. You'll need to first master linear algebra and abstract algebra. You should have a strong understanding of fields, groups, rings, vector spaces and modules. Someone else mentioned commutative algebra - that is more of a circular dependency with algebraic geometry than a hard one. It's good to have walking in, but realistically you can't master the subject without knowing algebraic geometry. You'll also need analysis, in particular complex analysis for curves. Real analysis and topology should also be covered but I suppose with tenacity you could get by without them. To translate these into concrete suggestions, in your position I'd try to work through the following, in order: 1. Linear Algebra Done Right (Axler) 2. Abstract Algebra (Dummit & Foot) 3. Complex Analysis (Ahlfors) 4. Algebraic Curves (Fulton) The last one is a standard upper undergraduate introduction to the subject. If possible you should organize a study group or take a class though, because trying to learn math on your own from a textbook is rough.

NiftyWalrus 8y ago
Could you describe what Grothendieck was doing in algerbiac geometry? Would studying the above get you up to the point of his work? If not, what are concrete suggestions that would get you there?
- obastani 8y ago
  The "Background and history" section of the following article gives a very high-level idea of Grothendieck's contribution: https://en.wikipedia.org/wiki/Weil_conjectures https://en.wikipedia.org/wiki/Weil_conjectures
  - NiftyWalrus 8y ago
    What is the modern path to study this area now? I’m sure it’s better understood now and one wouldn’t have to follow the historical approach to study the same concepts.

mcslearner 8y ago

Depends how much math you know, but Chapter 15 (Commutative Rings and Algebraic Geometry) of Dummit and Foote may be a good starting place.

mtzet 8y ago

I highly recommend the book "An Invitation to Algebraic Geometry" by Karen E. Smith et. al., as an introduction. It'll give you a taste of the field, without delving too deeply into technicalities.

vitriol83 8y ago

The difficulty with learning 'modern' algebraic geometry is not only is it very dense and general, but that means the original motivation can become lost. So I think understanding Weil conjectures are key for modern algebraic geometry. And it's always easier to understand algebraic curves (algebraic geometry with dimension 1) and their connection to Riemann surfaces (algebraic curves over the complex numbers with analytic rather then algebraic structure), as they provide motivation for many of the results and constructions. A good introduction to Algebraic Curves and the Weil conjectures I've found is following https://math.mit.edu/~poonen/papers/curves.pdf https://math.mit.edu/~poonen/papers/curves.pdf For general algebraic geometry, JS Milne's notes are rather good https://www.jmilne.org/math/CourseNotes/ag.html https://www.jmilne.org/math/CourseNotes/ag.html and for an introduction to commutative algebra Atiyah-Macdonald's book is great.

kaitai 8y ago

If you do have some math background, "An Invitation to Algebraic Geometry" by Karen Smith et al is a lovely, slim volume that gets to important ideas right away. "Ideals, Varieties, and Algorithms" is an "invitation to computational geometry" by Cox, Little, and O'Shea. I think parts of it might really appeal to HN readers and it's supposed to be for an undergrad math major audience.

johncarlosbaez 8y ago

My favorite intros to algebraic geometry are these: Igor R. Shafarevich, Basic Algebraic Geometry, two volumes, third edition, Springer, 2013. Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry, 1994. (Especially nice if you like complex analysis, differential geometry and de Rham theory.) David Eisenbud and Joseph Harris, The Geometry of Schemes, Springer, 2010. Currently at https://www.maths.ed.ac.uk/~v1ranick/papers/eisenbudharris.pdf https://www.maths.ed.ac.uk/~v1ranick/papers/eisenbudharris.p... But whenever I want to learn some new field of math I start with Wikipedia. This is a pretty good overview: https://en.wikipedia.org/wiki/Algebraic_geometry https://en.wikipedia.org/wiki/Algebraic_geometry