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Algebraic Geometry Help

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Hello MHB.

My masters thesis is on the subject of tensegreties (Most likely you don't know what that means.) Their study requires familiarity with Algebraic Geometry. I have no background in AG.

I need to understand this paper:
https://docs.google.com/file/d/0B77QF0wgZJZ7d2g2S3g1RVlQOE0/edit

Proposition 3.2 in the paper uses concept of an 'algebraic set', 'path selection results from AG' and 'curve selection result due to Milnor'.

At this point I have no idea what these are.

I have good understanding of UG algebra (esp groups and fields & Galois theory), Analysis and Point-set Topology.

Can somebody tell me what book(s) and topics I should cover so that I'd be in a position to understand the the paper at least partially?

I am allowed to do 3 months of reading bookish material and then go to reading papers.
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,705
Hello MHB.

My masters thesis is on the subject of tensegreties (Most likely you don't know what that means.) Their study requires familiarity with Algebraic Geometry. I have no background in AG.

I need to understand this paper:
https://docs.google.com/file/d/0B77QF0wgZJZ7d2g2S3g1RVlQOE0/edit

Proposition 3.2 in the paper uses concept of an 'algebraic set', 'path selection results from AG' and 'curve selection result due to Milnor'.

At this point I have no idea what these are.

I have good understanding of UG algebra (esp groups and fields & Galois theory), Analysis and Point-set Topology.

Can somebody tell me what book(s) and topics I should cover so that I'd be in a position to understand the the paper at least partially?

I am allowed to do 3 months of reading bookish material and then go to reading papers.
I know very little algebraic geometry, but if I had to study that paper on tensegrity frameworks, I would start by looking at its Reference 10: Singular points of complex hypersurfaces, by John Milnor. You can get an idea of the style of Milnor's monograph by looking at the Google Books extracts here. Milnor seems to start in a fairly gentle way, by defining an algebraic set. The proof of his curve selection lemma comes quite early in the monograph, so you probably would not need to read the whole thing.

If you find Milnor tough going, you need to go back further, to a gentle introduction to the basics of algebraic geometry. I recommend Miles Reid: Undergraduate algebraic geometry.
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
I know very little algebraic geometry, but if I had to study that paper on tensegrity frameworks, I would start by looking at its Reference 10: Singular points of complex hypersurfaces, by John Milnor. You can get an idea of the style of Milnor's monograph by looking at the Google Books extracts here. Milnor seems to start in a fairly gentle way, by defining an algebraic set. The proof of his curve selection lemma comes quite early in the monograph, so you probably would not need to read the whole thing.

If you find Milnor tough going, you need to go back further, to a gentle introduction to the basics of algebraic geometry. I recommend Miles Reid: Undergraduate algebraic geometry.
Thank you Opalg for this detailed post.

Umm.. I don't think Milnor defines an Algebraic Set in that monograph.

Would you suggest reading commutative ring theory separately? I may be able to cover decent material from Martin Isaac's Algebra: A Graduate Course Algebra (Graduate Studies in Mathematics): I. Martin Isaacs: 9780821847992: Amazon.com: Books
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,705
Umm.. I don't think Milnor defines an Algebraic Set in that monograph.
Umm.. it's the first definition in the monograph!

Screen shot 2013-09-06 at 17.20.46.png

Would you suggest reading commutative ring theory separately?
No, I doubt whether that would be very much help. My impression is that, for the application you are interested in, you do not need a "modern", commutative-algebra-based version of algebraic geometry, but a more traditional, geometric approach that emphasises curves and surfaces. That is why I think that Miles Reid's book would be a good starting point.​
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Umm.. it's the first definition in the monograph!


Ummsorry. I saw the first line in section 1 where the author uses the name algebraic set without defining it. That's why I thought it's not defined in it. :eek:
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Hello. Me again.

I have been reading the monograph by Milnor and Reid's book's chapter 2 seems to make things easy for me.

But I am having problems with things like 'smooth manifolds' and 'dimension of a manifold'. I know basic point set topology (as given in Part 1 of Simmon's Introduction to Topology and Modern Analysis) and I am comfortable with compactness, connectedness, Hausdorffness etc. But in Simmon's book manifolds weren't discussed at all. So that's why I need to read about them separately if I am to understand the curve selection lemma given in Milnor's monograph.

Can you suggest a good source for this. I wouldn't want to read something equivalent to a full course in these things since I don't have that much time. So please make suggestions keeping that in mind.

Thanks.
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,705
I am having problems with things like 'smooth manifolds' and 'dimension of a manifold'. I know basic point set topology (as given in Part 1 of Simmon's Introduction to Topology and Modern Analysis) and I am comfortable with compactness, connectedness, Hausdorffness etc. But in Simmon's book manifolds weren't discussed at all. So that's why I need to read about them separately if I am to understand the curve selection lemma given in Milnor's monograph.

Can you suggest a good source for this. I wouldn't want to read something equivalent to a full course in these things since I don't have that much time. So please make suggestions keeping that in mind.
If you just need definitions and basic properties then you can find these online. Wikipedia is probably as good as anything. Start on the manifolds page, and follow references from there.
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
If you just need definitions and basic properties then you can find these online. Wikipedia is probably as good as anything. Start on the manifolds page, and follow references from there.
Okay.. I'll give it a shot. Thanks. My problem is I myself don't know how much I need to know about manifolds in order to understand the curve selection lemma. But anyway, I wouldn't know without trying I guess.