Book recommendations about singular points of algebraic curves

In summary: The best book that I have read on singular points is Singular Points on Algebraic Curves by Geoffrey A. Shafarevich.
  • #1
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I'm not quite sure if this is an appropriate question in this forum, but here is the situation.

I have just finished my graduate studies. Now, I want to explore algebraic geometry. Precisely, I am interested in the following topics:
Singular points of algebraic curves;
General methods employed to determine the singular points of algebraic curves;
Classification of singular points of algebraic curves;

Based on your experience, what are the best books/references for self-study on those topics?
 
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  • #2
I know only a little about singular points, but I myself began first to have some grasp of them by reading chapter 3 of the well written and precise little book by Robert J. Walker, Algebraic Curves. Here is a cheap used copy:
https://www.abebooks.com/9780486603360/Algebraic-Curves-Walker-Robert-J-0486603369/plp

this one helped me personally the most. The others I have on my shelf are:

Shafarevich, Basic Algebraic Geometry, vol. I, 2nd edition, chapter IV.4.

I have not read the following as much, but hope to some day:

The wonderful book by Milnor: Singularities of complex hypersurfaces,
https://www.amazon.com/dp/0691080658/?tag=pfamazon01-20

and for surfaces only: Normal; two dimensional singularities, by Henry Laufer. (I have never gotten into this, but he is an expert.)

I have dipped into this next one with good results, especially (I think) its accounts of Milnor's results:
Introduction to singularities and deformations, by Greuel, Lossen and Shustin.

Another excellent one whose summaries of results have helped me is:
V. Arnol’d, S. Gusein-Zade, A. Varchenko, Singularities of Differentiable Maps,
vol.I, Monographs in Mathematics, Birkh¨auser, 1985.

So to get started, I suggest Walker. Oh yes, and you might take a look at chapter 3 of Plane algebraic curves, by Brieskorn and Knorrer. and the Shafarevich reference above.
 
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  • #3
mathwonk said:
I know only a little about singular points, but I myself began first to have some grasp of them by reading chapter 3 of the well written and precise little book by Robert J. Walker, Algebraic Curves. Here is a cheap used copy:
https://www.abebooks.com/9780486603360/Algebraic-Curves-Walker-Robert-J-0486603369/plp

this one helped me personally the most. The others I have on my shelf are:

Shafarevich, Basic Algebraic Geometry, vol. I, 2nd edition, chapter IV.4.

I have not read the following as much, but hope to some day:

The wonderful book by Milnor: Singularities of complex hypersurfaces,
https://www.amazon.com/dp/0691080658/?tag=pfamazon01-20

and for surfaces only: Normal; two dimensional singularities, by Henry Laufer. (I have never gotten into this, but he is an expert.)

I have dipped into this next one with good results, especially (I think) its accounts of Milnor's results:
Introduction to singularities and deformations, by Greuel, Lossen and Shustin.

Another excellent one whose summaries of results have helped me is:
V. Arnol’d, S. Gusein-Zade, A. Varchenko, Singularities of Differentiable Maps,
vol.I, Monographs in Mathematics, Birkh¨auser, 1985.

So to get started, I suggest Walker. Oh yes, and you might take a look at chapter 3 of Plane algebraic curves, by Brieskorn and Knorrer. and Shafarevich.
Many, many thanks for the suggestions!
 
  • #4
ok here is a comprehensive treatment by an expert, of the full range of ideas involved in studying singular points of plane curves. Unfortunately it is not cheap. I also have a (used) copy of this on my shelf and it looks quite promising, but I have not read it much yet. Singular points of plane curves, by C.T.C.Wall:
at least there is an affordable ecopy available and a used copy at half the exhorbitant new price: it should also be available in libraries. I would still start with Walker.

https://www.amazon.com/dp/0521839041/?tag=pfamazon01-20
 
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