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
V9999
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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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1. What are singular points of algebraic curves?

Singular points of algebraic curves are points on a curve where the curve is not smooth or differentiable. This means that the curve has a sharp point, cusp, or self-intersection at that point.

2. Why are singular points important in the study of algebraic curves?

Singular points are important because they give information about the behavior of the curve. They can help determine the degree of the curve and its intersection with other curves, and they can also reveal symmetries and other geometric properties of the curve.

3. What are some common techniques used to study singular points of algebraic curves?

Some common techniques used to study singular points of algebraic curves include computing the tangent lines at the point, finding the Jacobian matrix, using intersection theory, and analyzing the curve's parametrization.

4. Are there any recommended books that specifically focus on singular points of algebraic curves?

Yes, there are several recommended books that focus on singular points of algebraic curves, such as "Singular Points of Complex Hypersurfaces" by János Kollár and "Singular Points of Plane Curves" by C. T. C. Wall.

5. Can you recommend any online resources for learning about singular points of algebraic curves?

Yes, there are several online resources that provide information and tutorials on singular points of algebraic curves, such as the "Singular Points of Plane Curves" lecture notes by John Milne and the "Singular Points of Plane Curves" course by David Speyer on the website MathOverflow.

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