Walker for algebraic curves?

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I recently became interested in algebraic curves, specifically topics like parametrization and its links to differential equations. I read a number of papers but I'm looking for a good (introduction) textbook on (planar) algebraic curves that gives a solid background, not pure theoretical but with applications in mind. I'm interested in the introduction to basic things like how to classify and treat singularities, how to construct a parametrization, how to determine the genus. Walker seems to be the standard textbook on the matter, but I was wondering if there is a more recent book that is as good as Walker but treats a number of recent developments as well, like e.g. neighborhood graphs. As an alternative, I'm looking at the book of Sendra,Winkler and Perez-diaz, rational algebraic curves.

Any other recommendations?

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  • #2
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That book you mention by Sendra, et. al. looks very good for your stated purposes. As you may have noticed, it borrows heavily from Walker, but omits several lengthy proofs that Walker gives in full. Since you want the results and aplications more than than the proof details, it should serve you well. And when you want more details, your book tells you where to find them in Walker, or sometimes in Fulton. You probably know Fulton's book is available free on his webpage. As for applications to diff eq, e.g., your book is the only one I know of that discusses them.

On the issue of genus and parametrization, as your book probably states, the only curves that admit rational global parametrizations are those of genus zero. It so happens I have recently posted some notes which discuss how to compute the genus of a plane curve in this forum, in the math section, on a thread about maps from a torus to a plane cubic curve. especially post #13:


Maybe they will be useful. Briefly, if you have an irreducible projective plane curve of degree d, and if it has no singular points, then the genus is g = (1/2)(d-1)(d-2), so it is rational if and only if d ≤ 2. If it has some singular points, the method I discuss for computing the genus requires you to compute the number r of local branches at the singularity, as wlll as the "Milnor number" m of "vanishing cycles". Then the genus is smaller than that for a non singular curve, diminished by the number ∂ = (1/2)(m+r-1), computed for each singularity.

E.g. an irreducible plane cubic curve with one singularity which is an ordinary double point, looks like two smooth branches crossing at a point, and has r = 2, and m = 1, hence the drop in genus contribued by this singularity is 1, and so the actual genus of the cubic is (1/2)(3-1)(3-2) - (1/2)(1+2-1) = 1-1 = 0. so this curve is rational.

a cubic with one "cusp" has only one branch where m = 2, so the drop in genus is again (1/2)(2+1-1) = 1, so again the genus is zero and the curve is rational.

By the way, as Walker makes clear, an irreducible curve of degree d, with a single singular point of multiplicity d-1, is already rational, and can be parametrized using Bezout's theorem. Thus a cubic with a double point is always rational. I.e. a line through the singular point meets the cubic "twice" at the double point, and hence mets the curve elsewhere only once, since the total number of intersections is 3 by Bezout. Hence by letting the line revolve around the singular point, the extra intersection sweeps out the whole cubic curve, parametrizing each point by one of the lines. Hence the curve is parametrized by the 1 dimensional parameter space for the family, or "pencil" of lines through the singularity.

More concretely, choose another random line. Then each line through the singularity of the cubic meets the cubic once further and also meets the random line once. This sets up a one -one correspondence between the points of the cubic and the points of the random line. Technically there are a finite number of exceptions to the one-one correspondence, given by those lines that are tangent to one of the branches of the cubic. So a rational "parametrization" is generically a bijection with a line, but sometimes with a finite number of exceptions.

Anyway it is a lovely story. The Milnor number formula in my notes is probably not discussed in your book, nor in Walker. The harder part of using the Milnor number formula is calculating the number of branches as I recall. As to the Puiseux method in Walker, and Sendra, perhaps that is acomplished by a local factorization of a formal power series. I.e. my branches correspond to what Walker calls "places" and he seems to have a method of computing their number, or at least he has some exercises asking you to do so.

Another good book on curves, but not applications or computer algebra, is the one by Rick Miranda, where I learned a lot. Fulton is also a masterpiece, especially for intersection numbers, for which it is cited by your book, and the book by Gerd Fischer looks nice as well. Brieskorn and Knorrer is also rather impressive. But if you want a useful, finite introduction, rather than a life's project, I think you will be happy to start with Sendra et. al., as you suggested.
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  • #3
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I can also recommend Fulton. I was able to read (a great part of) it with no background in algebraic geometry. You just need basic knowledge about abstract algebra.
  • #4
Thanks to both of you for your replies! I will now definitely get the book of Sendra et al. I indeed found the book of Fulton online, I will also study that in more detail. I will leave the purchase of Walker for another time then - when I can find it for a good price.
Most online lecture notes on the topic mention the genus formula and sometimes the milnor number, and they talk about singularities and multiplicities, but it is not enough for me to really say e.g. "ok, now I can take an algebraic curve and determine its genus, no matter what type the singularity is". I think a good textbook will clear up some things for me.

I will read your other post as well, @mathwonk , good to know that there is an expert in the room!
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  • #6
That's not too bad actually, but with shipping to the Netherlands it jumps to $50. But I see some european sellers as well, I might buy this one as well... plenty of time now with all those mandatory holidays.

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