# Books for (Complex) Algebraic Curves

I would like to suggest me some good books on (Complex) Algebraic Curves.

I am searching a book for Undergraduate-Begginer Level in this part of mathematics, the algebraic curves. I found some books like "Plane Algebraic Curves" from Gerd Fischer, "Complex Algebraic Curves" from Frances Kirwan, "Elementary Geometry of Algebraic Curves: An Undergraduate Introduction" from Gibson but these were too difficult for my level.

Also, please suggest my a book (or combine chapters of books or notes), which covers the following topics:

1. Plane Curves; The set of points $V(f)$ of a plane curve, Transformations in $\mathbb{C}^2$, Conics, Intersection number, Isolated (abnormal) points, tangent lines, rational curves

2. Intersection Of Curves; The projective space, Lines in projective space, Projective transformations, Εquivalence of projective curves, Intersection number, Irregular poits, Tangent Lines, Flex points, Rational Projective Curves

3. Linear Systems of Curves; Intersection Points, Bézout's theorem

• William Fulton has a book on this topic. math.lsa.umich.edu/~wfulton/CurveBook.pdf – Xuqiang QIN Nov 12 '16 at 1:32
• I really enjoyed reading Plane algebraic curves by Brieskorn/Knörrer. – Takumi Murayama Nov 12 '16 at 2:02
• @Xiqiang QIN I found Fulton too difficult for undergraduate. Could you suggest me something else? Thank you. – Chris Nov 12 '16 at 2:09
• Gibson's book is very elementary; he assumes only linear algebra and several variable calculus. If you don't have that background, you would be better off to learn some linear algebra and some several variable calculus first before trying to learn about algebraic curves. – Ben McKay Nov 12 '16 at 12:03
• H. Clemens: A scrapbook of complex curve theory. – David Lehavi Jan 30 '18 at 12:33

A good classical book is Walker, Algebraic curves, Princeton, 1950. A more modern one on the same elementary level is Gerd Fischer, Plane algebraic curves, AMS, 2001.

Both books a small and elementary, ideal for the first introduction.

• Does Walker use field extensions? Thank you. – Chris Nov 13 '16 at 21:06
• @Chris: Yes, he does, but not much. Beginning from Chap V, section 2. – Alexandre Eremenko Nov 14 '16 at 13:41
• Do you have something on your mind without extensions? Walker looks great, but I can't use extensions yet. – Chris Nov 14 '16 at 22:24
• @Chris: Your strange requirement essentially puts us back to 19th century. Try Salmon, A treatise on the higher plane curves, but this can be really hard reading, unlike modern books that I recommended. – Alexandre Eremenko Nov 15 '16 at 22:06
• You are right. I had a look in these books and I will prefer the mordern books which do not use too much extensions. I think the best for my case is Gibson's. Something else in Gibson's spirit? Thank you very much for your time. – Chris Nov 16 '16 at 19:54

When I was a student I liked a lot the book by Egbert Brieskorn and Horst Knörrer: Plane Algebraic Curves. I can recommend it.

• Thank you for your answer. As far as I saw this book, it was well written and clear. – Chris Jan 30 '18 at 14:27

Rick Miranda's Algebraic Curves and Riemann Surfaces is a great place to look for a more complex analytic point of view. I think it starts from very little and only asks you know a bit of complex analysis. See here for a review of this book by Gunning.

As David Lehavi has already recommended in the comments, Herbert Clemens's A Scrapbook of Complex Curve Theory is a beautiful panorama into all the topics you mention. The treatment of technical material is perhaps a bit brief, but it provides marvelous intuition as well as a coherent picture as to how everything fits together.

Finally, Miles Reid's Undergraduate Algebraic Geometry, again although somewhat informal, is simply a fascinating read, if nothing other than for the Woffle that begins the book. See here for a pretty $\TeX$ version by Reid himself.

Keith Kendig's A guide to plane algebraic curves is quite elementary and readable.

• Thank you for your answer. I ll have a look! – Chris Nov 18 '18 at 17:50

Another beginner-friendly book is P.A. Griffiths, Introduction to algebraic curves.

• Thank you for your answer. This book is really useful! – Chris Nov 24 '18 at 21:36