# What is algebraic curves: Definition and 16 Discussions

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation h(x, y, t) = 0 can be restricted to the affine algebraic plane curve of equation h(x, y, 1) = 0. These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.
More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a birational equivalence.
These birational equivalences reduce most of the study of algebraic curves to the study of algebraic plane curves. However, some properties are not kept under birational equivalence and must be studied on non-plane curves. This is, in particular, the case for the degree and smoothness. For example, there exist smooth curves of genus 0 and degree greater than two, but any plane projection of such curves has singular points (see Genus–degree formula).
A non-plane curve is often called a space curve or a skew curve.

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1. ### Book recommendations about singular points of algebraic curves

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...
2. ### I Questions about algebraic curves and homogeneous polynomial equations

It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~[1]. In addition, if...
3. ### I Finding intersection of two algebraic curves

Given two algebraic curves: ##f_1(z,w)=a_0(z)+a_1(z)w+\cdots+a_n(z)w^n=0## ##f_2(z,w)=b_0(z)+b_1(z)w+\cdots+b_k(z)w^k=0## Is there a general, numeric approach to finding where the first curve ##w_1(z)## intersects the second curve ##w_2(z)##? I know for low degree like quadratic or cubics...
4. ### Geometry Is Walker's Textbook the Best Resource for Algebraic Curves?

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...
5. ### MHB Real Algebraic Curves: Solving Example 1.4 from C.G. Gibson's Book

I am reading C. G. Gibson's book: Elementary Geometry of Algebraic Curves. I need some help with aspects of Example 1.4 The relevant text from Gibson's book is as follows: Question 1In the above text, Gibson writes the following: " ... ... Then a brief calculation verifies that any point p +...
6. ### MHB Affine Algebraic Curves - Kunz - Theorem 1.3

I am reading Ernst Kunz book, "Introduction to Plane Algebraic Curves" I need help with some aspects of Kunz' proof of Theorem 1.3 ... The relevant text from Kunz is as follows:http://mathhelpboards.com/attachment.php?attachmentid=4559&stc=1In the above text we read the following: " ... ...
7. ### MHB Affine Algebraic Curves - Kunz - Definition 1.1

I am reading Ernst Kunz book, "Introduction to Plane Algebraic Curves" I need help with some aspects of Kunz' Definition 1.1. The relevant text from Kunz' book is as follows:In the above text, Kunz writes the following: " ... ... If K_0 \subset K is a subring and \Gamma = \mathscr{V} (f) for...
8. ### Affine Algebraic Curves - Kunz - Exercise 1 - Chapter 1

I am reading Ernst Kunz book, "Introduction to Plane Algebraic Curves" I need help with Exercise 1, Chapter 1 ... Indeed ... I am a bit overwhelmed by this problem .. Exercise 1 reads as follows: Hope someone can help ... ...To give a feel for the context and notation I am providing the...
9. ### MHB Affine Algebraic Curves - Kunz - Exercise 1 - Chapter 1

I am reading Ernst Kunz book, "Introduction to Plane Algebraic Curves" I need help with Exercise 1, Chapter 1 ... Indeed ... I am a bit overwhelmed by this problem .. Exercise 1 reads as follows:https://www.physicsforums.com/attachments/4549Hope someone can help ... ...To give a feel for the...
10. ### MHB How to Interpret and Visualize Affine Algebraic Curves in Ernst Kunz's Book?

I am reading Ernst Kunz book, "Introduction to Plane Algebraic Curves" I need help with interpreting Example 1.2. The relevant text pertaining to Example 1.2 is as follows: https://www.physicsforums.com/attachments/4548 Question 1 In Example 1.2 above how do we interpret aX + bY + c = 0? ...
11. ### MHB Kunz - Plane Algebraic Curves - Theorem 1.3 - basic question

I am reading the book, "Introduction to Plane Algebraic Curves" by Ernst Kunz - which the author claims gives a basic introduction to the elements of algebraic geometry. I need help with an apparently simple statement that I find confusing and puzzling. Theorem 1.3 and its proof reads as...
12. ### MHB Plane algebraic curves - basic definition of affine plane

I am reading the book, "Introduction to Plane Algebraic Curves" by Ernst Kunz - which the author claims gives a basic introduction to the elements of algebraic geometry. The opening few paragraph of Kunz' text reads as follows:I am puzzled by Kunz statement: " \mathbb{A} (K) := K^2 denotes...
13. ### Geometry Algebraic Curves and Riemann Surfaces by Miranda

Author: Rick Miranda Title: Algebraic Curves and Riemann Surfaces Amazon Link: https://www.amazon.com/dp/0821802682/?tag=pfamazon01-20 Prerequisities: Complex Analysis, Differential Geometry, Abstract Algebra Level: Grad Table of Contents: Preface Riemann Surfaces: Basic Definitions...
14. ### Intersection of Algebraic Curves P & Q at p - Proof

Hi I am pretty stuck on a proof so any help would be great: Let P and Q be two projective curves, and let p belong to both of them. Show that the intersection number of P and Q at p is equal to one iff the tangent lines to p of P and Q are distinct NB-we have defined intersection numbers...
15. ### Algebraic Curves ( on understanding this proof)

I don't get why G=0 is a contradiction. Does it imply F=0, which cannot be true since the question stated F is non constant? Can anyone give me another proof for this first part please? As the step he made to get G would have been something I would never have thought of. By the way problem 1.4...
16. ### Advanced Algebraic Curves problem

[Question] Let p1, p2 and p3 be 3 distinct points in PC2( Projective space, ie (z0,z1,z2) belong to PC2) Find the dimension of the linear system of cubics containing these 3 points. I have solved it for the non collinear case, by taking a projective transformation of the 3 points to...