algebraic curves Definition and 11 Threads

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...

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...

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...

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...

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...

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}...

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...

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: "$$ \mathbb{A} (K) := K^2 $$...

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...

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...

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...