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.

theory of algebraic curves from the viewpoint of modern algebraic geometry, but without excessive prerequisites. We have assumed that the reader is familiar with some basic properties of rings, ideals, and polynomials, such as is often covered in a one-semester course in mod-ern algebra; additional commutative algebra is developed in later ...

Dec 17, 2019 · An algebraic curve is the most frequently studied object in algebraic geometry. In the sequel, an algebraic curve means an irreducible algebraic curve over an algebraically closed field. The simplest and clearest concept is that of a plane affine algebraic curve.

algebraic curves as a smooth projective curve (given by explicit equations), and an explicit divisor, there is an algorithm to determine the space L(D). We’ll do that in a week or two. The one thing to observe is that there’s a certain redundancy here, in this problem.

curves. 0BXY Lemma 2.1. Let kbe a field. LetX be a curve and Y a proper variety. Let U ⊂X be a nonempty open and let f: U →Y be a morphism. If x∈X is a closed point such that O X,x is a discrete valuation ring, then there exist an open U⊂U′⊂Xcontainingxandamorphismofvarietiesf′: U′→Y extendingf. Proof ...

5 days ago · An algebraic curve over a field K is an equation f(X,Y)=0, where f(X,Y) is a polynomial in X and Y with coefficients in K. A nonsingular algebraic curve is an algebraic curve over K which has no singular points over K. A point on an algebraic curve is simply a solution of the equation of the curve.

We show one can construct algebraic curves passing through a given finite collection of points in the plane (interpolation). We look at other ways in which algebraic curves can arise, through rational or trigono-metric parametrizations. (19a) Definition and first examples.

An algebraic curve is a curve which is described by a polynomial equation: f(x,y) = X aijx iyj= 0 in xand y. The degree of the curve is the degree of the polynomial f(x,y). A parametric curve α(t) = (p(t),q(t)), where p(t) and q(t) are polynomials in t, is an algebraic curve. This is because

An algebraic curve is an algebraic variety each of whose irreducible components has dimension one; a plane algebraic curve is an algebraic curve of codimension one, i.e.

We see some of the purely algebraic results reappearing topologically. The last section contains some exercises about rational curves. 1. Families of Plane Curves. Formula 1.1. The projective space V (d) of all plane curves of degree d has dimen-sion d(d + 3)=2.