Plane algebraic curve | Wikipedia audio article
Channel: wikipedia tts
Date: 29/05/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Algebraic_curve


00:01:43 1 In Euclidean geometry
00:04:35 2 Plane projective curves
00:06:10 3 Remarkable points of a plane curve
00:10:17 3.1 Intersection with a line
00:11:05 3.2 Tangent at a point
00:12:18 3.3 Asymptotes
00:12:38 3.4 Singular points
00:13:19 4 Analytic structure
00:18:02 5 Non-plane algebraic curves
00:18:10 6 Algebraic function fields
00:24:04 7 Complex curves and real surfaces
00:28:54 7.1 Compact Riemann surfaces
00:38:55 8 Singularities
00:43:43 8.1 Classification of singularities
00:44:30 9 Examples of curves
00:45:27 9.1 Rational curves
00:46:24 9.2 Elliptic curves
00:47:40 9.3 Curves of genus greater than one
00:48:46 10 See also
00:50:04 10.1 Classical algebraic geometry
00:53:20 10.2 Modern algebraic geometry
00:56:25 10.3 Geometry of Riemann surfaces
00:58:15 11 Notes
00:58:24 12 References
01:00:08 1, where (−1, 0) is a rational point. Drawing a line with slope t from (−1,0), y
01:04:01 Elliptic curves
01:07:40 Curves of genus greater than one
01:08:14 zn when n is greater than three.
01:08:19 See also
01:08:28 === Classical algebraic geometry
01:08:42 === Modern algebraic geometry
01:08:56 === Geometry of Riemann surfaces
01:09:10 



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SUMMARY
=======
In mathematics, a 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 can be restricted to an affine algebraic plane curve by replacing by one some indeterminate of the defining homogeneous polynomial. As these two operations are each inverse to the other, 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 allow reducing most of the study of algebraic curves to the study of algebraic plane curve. However some properties are not kept under birational equivalence and must be studied on non-plane curves (often called space curves or skew curves). This is, in particular the case of smoothness, since many non-singular algebraic curves are not birationally equivalent to any algebraic plane curve (this is the case of all curves of positive genus).