Some simple types of curves

Egbert Brieskorn and Horst Knörrer
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Egbert Brieskorn: Universität Bonn, Mathematisches Institut
Horst Knörrer: Universität Bonn, Mathematisches Institut

Chapter 7 in Plane Algebraic Curves, 1986, pp 278-322 from Springer

Abstract: Abstract Next to lines, quadrics are the simplest plane curves. From the complex-projective standpoint they are the analogues of the conic sections of antiquity. If one also admits curves with multiple components, and thus understands the quadrics to include all “curves” with equations of degree 2, then a quadric is just a curve with a homogeneous equation ∑ i , j = 0 2 a i j x i x j = 0 , $$\sum\limits_{i,j = 0}^2 {{a_{ij}}{x_i}{x_j} = 0,} $$ where one can assume without loss of generality that aij = aji. The polynomial Σaijxixj is a form of degree 2, a quadratic form. If A is the matrix (aij), then one can write this form in matrix fashion as follows: x t A x . $${x^t}Ax.$$

Keywords: Normal Form; Inflection Point; Complex Torus; Intersection Multiplicity; Complex Projective Plane (search for similar items in EconPapers)
Date: 1986
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DOI: 10.1007/978-3-0348-5097-1_7

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