Some simple types of curves

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

$$\sum\limits_{i,j = 0}^2 {{a_{ij}}{x_i}{x_j} = 0,} $$

where one can assume without loss of generality that a_ij = a_ji. The polynomial Σa_ijx_ix_j is a form of degree 2, a quadratic form. If A is the matrix (a_ij), then one can write this form in matrix fashion as follows:

$${x^t}Ax.$$