Plane curves

Abstract

This section is devoted to projective spaces. The question of defining “projective space” is akin to defining “sphere” in that both terms are used at several different levels. By “sphere” one can mean a topological sphere, or sphere as a differential manifold, or, more strictly, as the set V(X² + Y² + Z² − 1) in ℝ³. This last object has the most structure of all; for instance, it is also a real algebraic variety. In turn, one can take this specific object, with its multitude of properties, and isolate certain of its properties to get other notions of “sphere.” Its equation is defined by a sum of squares, so one has the n-dimensional real sphere V(X₁ ² + … + X_n ² − 1). We also have, analogously, complex n-spheres, when k = ℂ; ℚ-spheres, when k = ℚ; and so on.