Transcription of Chapter 5 Basics of Projective Geometry
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Chapter 5 Basics of Projective Geometry
Chapter 5 Basics of Projective GeometryThink geometrically, prove algebraically. John Why Projective Spaces?For a novice, Projective Geometry usually appears to be a bitodd, and it is notobvious to motivate why its introduction is inevitable and motivations arises from algebraic main goal of algebraic Geometry is to study the propertiesofgeometricob-jects, such as curves and surfaces, defined implicitly in terms of algebraic instance, the equationx2+y2 1=0defines a circle ,wecanconsiderthecurvesdefinedbygenerale quationsax2+by2+cxy+dx+ey+f=0of degree 2, known curves according to their generic geometric shape. This is indeed for so-called singular cases, we get ellipses, parabolas, and hyperbolas. Thesame question can be asked for surfaces defined by quadratic equations, knownasquadrics,andagain, ,theseclassificationsare a bit artificial. For example, an ellipse and a hyperbola differ by the fact thatahyperbolahaspointsatinfinity,andyet ,theirgeometricproperties are identical,provided that points at infinity are handled important problem is the study of intersection of geometric objects (de-fined algebraically).
nel [14], Sidler [24], Tisseron [26], Lehmann and Bkouche [20], Vienne [30], and the classical treatise by Veblen and Young [28, 29], which, although slightly old-fashioned, is definitely worth reading. Emil Artin’s famous book [1] contains, among other things, an axiomatic presentation of projectivegeometry,andawealth
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