The structure of algebraic threefolds: an introduction to Mori’s program

25 per page 211 2. What is algebraic geometry? 2.1. STARTING POINT. After Descartes introduced coordinates on the plane, it became clear that simple and much studied geometric objects (e.g., lines and conies) can be defined by simple polynomial equations (linear, resp. quadratic). This suggests that as a next step one could try to study curves defined by higher-degree polynomials. Already Newton studied plane cubics in depth. Two problems, however, tended to make results cumbersome. The first one is the missing infinity. Two different lines mostly intersect in a point, but sometimes they are parallel. It turns out to be very convenient to claim that they intersect at infinity. This leads to the introduction of the projective plane RP 2 . The other problem is apparent already with one-variable polynomials: the roots of a decent-looking polynomial can be lurking in the complex plane far away from the reals. Even when the real picture seems good, the explanation of certain phenomena might be seen only by studying them over C: For example, why does the Taylor series of (1 + JC 2 )" 1 refuse to converge everywhere? Therefore we replace R by C and get CP 2 . There is no reason to stop with dimension 2 and so we introduce: 2.2. DEFINITION of CP n . As a point set this consists of (n + l)-tuples (x 0 :. ..:x n ) such that (x 0 :...:x n )