Page 68 -

2

Plane Algebraic Curves

In the Introduction and Chapter 1 we considered several ways of studying properties
of rational points on curves using intersection theory, and we saw how modifications
had to be made for singular points. In this chapter we develop background material
on projective spaces, plane curves, and to a limited extent on hypersurfaces, and in
the next chapter we apply these results to cubic curves.
Elementary intersection theory and the theory of singular points, as given here,
is based on the resultant of two polynomials. This is a very classical and elementary
approach. In the Appendix to this chapter the theory of the resultant is worked out
and other background in algebra is supplied. In particular many aspects related to
foundational questions in algebraic geometry are not made explicit. Frequently the
same symbol is used for coordinates and for variables.
This chapter is used to give some details left open concerning the group law and
singular points on cubics. It will be used also in Chapter 5 for the reduction modulo
p of curves defined over the rational numbers.

§1. Projective Spaces

In the Introduction we considered the projective plane in order to have a satisfactory
intersection theory of lines in the plane. The result of the basic geometric assertion:

(P) Two distinct points determine, i.e., lie on, a unique line, and two distinct lines
determine, i.e., intersect at, a unique point.

The projective plane was modeled on the set of one-dimentional subspaces of
3
the vector space k over the field k. The lines in the projective plane were sets of
all one-dimentional subspaces contained in a given two-dimentional subspace. In
this context projective transformations are just induced by linear automorphisms by
taking direct image.
It will be useful to have higher-dimentional projective spaces, as we did in the
Introduction, in order to speak of the space of all cubic curves.