Bezout Curves in the Plane

First, recall how Bezout’s Theorem can be made more precise by specifying the meaning of the word “generically” used above, using elementary scheme-theoretic language ([H, II]). Fix an algebraically closed field k. There is little loss in assuming k = C (until Section 3). Since we work over an algebraically closed field, we often identify reduced algebraic sets with their k-valued points. In this paper, a plane curve over k is a purely 1-dimensional closed subscheme of P2 (not necessarily reduced or irreducible, but projective by definition). Such a scheme is always of the form Proj(k[X,Y, Z]/(f)) for some homogeneous polynomial f ∈ k[X,Y, Z], uniquely determined up to scalars ([H, I.1.13]), hence curves of a given degree d > 1 are naturally parameterized by an irreducible algebraic variety Hd over k, namely the projective space PN−1 k associated with the N = (d + 1)(d + 2)/2 coefficients of a homogeneous polynomial P ∈ k[X,Y, Z] of degree d (which gives the equation for the curve). We also consider subsets S ⊂ P2(k) of the set of closed points of P2, and say that such an S is a curve when S = C(k) for some plane curve C, and C is unambiguously defined if we ask that it be reduced. Similarly the Zariski closure of a subset S ⊂ P2(k) is the smallest reduced closed subscheme S with S ⊂ S(k). Now a more precise form of Bezout’s Theorem is: let C/k be a reduced curve of degree c. Then for any d > 1, there exists a Zariski open subset Ud ⊂ Hd such that for all curves D ∈ Ud, and all x ∈ C ∩D, i(C,D;x) = 1. Hence, for D outside a closed subset of Hd, the intersection C ∩D is a finite set with cd elements.