CHAPTER 1

Intersections of Hypersurfaces

1.1. Early history (Bezout, Poncelet)

Á most basic question in intersection theory is to describe the intersection of

several algebraic hypersurfaces in n-spaee, i.e., the common Solutions of several

polynomials in ç variables. The ancients certainly knew about the possible inter-

sections of lines and conics in the plane, and they also knew that rational Solutions

of two quadric equations in three variables behaved like Solutions of one cubic equa-

tion in two variables [61].

We do not know who first observed that two plane curves of degrees ñ and q

should intersect in pq points. By 1680 Newton [48] had developed an elimination

theory for two such equations. This produced a resultant, which was a polynomial in

one variable of degree pq whose Solutions gave an abscissa of the intersection points

of the two curves. The corresponding construction and assertion for ç equations in

ç variables were made in 1764 by Bezout [5, 6]. Bezout's treatment was entirely

algebraic, although he briefly interpreted his result for ç = 2 and ç = 3: the

number of intersections of two plane curves (or three surfaces in space) is at most

the products of their degrees.

By referring to the resultants, which are polynomials in one variable, one can

also discuss the possibilities of nonreal Solutions, asymptotic Solutions, and multi-

ple solutions. As geometry developed, the first two of these situations were sub-

sumed by considering intersections of hypersurfaces H\,... , Hn in complex projec-

tive space Pg. Now we assign an intersection multiplicity

i(P)=i(P,H1.....Hn)

to a point Ñ of the intersection f) Ç÷\ if the Hi do not meet transversally at P , this

multiplicity will be greater than one.

Although there was little early discussion of this multiplicity, the governing

principle of continuity was well understood, at least since Poncelet [51]. If the Hi

vary in families Hi(t), with Hi(0) = Hi, and Pi(t),... , Pr(t) are the points of the

general intersection f]Hi(t) which approach Ñ as t — 0, then

r

i(P,£Ti · ... · Hn) = Óé(Ñ&), H^t) · ... · Hn(t)).

3 = 1

Varying the Hi so that the Hi(t) meet transversally, this determines the multiplicity

i ( P , f f i . . . . . f f

n

) .

In all the above discussion, it is assumed that the intersection of the hypersur-

faces is a finite set, or at least that Ñ is an isolated point ïú f)Hi.

http://dx.doi.org/10.1090/cbms/054/01