intersection theory

In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand the topological theory more quickly reached a definitive form.

Topological intersection form

For a connected oriented manifold M of dimension 2n the intersection form is defined on the n^th cohomology group (what is usually called the 'middle dimension') in the form of the cup product (in what follows, we can drop the orientability condition and work with coefficients). Stated precisely, there is a bilinear form

given by

This is a quadratic form for n even, and an alternating form for n odd, because of the graded-commutative nature of the cohomology ring.

These forms are important topological invariants. For example, a theorem of Michael Freedman states that simply connected compact 4-manifolds are (almost) determined by their intersection forms up to homeomorphism.

By Poincaré duality, it turns out that there is a way to think of this geometrically. Choose representative n-dimensional submanifolds A, B for the Poincaré duals of a and b. Then Q_M(a, b) is the oriented intersection number of A and B, which is well-defined because of the dimensions of A and B. This explains the terminology intersection form.

Intersection theory in algebraic geometry

William Fulton in Intersection Theory (1984) writes

... if A and B are subvarieties of a non-singular variety X, the intersection product A.B should be an equivalence class of algebraic cycles closely related to the geometry of how A∩B, A and B are situated in X. Two extreme cases have been most familiar. If the intersection is proper, i.e. dim(A∩B) = dim A + dim B − dim X, then A.B is a linear combination of the irreducible components of A∩B, with coefficients the intersection multiplicities. At the other extreme, if A = B is a non-singular subvariety, the self-intersection formula says that A.B is represented by the top Chern class of the normal bundle of A in X.

To give a definition, in the general case, of the intersection multiplicity was the major concern of André Weil's 1946 book Foundations of Algebraic Geometry. Work in the 1920s of B. L. van der Waerden had already addressed the question; in the Italian school of algebraic geometry the ideas were well known, but foundational questions were not addressed in the same spirit.

Moving cycles

A well-working machinery of intersecting algebraic cycles V and W requires more than taking just the set-theoretic intersection of the cycles in question. Certainly, the intersection V ∩ W or, more commonly called intersection product, denoted V · W, should consist of the set-theoretic intersection of the two subvarieties. However it occurs that cycles are in bad position, e.g. two parallel lines in the plane, or a plane containing a line (intersecting in 3-space). In both cases the intersection should be a point, because, again, if one cycle is moved, this would be the intersection. The intersection of two cycles V and W is called proper if the codimension of the (set-theoretic) intersection V ∩ W is the sum of the codimensions of V and W, respectively, i.e. the "expected" value.

Therefore the concept of moving cycles using appropriate equivalence relations on algebraic cycles is used. The equivalence must be broad enough that given any two cycles V and W, there are equivalent cycles V' and W' such that the intersection V' ∩ W' is proper. Of course, on the other hand, for a second equivalent V" and W", V' ∩ W' needs to be equivalent to V" ∩ W".

For the purposes of intersection theory, rational equivalence is the most important one. Briefly, two r-dimensional cycles on a variety X are rationally equivalent if there is a rational function f on a (k+1)-dimensional subvariety Y, i.e. an element of the function field k(Y) or equivalently a function f : Y → P¹, such that V - W = f^-1(0) - f^-1(∞), where f^-1(-) is counted with multiplicities. Rational equivalence accomplishes the needs sketched above.

Intersection multiplicities

Intersection of lines and parabola

The guiding principle in the definition of intersection multiplicities of cycles is continuity in a certain sense. Consider the following elementary example: the intersection of a parabola y = x² and an axis y=0 should be 2·(0,0), because if one of the cycles moves (yet in an undefined sense), there are precisely two intersection points which both converge to (0,0) when the cycles approach the depicted position. (The picture is misleading insofar as the apparently empty intersection of the parabola and the line y=-3 is empty, because only the real solutions of the equations are depicted).

The first fully satisfactory definition of intersection multiplicities was given by Serre: Let the ambient variety X be smooth (or all local rings regular). Further let V and W be two (irreducible reduced closed) subvarieties, such that their intersection is proper. The construction is local, therefore the varieties may be represented by two ideals I and J in the coordinate ring of X. Let Z be an irreducible component of the set-theoretic intersection V ∩ W and z its generic point. The multiplicity of Z in the intersection product V · W is defined by

,

the alternating sum over the length over the local ring of X in z of torsion groups of the factor rings corresponding to the subvarieties. This expression is sometimes referred to as Serre's Tor-formula.

Remarks:

• The first summand, the length of is the "naive" guess of the multiplicity; however, as Serre shows, it is not sufficient.
• The sum is finite, because the regular local ring has finite Tor-dimension.
• If the intersection of V and W is not proper, the above multiplicity will be zero. If it is proper, it is strictly positive. (Both statements are not obvious from the definition).
• Using a spectral sequence argument, it can be shown that μ(Z;V,W) = μ(Z;W,V).

The Chow ring

The Chow ring is the group of algebraic cycles modulo rational equivalence together with the following commutative intersection product:

where V ∩ W = ∪︀ Z_i is the decomposition of the set-theoretic intersection into irreducible components.

Self-intersection

Self-intersection is a key idea, for example in birational geometry. There on an algebraic surface S, blowing up creates from a point a curve C recognisable by its genus, which is 0, and its self-intersection number, which is −1. There is no paradox, since while C∩C is C as a set, C.C does not mean that set-theoretic intersection. Consider a line L in the projective plane: it has self-intersection number 1 since all other lines cross it once. A line on a non-singular quadric Q in P³ has self-intersection 0, since Q is also P¹×P¹ and a line P¹ can be moved off itself. The quadric Q projects to the plane by means of lines through a fixed point on it. In terms of intersection forms the plane has one of type x² while the quadric one of type XY. This happens on the addition of a negative term to x² − y², and then a change of basis.

References

• (1998), Intersection theory, vol. 2, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Berlin, New York:, MR1644323, ISBN 978-3-540-62046-4; 978-0-387-98549-7
• (1965), Algèbre locale. Multiplicités, vol. 11, Cours au Collège de France, 1957--1958, rédigé par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Mathematics, Berlin, New York:, MR0201468

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