Intersection number

This article is about algebraic geometry. For the concept in graph theory, see Intersection number (graph theory).

In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem.

The intersection number is obvious in certain cases, such as the intersection of x- and y-axes which should be one. The complexity enters when calculating intersections at points of tangency and intersections along positive dimensional sets. For example if a plane is tangent to a surface along a line, the intersection number along the line should be at least two. These questions are discussed systematically in intersection theory.

Definition for Riemann surfaces

Let X be a Riemann surface. Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function $c : S^1 \to X$), we can associate a differential form $\eta_c$ with the pleasant property that integrals along c can be calculated by integrals over X:

$\int_c \alpha = -\int \int_X \alpha \wedge \eta_c = (\alpha, *\eta_c)$, for every closed (1-)differential $\alpha$ on X,

where $\wedge$ is the wedge product of differentials, and $*$ is the hodge star. Then the intersection number of two closed curves, a and b, on X is defined as

$a \cdot b := \int \int_X \eta_a \wedge \eta_b = (\eta_a, -*\eta_b) = -\int_b \eta_a$.

The $\eta_c$ have an intuitive definition as follows. They are a sort of dirac delta along the curve c, accomplished by taking the differential of a unit step function that drops from 1 to 0 across c. More formally, we begin by defining for a simple closed curve c on X, a function f_c by letting $\Omega$ be a small strip around c in the shape of an annulus. Name the left and right parts of $\Omega \setminus c$ as $\Omega^{+}$ and $\Omega^{-}$. Then take a smaller sub-strip around c, $\Omega_0$, with left and right parts $\Omega_0^{-}$ and $\Omega_0^{+}$. Then define f_c by

$f_c(x) = \begin{cases} 1, & x \in \Omega_0^{-} \\ 0, & x \in X \setminus \Omega^{-} \\ \mbox{smooth interpolation}, & x \in \Omega^{-} \setminus \Omega_0^{-} \end{cases}$.

The definition is then expanded to arbitrary closed curves. Every closed curve c on X is homologous to $\sum_{i=1}^N k_i c_i$ for some simple closed curves c_i, that is,

$\int_c \omega = \int_{\sum_i k_i c_i} \omega = \sum_{i=1}^N k_i \int_{c_i} \omega$, for every differential $\omega$.

Define the $\eta_c$ by

$\eta_c = \sum_{i=1}^N k_i \eta_{c_i}$.

Definition for algebraic varieties

The usual constructive definition in the case of algebraic varieties proceeds in steps. The definition given below is for the intersection number of divisors on a nonsingular variety X.

1. The only intersection number that can be calculated directly from the definition is the intersection of hypersurfaces (subvarieties of X of codimension one) that are in general position at x. Specifically, assume we have a nonsingular variety X, and n hypersurfaces Z₁, ..., Z_n which have local equations f₁, ..., f_n near x for polynomials f_i(t₁, ..., t_n), such that the following hold:

• $n = \dim_k X$.
• $f_i(x) = 0$ for all i. (i.e., x is in the intersection of the hypersurfaces.)
• $\dim_x \cap_{i=1}^n Z_i = 0$ (i.e., the divisors are in general position.)
• The $f_i$ are nonsingular at x.

Then the intersection number at the point x is

$(Z_1 \cdots Z_n)_x := \dim_k \mathcal{O}_{X, x} / (f_1, \dots, f_n)$,

where $\mathcal{O}_{X, x}$ is the local ring of X at x, and the dimension is dimension as a k-vector space. It can be calculated as the localization $k[U]_{\mathfrak{m}_x}$, where $\mathfrak{m}_x$ is the maximal ideal of polynomials vanishing at x, and U is an open affine set containing x and containing none of the singularities of the f_i.

2. The intersection number of hypersurfaces in general position is then defined as the sum of the intersection numbers at each point of intersection.

$(Z_1 \cdots, Z_n) = \sum_{x \in \cap_i Z_i} (Z_1 \cdots Z_n)_x$

3. Extend the definition to effective divisors by linearity, i.e.,

$(n Z_1 \cdots Z_n) = n(Z_1 \cdots Z_n)$ and $((Y_1 + Z_1) Z_2 \cdots Z_n) = (Y_1 Z_2 \cdots Z_n) + (Z_1 Z_2 \cdots Z_n)$.

4. Extend the definition to arbitrary divisors in general position by noticing every divisor has a unique expression as D = P - N for some effective divisors P and N. So let D_i = P_i - N_i, and use rules of the form

$((P_1 - N_1) P_2 \cdots P_n) = (P_1 P_2 \cdots P_n) - (N_1 P_2 \cdots P_n)$

to transform the intersection.

5. The intersection number of arbitrary divisors is then defined using a "Chow's moving lemma" that guarantees we can find linearly equivalent divisors that are in general position, which we can then intersect.

Note that the definition of the intersection number does not depend on the order of the divisors.

Further definitions

The definition can be vastly generalized, for example to intersections along subvarieties instead of just at points, or to arbitrary complete varieties.

In algebraic topology, the intersection number appears as the Poincaré dual of the cup product. Specifically, if two manifolds, X and Y, intersect transversely in a manifold M, the homology class of the intersection is the Poincaré dual of the cup product $D_M X \smile D_M Y$ of the Poincaré duals of X and Y.

Intersection multiplicities for plane curves

There is a unique function assigning to each triplet $(P,Q,p)$ consisting of a pair of projective curves, $P$ and $Q$, in $K[x,y]$ and a point $p \in K^2$, a number $I_p(P,Q)$ called the intersection multiplicity of $P$ and $Q$ at $p$ that satisfies the following properties:

1. $I_p(P,Q) = I_p(Q,P)$
2. $I_p(P,Q) = \infty$ if and only if $P$ and $Q$ have a common factor that is zero at $p$
3. $I_p(P,Q) = 0$ if and only if one of $P(p)$ or $Q(p)$ is non-zero (i.e. the point $p$ is off one of the curves)
4. $I_p(x,y) = 1$ where $p = (0,0)$
5. $I_p(P,Q_1Q_2) = I_p(P,Q_1) + I_p(P,Q_2)$
6. $I_p(P + QR,Q) = I_p(P,Q)$ for any $R \in K[x,y]$

Although these properties completely characterize intersection multiplicity, in practice it is realised in several different ways.

One realization of intersection multiplicity is through the dimension of a certain quotient space of the power series ring $K[[x,y]]$. By making a change of variables if necessary, we may assume that $p = (0,0)$. Let $P(x,y)$ and $Q(x,y)$ be the polynomials defining the algebraic curves we are interested in. If the original equations are given in homogeneous form, these can be obtained by setting $z = 1$. Let $I = (P,Q)$ denote the ideal of $K[[x,y]]$ generated by $P$ and $Q$. The intersection multiplicity is the dimension of $K[[x,y]]/I$ as a vector space over $K$.

Another realization of intersection multiplicity comes from the resultant of the two polynomials $P$ and $Q$. In coordinates where $p = (0,0)$, the curves have no other intersections with $y = 0$, and the degree of $P$ with respect to $x$ is equal to the total degree of $P$, $I_p(P,Q)$ can be defined as the highest power of $y$ that divides the resultant of $P$ and $Q$ (with $P$ and $Q$ seen as polynomials over $K[x]$).

Intersection multiplicity can also be realised as the number of distinct intersections that exist if the curves are perturbed slightly. More specifically, if $P$ and $Q$ define curves which intersect only once in the closure of an open set $U$, then for a dense set of $(\epsilon,\delta) \in K^2$, $P - \epsilon$ and $Q - \delta$ are smooth and intersect transversally (i.e. have different tangent lines) at exactly some number $n$ points in $U$. We say then that $I_p(P,Q) = n$.

Example

Consider the intersection of the x-axis with the parabola

$y = x^2.\$

Then

$P = y,\$

and

$Q = y - x^2,\$

so

$I_p(P,Q) = I_p(y,y - x^2) = I_p(y,x^2) = I_p(y,x) + I_p(y,x) = 1 + 1 = 2.\,$

Thus, the intersection degree is two; it is an ordinary tangency.

Self-intersections

Some of the most interesting intersection numbers to compute are self-intersection numbers. This should not be taken in a naive sense. What is meant is that, in an equivalence class of divisors of some specific kind, two representatives are intersected that are in general position with respect to each other. In this way, self-intersection numbers can become well-defined, and even negative.

Applications

The intersection number is partly motivated by the desire to define intersection to satisfy Bézout's theorem.

The intersection number arises in the study of fixed points, which can be cleverly defined as intersections of function graphs with a diagonals. Calculating the intersection numbers at the fixed points counts the fixed points with multiplicity, and leads to the Lefschetz fixed point theorem in quantitative form.

References

• Appendix A.
• Algebraic Curves: An Introduction To Algebraic Geometry, by William Fulton with Richard Weiss. New York: Benjamin, 1969. Reprint ed.: Redwood City, CA, USA: Addison-Wesley, Advanced Book Classics, 1989. ISBN 0-201-51010-3. Full text online.