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 the x- and y-axes in a plane, which should be one. The complexity enters when calculating intersections at points of tangency, and intersections which are not just points, but have higher dimension. 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.

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 ${\displaystyle c:S^{1}\to X}$), we can associate a differential form ${\displaystyle \eta _{c}}$ of compact support, the Poincaré dual of c, with the property that integrals along c can be calculated by integrals over X:

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

The ${\displaystyle \eta _{c}}$ 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 ${\displaystyle \Omega }$ be a small strip around c in the shape of an annulus. Name the left and right parts of ${\displaystyle \Omega \setminus c}$ as ${\displaystyle \Omega ^{+}}$ and ${\displaystyle \Omega ^{-}}$. Then take a smaller sub-strip around c, ${\displaystyle \Omega _{0}}$, with left and right parts ${\displaystyle \Omega _{0}^{-}}$ and ${\displaystyle \Omega _{0}^{+}}$. Then define f_c by

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

Define the ${\displaystyle \eta _{c}}$ by

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.