Degree of an algebraic variety

The degree of an algebraic variety in mathematics is defined, for a projective variety V, by an elementary use of intersection theory.

Definition

For V embedded in a projective space Pⁿ and defined over some algebraically closed field K, the degree d of V is the number of points of intersection of V, defined over K, with a linear subspace L in general position, when

$dim(V) + dim(L) = n.$

Here dim(V) is the dimension of V, and the codimension of L will be equal to that dimension. The degree d is an extrinsic quantity, and not intrinsic as a property of V. For example the projective line has an (essentially unique) embedding of degree n in Pⁿ.

Properties

The degree of a hypersurface F = 0 is the same as the total degree of the homogeneous polynomial F defining it (granted, in case F has repeated factors, that intersection theory is used to count intersections with multiplicity, as in Bézout's theorem).

Other approaches

For a more sophisticated approach, the linear system of divisors defining the embedding of V can be related to the line bundle or invertible sheaf defining the embedding by its space of sections. The tautological line bundle on Pⁿ pulls back to V. The degree determines the first Chern class. The degree can also be computed in the cohomology ring of Pⁿ, or Chow ring, with the class of a hyperplane intersecting the class of V an appropriate number of times.

Extending Bézout's theorem

The degree can be used to generalize Bézout's theorem in an expected way to intersections of n hypersurfaces in Pⁿ.