Degree of an algebraic variety

The degree is not an intrinsic property of the variety, as it depends on a specific embedding of the variety in an affine or projective space.

The degree of a hypersurface is equal to the total degree of its defining equation. A generalization of Bézout's theorem asserts that, if an intersection of n projective hypersurfaces has codimension n, then the degree of the intersection is the product of the degrees of the hypersurfaces.

The degree of a projective variety is the evaluation at 1 of the numerator of the Hilbert series of its coordinate ring. It follows that, given the equations of the variety, the degree may be computed from a Gröbner basis of the ideal of these equations.

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, such that

${\displaystyle \dim (V)+\dim (L)=n.}$

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.