For (irreducible) varieties, if one takes into account the multiplicities and, in the affine case, the points at infinity, the hypothesis of general position may be replaced by the much weaker condition that the intersection of the variety has the dimension zero (that is, consists of a finite number of points).
The degree of a projective variety is the evaluation at 1 of the numerator of the Hilbert series of its coordinate ring.
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 Here dim(V) is the dimension of V, and the codimension of L will be equal to that dimension.
The tautological line bundle on Pn pulls back to V. The degree determines the first Chern class.
The degree can be used to generalize Bézout's theorem in an expected way to intersections of n hypersurfaces in Pn.