Degree of an algebraic variety

In mathematics, the degree of an affine or projective variety of dimension n is the number of intersection points of the variety with n hyperplanes in general position (for an algebraic set, the intersection points must be counted with their intersection multiplicity) 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 an 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. For details, see Hilbert series and Hilbert polynomial § Degree of a projective variety and Bézout's theorem.

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

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