 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.
For V embedded in a projective space P^{n} 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 P^{n}.
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).
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 P^{n} pulls back to V. The degree determines the first Chern class. The degree can also be computed in the cohomology ring of P^{n}, or Chow ring, with the class of a hyperplane intersecting the class of V an appropriate number of times.
The degree can be used to generalize Bézout's theorem in an expected way to intersections of n hypersurfaces in P^{n}.
Categories: Algebraic varieties
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