Degree of an algebraic variety

In mathematics, the degree of an algebraic variety 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 an algebraic variety 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 embedding (essentially unique) of degree n − 1 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