Chow ring
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.