Further Examples and Applications of Degree

Abstract

Since finding out that a quadric surface Q ⊂ ℙ³ is abstractly isomorphic to the product ℙ¹ × ℙ¹ we have observed a number of times that, in describing a curve C ⊂ Q, it is much more useful to give its bidegree (a, b) in ℙ¹ × ℙ¹ (that is, the bidegree of the bihomogeneous polynomial F defining it as a subvariety of ℙ¹ ⊂ ℙ¹) than to give just its degree as a curve in P³ (the reader can check this is just a + b). We ask now what are the analogous numerical invariants of a k-dimensional subvariety X ⊂ ℙ^m × ℙⁿ in general.