Further Examples and Applications of Degree
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 × ℙ n in general.
KeywordsInverse Image Plane Curve Hyperplane Section Determinantal Variety Fundamental Class
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