Further Examples and Applications of Degree

  • Joe Harris
Part of the Graduate Texts in Mathematics book series (GTM, volume 133)


Since finding out that a quadric surface Q ⊂3 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.


Inverse Image Plane Curve Hyperplane Section Determinantal Variety Fundamental Class 
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Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Joe Harris
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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