LetX be a smooth variety, andE,F vector bundles onX of rankse, f respec-tively. Assume there is a mapφ:E → F. The Porteous (or Thom-Porteous) formula applies in this case, and describes the Chow class of thedegeneracy loci Mk(φ). Here,Mk(φ) is the locus onX whereφhas rank≤k.
Theorem B.3.1 ([3264, see chapter 12]). Assume thatMk(φ)has codimension (e−k)(f−k), andk=e−1< f. Then
The fraction of Chern classes is purely formal. The Porteous formula is actually a large generalization of this, but the above version is all we need.
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