Chern-schwartz-macpherson Classes and the Euler Characteristic of Degeneracy Wci and Special Divisors

This concept overlaps a large family of interesting varieties (for example, the varieties of special divisors studied in Section 3). Several authors have worked out explicit formulas for the Euler characteristic of D r( rp) in terms of different cohomological and numerical invariants under the assumption that X is nonsingular and rp is appropriately "general". For instance, if rp is a section of a vector bundle, then the formulas for the Euler characteristic X(Do(rp» were given by Hirzebruch [H] and Navarro-Aznar [N]. If Dr(rp) is a curve or a surface in a nonsingular X, some explicit formulas for X (Dr ( rp » were given by Harris and Tu [H-T] in terms of the Chern classes of E, F and X, but under the extra assumption Dr_I (rp) = 0 (which implies that Dr(rp) is nonsingular). In loc.cit. the authors also posed the problem of finding a general formula for X(Dr(rp»-if such exists!-under the assumption Dr _ 1(rp) = 0 or, even stronger, without this assumption. The first problem was solved positively by the second named author [PrI, Proposition 5.7], by the use of polynomials universally supported on degeneracy