 Degeneracy Loci and GrassmanniansWilliam Fulton ()
Chapter Chapter 14 in Intersection Theory, 1998, pp 242-279 from Springer Abstract: Abstract Let σ: E → F be a homomorphism of vector bundles of ranks e and f on a variety X, and let k min (e,f). The degeneracy locus $$ {D_k}\left(\sigma \right) = \left\{{x \in X\left| {rank} \right.\left({\sigma \left(x \right)} \right) \leqq k} \right\} $$ has codimension at most (e - k) (f - k) in X, if it is not empty. We construct a class $$ {\mathbb{D}_k}\left(\sigma \right) \in {A_m}\left({{D_k}\left(\sigma \right)} \right) $$ m = dim (X) - (e - k) (f - k), whose image in A m (X) is given by the Thom-Porteous formula: $$ {\mathbb{D}_k}\left(\sigma \right) = \Delta _{f - k}^{\left({e - k} \right)}\left({c\left({F - K} \right)} \right) \cap \left[X \right] $$ Here Δ q (p) (c) denotes the determinant of the p by p matrix $$ {\left({{c_{q + j - i}}} \right)_{1 \leqq i,j \leqq p}} $$ If dim (D k (σ) = m, and X is non-singular, or, more generally, if a suitable depth condition is satisfied, then $$ {\mathbb{D}_k}\left(\sigma \right) $$ is the m-cycle determined by the natural scheme structure on $$ {\mathbb{D}_k}\left(\sigma \right) $$ In general the formation of $$ {\mathbb{D}_k}\left(\sigma \right) $$ commutes with other intersection operations. These properties determine $$ {\mathbb{D}_k}\left(\sigma \right) $$ in case dim $$ \left({{D_k}\left(\sigma \right)} \right) > m $$ . If $$ {A_1} \subset \ldots \subset {A_d} \subset E $$ is a flag of sub-bundles of E, the determinantal locus is $$ \Omega \left({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{A} ;\sigma} \right) = \left\{{x \in X\left| {\dim \left({Ker\left({\sigma \left(x \right)} \right) \cap {A_i}\left(x \right)} \right) \geqq ifor1 \leqq i \leqq d} \right.} \right\} $$ Similarly, there are classes $$ \Omega \left({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{A} ;\sigma} \right) $$ in $$ {A_ *}\left({\Omega \left({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{A} ;\sigma } \right)} \right) $$ , whose images in A * (X) are given by certain determinants in Chern classes. If $$ c\left({E/{A_i}} \right) = 1 $$ , the formula is $$ \Omega \left({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{A} ;\sigma } \right) = {\Delta _\lambda }\left({c\left({F - E} \right)} \right) \cap \left[X \right]$$ where $$ \lambda = \left({{\lambda _i}, \ldots,{\lambda _d}} \right)$$ , $$ {\lambda _i} = f - rank\left({{A_i}} \right) + i$$ , and Δλis the Schur polynomial $$ {\Delta _{\lambda \left(c \right)}} = \det \left({\begin{array}{*{20}{c}} {{c_{{\lambda _1}}}}{{c_{{\lambda _1} + 1}} \ldots {c_{{\lambda _1} + d - 1}}} \\ {{c_{{\lambda _2} - 1}}}{{c_{{\lambda _2}}} \ldots {c_{{\lambda _2} + d - 2}}} \\ {{c_{{\lambda _d} - d + 1}}}{{c_{{\lambda _d}}}} \end{array}} \right)$$ A special case of degeneracy locus is the zero set of a section of a vector bundle. In this case the degeneracy class localizes the top Chern class of the bundle (§ 14.1). The construction of general degeneracy loci is reduced to the case of sections of bundles on Grassmannians; proving the formulas requires some Gysin computations (§ 14.2). Formal identities among Schur polynomials determine formulas for intersecting determinantal classes. When applied to Grassmann bundles, these formulas yield generalizations of classical formulas of Schubert calculus: the basis theorem, duality, Pieri's formula, and Giambelli's formula. Notation. The fibre of a vector bundle E over a scheme X at a point x ∈ X is denoted E (x); it is a vector space over $$\kappa (x) $$ . If σ: E→ F is a vector bundle homomorphism, ∧k σ denotes the induced homomorphism on k th exterior powers. If σ: E → F is a homomorphism of vector bundles on a scheme X, the zero scheme of σ will be denoted Z (σ). On an affine open set U where E and F are trivial, σ is defined by a matrix of elements in the coordinate ring of U, which generate the ideal of Z (σ) on U. In particular, if s is a section of a bundle E, i.e., a homomorphism from the trivial line bundle to E, its zero scheme is denoted Z(s) (cf. Appendix B.3.2). More generally, for a non-negative integer k, we have the k th degeneracy locus $$ {D_k}(\sigma ) = \{ x \in X|rank(\sigma (x)) \leqslant k\} = Z({ \wedge ^{k + 1}}(\sigma ))$$ The second description determines the scheme structure on D k (σ): locally its ideal is generated by (k +1)-minors of a matrix for σ. Let A be a flag of subbundles of E: $$ 0 \subsetneqq {A_1} \subsetneqq ... \subsetneqq {{\rm A}_d} \subset E$$ Given a: E → F, set $$ \Omega (\underline A ;\sigma ) = \{ x \in X|\dim (Ker(\sigma (x)) \cap {A_i}(x)) \geqq i,1 \leqq i \leqq d\} = \mathop \cap \limits_{i = 1}^d Z({ \wedge ^{{a_i} - i + 1}}({\sigma _i})).$$ Here σi:A i → Fis the restriction of a to i and a i is the rank of A i The second description determines the scheme structure. For vector bundles E, F on X, set $$ c(F - E) = c(F)/c(E) = 1 + ({c_1}(F) - {c_1}(E) + ...)$$ and let c i (F— E) be the term of degree i in this expansion. If λ1,…,λd are integers, and c (1) …, c (d) are formal sums: $$ {c^{(i)}} = \sum\limits_j {c_j^{(i)}}$$ with c j (i) of degree j, $$ j \in \mathbb{Z}$$ , Set $$ {\Delta _{{\lambda _{1,}}...,{\lambda _d}}}({c^{(1)}},...,{c^{(d)}}) = \det \left\{ \begin{gathered} c_{{\lambda _1}}^{(1)}c_{{\lambda _{1 + 1}}}^{(1)}...c_{{\lambda _{1 + d - 1}}}^{(1)} \hfill \\ c_{{\lambda _2}}^{(2)} \hfill \\ c_{{\lambda _{d - d + 1}}}^{(d)}c_{{\lambda _d}}^{(d)} \hfill \\ \end{gathered} \right\}$$ If c (1) =… = c (d) = c, we denote this Δ λ1,…,λd (c), or Δ λ (c), i.e. Δ λ (c)=|c λi+j-i |. If, in addition, λ1 =… =λd = e, this is Δ e (d) (c). Keywords: Vector Bundle; Line Bundle; Young Diagram; Chern Class; Schubert Variety (search for similar items in EconPapers) There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. 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