In mathematics, the Segre class is a characteristic class used in the study of cones, a generalization of vector bundles. For vector bundles the total Segre class is inverse to the total Chern class, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to more general cones, while the Chern class does not. The Segre class was introduced in the non-singular case by Beniamino Segre (1953). In the modern treatment of intersection theory in algebraic geometry, as developed e.g. in the definitive book of Fulton (1998), Segre classes play a fundamental role.
Definition Suppose that C {\displaystyle C} is a cone over X {\displaystyle X} , that q {\displaystyle q} is the projection from the projective completion P ( C ⊕ 1 ) {\displaystyle \mathbb {P} (C\oplus 1)} of C {\displaystyle C} to X {\displaystyle X} , and that O ( 1 ) {\displaystyle {\mathcal {O}}(1)} is the anti-tautological line bundle on P ( C ⊕ 1 ) {\displaystyle \mathbb {P} (C\oplus 1)} . Viewing the Chern class c 1 ( O ( 1 ) ) {\displaystyle c_{1}({\mathcal {O}}(1))} as a group endomorphism of the Chow group of P ( C ⊕ 1 ) {\displaystyle \mathbb {P} (C\oplus 1)} , the total Segre class of C {\displaystyle C} is given by:
s ( C ) = q ∗ ( ∑ i ≥ 0 c 1 ( O ( 1 ) ) i [ P ( C ⊕ 1 ) ] ) . {\displaystyle s(C)=q_{*}\left(\sum _{i\geq 0}c_{1}({\mathcal {O}}(1))^{i}[\mathbb {P} (C\oplus 1)]\right).}
The i {\displaystyle i} th Segre class s i ( C ) {\displaystyle s_{i}(C)} is simply the i {\displaystyle i} th graded piece of s ( C ) {\displaystyle s(C)} . If C {\displaystyle C} is of pure dimension r {\displaystyle r} over X {\displaystyle X} then this is given by:
s i ( C ) = q ∗ ( c 1 ( O ( 1 ) ) r + i [ P ( C ⊕ 1 ) ] ) . {\displaystyle s_{i}(C)=q_{*}\left(c_{1}({\mathcal {O}}(1))^{r+i}[\mathbb {P} (C\oplus 1)]\right).}
The reason for using P ( C ⊕ 1 ) {\displaystyle \mathbb {P} (C\oplus 1)} rather than P ( C ) {\displaystyle \mathbb {P} (C)} is that this makes the total Segre class stable under addition of the trivial bundle O {\displaystyle {\mathcal {O}}} . If Z is a closed subscheme of an algebraic scheme X, then s ( Z , X ) {\displaystyle s(Z,X)} denote the Segre class of the normal cone to Z ↪ X {\displaystyle Z\hookrightarrow X} .
Relation to Chern classes for vector bundles For a holomorphic vector bundle E {\displaystyle E} over a complex manifold M {\displaystyle M} a total Segre class s ( E ) {\displaystyle s(E)} is the inverse to the total Chern class c ( E ) {\displaystyle c(E)} , see e.g. Fulton (1998). Explicitly, for a total Chern class
c ( E ) = 1 + c 1 ( E ) + c 2 ( E ) + ⋯ {\displaystyle c(E)=1+c_{1}(E)+c_{2}(E)+\cdots \,}
one gets the total Segre class
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