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 Segre (1953).[1] 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.[2]
Suppose C {\displaystyle C} is a cone over X {\displaystyle X} , 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 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 [ edit ] 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).[3]
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
s ( E ) = 1 + s 1 ( E ) + s 2 ( E ) + ⋯ {\displaystyle s(E)=1+s_{1}(E)+s_{2}(E)+\cdots \,} where
c 1 ( E ) = − s 1 ( E ) , c 2 ( E ) = s 1 ( E ) 2 − s 2 ( E ) , … , c n ( E ) = − s 1 ( E ) c n − 1 ( E ) − s 2 ( E ) c n − 2 ( E ) − ⋯ − s n ( E ) {\displaystyle c_{1}(E)=-s_{1}(E),\quad c_{2}(E)=s_{1}(E)^{2}-s_{2}(E),\quad \dots ,\quad c_{n}(E)=-s_{1}(E)c_{n-1}(E)-s_{2}(E)c_{n-2}(E)-\cdots -s_{n}(E)} Let x 1 , … , x k {\displaystyle x_{1},\dots ,x_{k}} be Chern roots, i.e. formal eigenvalues of i Ω 2 π {\displaystyle {\frac {i\Omega }{2\pi }}} where Ω {\displaystyle \Omega } is a curvature of a connection on E {\displaystyle E} .
While the Chern class c(E) is written as
c ( E ) = ∏ i = 1 k ( 1 + x i ) = c 0 + c 1 + ⋯ + c k {\displaystyle c(E)=\prod _{i=1}^{k}(1+x_{i})=c_{0}+c_{1}+\cdots +c_{k}\,} where c i {\displaystyle c_{i}} is an elementary symmetric polynomial of degree i {\displaystyle i} in variables x 1 , … , x k {\displaystyle x_{1},\dots ,x_{k}}
the Segre for the dual bundle E ∨ {\displaystyle E^{\vee }} which has Chern roots − x 1 , … , − x k {\displaystyle -x_{1},\dots ,-x_{k}} is written as
s ( E ∨ ) = ∏ i = 1 k 1 1 − x i = s 0 + s 1 + ⋯ {\displaystyle s(E^{\vee })=\prod _{i=1}^{k}{\frac {1}{1-x_{i}}}=s_{0}+s_{1}+\cdots } Expanding the above expression in powers of x 1 , … x k {\displaystyle x_{1},\dots x_{k}} one can see that s i ( E ∨ ) {\displaystyle s_{i}(E^{\vee })} is represented by a complete homogeneous symmetric polynomial of x 1 , … x k {\displaystyle x_{1},\dots x_{k}}
Here are some basic properties.
For any cone C (e.g., a vector bundle), s ( C ⊕ 1 ) = s ( C ) {\displaystyle s(C\oplus 1)=s(C)} .[4] For a cone C and a vector bundle E , c ( E ) s ( C ⊕ E ) = s ( C ) . {\displaystyle c(E)s(C\oplus E)=s(C).} [5] If E is a vector bundle, then[6] s i ( E ) = 0 {\displaystyle s_{i}(E)=0} for i < 0 {\displaystyle i<0} . s 0 ( E ) {\displaystyle s_{0}(E)} is the identity operator. s i ( E ) ∘ s j ( F ) = s j ( F ) ∘ s i ( E ) {\displaystyle s_{i}(E)\circ s_{j}(F)=s_{j}(F)\circ s_{i}(E)} for another vector bundle F . If L is a line bundle, then s 1 ( L ) = − c 1 ( L ) {\displaystyle s_{1}(L)=-c_{1}(L)} , minus the first Chern class of L .[6] If E is a vector bundle of rank e + 1 {\displaystyle e+1} , then, for a line bundle L , s p ( E ⊗ L ) = ∑ i = 0 p ( − 1 ) p − i ( e + p e + i ) s i ( E ) c 1 ( L ) p − i . {\displaystyle s_{p}(E\otimes L)=\sum _{i=0}^{p}(-1)^{p-i}{\binom {e+p}{e+i}}s_{i}(E)c_{1}(L)^{p-i}.} [7] A key property of a Segre class is birational invariance: this is contained in the following. Let p : X → Y {\displaystyle p:X\to Y} be a proper morphism between algebraic schemes such that Y {\displaystyle Y} is irreducible and each irreducible component of X {\displaystyle X} maps onto Y {\displaystyle Y} . Then, for each closed subscheme W ⊂ Y {\displaystyle W\subset Y} , V = p − 1 ( W ) {\displaystyle V=p^{-1}(W)} and p V : V → W {\displaystyle p_{V}:V\to W} the restriction of p {\displaystyle p} ,
p V ∗ ( s ( V , X ) ) = deg ( p ) s ( W , Y ) . {\displaystyle {p_{V}}_{*}(s(V,X))=\operatorname {deg} (p)\,s(W,Y).} [8] Similarly, if f : X → Y {\displaystyle f:X\to Y} is a flat morphism of constant relative dimension between pure-dimensional algebraic schemes, then, for each closed subscheme W ⊂ Y {\displaystyle W\subset Y} , V = f − 1 ( W ) {\displaystyle V=f^{-1}(W)} and f V : V → W {\displaystyle f_{V}:V\to W} the restriction of f {\displaystyle f} ,
f V ∗ ( s ( W , Y ) ) = s ( V , X ) . {\displaystyle {f_{V}}^{*}(s(W,Y))=s(V,X).} [9] A basic example of birational invariance is provided by a blow-up. Let π : X ~ → X {\displaystyle \pi :{\widetilde {X}}\to X} be a blow-up along some closed subscheme Z . Since the exceptional divisor E := π − 1 ( Z ) ↪ X ~ {\displaystyle E:=\pi ^{-1}(Z)\hookrightarrow {\widetilde {X}}} is an effective Cartier divisor and the normal cone (or normal bundle) to it is O E ( E ) := O X ( E ) | E {\displaystyle {\mathcal {O}}_{E}(E):={\mathcal {O}}_{X}(E)|_{E}} ,
s ( E , X ~ ) = c ( O E ( E ) ) − 1 [ E ] = [ E ] − E ⋅ [ E ] + E ⋅ ( E ⋅ [ E ] ) + ⋯ , {\displaystyle {\begin{aligned}s(E,{\widetilde {X}})&=c({\mathcal {O}}_{E}(E))^{-1}[E]\\&=[E]-E\cdot [E]+E\cdot (E\cdot [E])+\cdots ,\end{aligned}}} where we used the notation D ⋅ α = c 1 ( O ( D ) ) α {\displaystyle D\cdot \alpha =c_{1}({\mathcal {O}}(D))\alpha } .[10] Thus,
s ( Z , X ) = g ∗ ( ∑ k = 1 ∞ ( − 1 ) k − 1 E k ) {\displaystyle s(Z,X)=g_{*}\left(\sum _{k=1}^{\infty }(-1)^{k-1}E^{k}\right)} where g : E = π − 1 ( Z ) → Z {\displaystyle g:E=\pi ^{-1}(Z)\to Z} is given by π {\displaystyle \pi } .
Let Z be a smooth curve that is a complete intersection of effective Cartier divisors D 1 , … , D n {\displaystyle D_{1},\dots ,D_{n}} on a variety X . Assume the dimension of X is n + 1. Then the Segre class of the normal cone C Z / X {\displaystyle C_{Z/X}} to Z ↪ X {\displaystyle Z\hookrightarrow X} is:[11]
s ( C Z / X ) = [ Z ] − ∑ i = 1 n D i ⋅ [ Z ] . {\displaystyle s(C_{Z/X})=[Z]-\sum _{i=1}^{n}D_{i}\cdot [Z].} Indeed, for example, if Z is regularly embedded into X , then, since C Z / X = N Z / X {\displaystyle C_{Z/X}=N_{Z/X}} is the normal bundle and N Z / X = ⨁ i = 1 n N D i / X | Z {\displaystyle N_{Z/X}=\bigoplus _{i=1}^{n}N_{D_{i}/X}|_{Z}} (see Normal cone#Properties ), we have:
s ( C Z / X ) = c ( N Z / X ) − 1 [ Z ] = ∏ i = 1 d ( 1 − c 1 ( O X ( D i ) ) ) [ Z ] = [ Z ] − ∑ i = 1 n D i ⋅ [ Z ] . {\displaystyle s(C_{Z/X})=c(N_{Z/X})^{-1}[Z]=\prod _{i=1}^{d}(1-c_{1}({\mathcal {O}}_{X}(D_{i})))[Z]=[Z]-\sum _{i=1}^{n}D_{i}\cdot [Z].} The following is Example 3.2.22. of Fulton (1998).[2] It recovers some classical results from Schubert's book on enumerative geometry .
Viewing the dual projective space P 3 ˘ {\displaystyle {\breve {\mathbb {P} ^{3}}}} as the Grassmann bundle p : P 3 ˘ → ∗ {\displaystyle p:{\breve {\mathbb {P} ^{3}}}\to *} parametrizing the 2-planes in P 3 {\displaystyle \mathbb {P} ^{3}} , consider the tautological exact sequence
0 → S → p ∗ C 3 → Q → 0 {\displaystyle 0\to S\to p^{*}\mathbb {C} ^{3}\to Q\to 0} where S , Q {\displaystyle S,Q} are the tautological sub and quotient bundles. With E = Sym 2 ( S ∗ ⊗ Q ∗ ) {\displaystyle E=\operatorname {Sym} ^{2}(S^{*}\otimes Q^{*})} , the projective bundle q : X = P ( E ) → P 3 ˘ {\displaystyle q:X=\mathbb {P} (E)\to {\breve {\mathbb {P} ^{3}}}} is the variety of conics in P 3 {\displaystyle \mathbb {P} ^{3}} . With β = c 1 ( Q ∗ ) {\displaystyle \beta =c_{1}(Q^{*})} , we have c ( S ∗ ⊗ Q ∗ ) = 2 β + 2 β 2 {\displaystyle c(S^{*}\otimes Q^{*})=2\beta +2\beta ^{2}} and so, using Chern class#Computation formulae ,
c ( E ) = 1 + 8 β + 30 β 2 + 60 β 3 {\displaystyle c(E)=1+8\beta +30\beta ^{2}+60\beta ^{3}} and thus
s ( E ) = 1 + 8 h + 34 h 2 + 92 h 3 {\displaystyle s(E)=1+8h+34h^{2}+92h^{3}} where h = − β = c 1 ( Q ) . {\displaystyle h=-\beta =c_{1}(Q).} The coefficients in s ( E ) {\displaystyle s(E)} have the enumerative geometric meanings; for example, 92 is the number of conics meeting 8 general lines.
Let X be a surface and A , B , D {\displaystyle A,B,D} effective Cartier divisors on it. Let Z ⊂ X {\displaystyle Z\subset X} be the scheme-theoretic intersection of A + D {\displaystyle A+D} and B + D {\displaystyle B+D} (viewing those divisors as closed subschemes). For simplicity, suppose A , B {\displaystyle A,B} meet only at a single point P with the same multiplicity m and that P is a smooth point of X . Then[12]
s ( Z , X ) = [ D ] + ( m 2 [ P ] − D ⋅ [ D ] ) . {\displaystyle s(Z,X)=[D]+(m^{2}[P]-D\cdot [D]).} To see this, consider the blow-up π : X ~ → X {\displaystyle \pi :{\widetilde {X}}\to X} of X along P and let g : Z ~ = π − 1 Z → Z {\displaystyle g:{\widetilde {Z}}=\pi ^{-1}Z\to Z} , the strict transform of Z . By the formula at #Properties ,
s ( Z , X ) = g ∗ ( [ Z ~ ] ) − g ∗ ( Z ~ ⋅ [ Z ~ ] ) . {\displaystyle s(Z,X)=g_{*}([{\widetilde {Z}}])-g_{*}({\widetilde {Z}}\cdot [{\widetilde {Z}}]).} Since Z ~ = π ∗ D + m E {\displaystyle {\widetilde {Z}}=\pi ^{*}D+mE} where E = π − 1 P {\displaystyle E=\pi ^{-1}P} , the formula above results.
Multiplicity along a subvariety [ edit ] Let ( A , m ) {\displaystyle (A,{\mathfrak {m}})} be the local ring of a variety X at a closed subvariety V codimension n (for example, V can be a closed point). Then length A ( A / m t ) {\displaystyle \operatorname {length} _{A}(A/{\mathfrak {m}}^{t})} is a polynomial of degree n in t for large t ; i.e., it can be written as e ( A ) n n ! t n + {\displaystyle {e(A)^{n} \over n!}t^{n}+} the lower-degree terms and the integer e ( A ) {\displaystyle e(A)} is called the multiplicity of A .
The Segre class s ( V , X ) {\displaystyle s(V,X)} of V ⊂ X {\displaystyle V\subset X} encodes this multiplicity: the coefficient of [ V ] {\displaystyle [V]} in s ( V , X ) {\displaystyle s(V,X)} is e ( A ) {\displaystyle e(A)} .[13]
^ Segre 1953 ^ a b Fulton 1998 ^ Fulton 1998 , p.50. ^ Fulton 1998 , Example 4.1.1. ^ Fulton 1998 , Example 4.1.5. ^ a b Fulton 1998 , Proposition 3.1. ^ Fulton 1998 , Example 3.1.1. ^ Fulton 1998 , Proposition 4.2. (a) ^ Fulton 1998 , Proposition 4.2. (b) ^ Fulton 1998 , § 2.5. ^ Fulton 1998 , Example 9.1.1. ^ Fulton 1998 , Example 4.2.2. ^ Fulton 1998 , Example 4.3.1. Fulton, William (1998), Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete . 3. Folge., vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag , ISBN 978-3-540-62046-4 , MR 1644323 Segre, Beniamino (1953), "Nuovi metodi e resultati nella geometria sulle varietà algebriche", Ann. Mat. Pura Appl. (in Italian), 35 (4): 1–127, MR 0061420