FACTORING THE JACOBIAN

9

9. Factorization of the Jacobian

Finally we are ready to state our jacobian factorization as a Theorem; it cor-

responds to Theorem (JF1) on page 160 of [AA].

THEOREM.

LetT

=

T(FG) where F

E

R \ k( (X)) is devoid of multiple factors

in R, and 0

=I=

GER. Then we have the following.

(I) If BET is such that nB(G)

=

1 and S(G, B)

=I=

0 then we have

degyf2B(J(F,G))

=

degyf2B((FG)y)

=

degyf2B(Fy)

=

D"(B)

and

degyf2~(J(F,G))

=

degyf2~((FG)y)

=

degyf2~(Fy)

=

D'(B)

and

nB(J(F,G))

=

n~(J(F,G))

II

n~,(J(F,G))

B1 E1r(T,B)

where for every B'

E

rr(T, B) we have nB'(G)

=

1 and S(G, B')

=I=

0 and

and

and

and

degyf2

8

,(J(F,G))

=

degyf2

8

,((FG)y)

=

degyf2

8

,(Fy)

=

D"(B')

degyf2~,(J(F,G))

=

degyn~,((FG)y)

=

degyf2~,(Fy)

=

D'(B').

{II) If BET is such that f2B(G)

=

1 then we have

degyf2B((FG)y)

=

degyf2B(Fy)

=

D"(B)

degyf2~((FG)y)

=

degyf2~(Fy)

=

D'(B)

nB((FG)y)

=

n~((FG)y)

II

n~,((FG)y)

B'E1r(T,B)

where for every B'

E

rr(T, B) we have nB' (G)

=

1 and

degyf2B'((FG)y)

=

degyf2B'(Fy)

=

D"(B')

and

degyf2~,((FG)y) = degyf2~,(Fy) =

D'(B').

NOTE.

Upon letting

H

=

J(F, G), let us factor

H

into irreducible components

as in Section 7. Let B

E

T

=

T(FG). Then a(B) is a set of certain irreducible

components of FG. Moreover f2B(H) is the product of those irreducible components

of

H

whose normalized contact with every member of a(B) is at least .A(B); amongst

these, only those irreducible components are kept in

n~(H)

whose normalized

contact with every member of a(B) is exactly .A( B). Finally, the NOTE in Section

7 tells us how to calculate the relevant intersection multiplicities.