20 the concept of a height [Chap. I
2.8. For most theoretical investigations and, perhaps, some practical ones,
it is actually better to work with the logarithm of the absolute height.
2.9. Definition. For (x0 : . . . : xn) Pn( ), put
hnaive(x0 : . . . : xn) := log Hnaive(x0 : . . . : xn) .
The height function hnaive is called the logarithmic height.
ii. Application: The height defined by an ample invertible sheaf.
2.10. Definition. Let X be a projective variety over a number field K, and
let L be an ample invertible sheaf on X.
Then a height function hL induced by L is given as follows. Let m be such
that L
⊗m
is very ample. Put
hL (x) :=
1
m
hnaive
(
iL
⊗m
(x)
)
for x P a closed point. Here, iL
⊗m
: P
PN
denotes a closed embedding defined
by L
⊗m.
2.11. Lemma. Let X be a projective variety over a number field K, and let
L be an ample invertible sheaf on X. If hL
(1)
and hL
(2)
are two height functions
induced by L , then there is a constant C such that
| hL
(1)
(x) hL
(2)
(x)| C
for every x X( ).
Proof. First, it is clear that the k-tuple embedding fulfills
hnaive
(
ρk(x)
)
= k·hnaive(x)
for every k and x
PN
( ). Hence, if hL is defined using L
⊗m,
then the
same height function may be defined using L ⊗km.
We may therefore assume that hL
(1)
and hL
(2)
are defined using the same tensor
power of L . Then the two embeddings
i(1)⊗m
L
and
i(2)
L ⊗m
differ by an automorphism
ι:
PN

PN
.
We have to verify that
| hnaive
(
ι(x)
)
hnaive(x)| C
for every x PN ( ). ι is explicitly given by
(x0 : . . . : xN ) (x0 : . . . : xN )
:= ((a00x0 + . . . + a0N xN ) : . . . : (aN0x0 + . . . + aNN xN ))
for some a = (aij)ij MN+1(K).
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