- 11 -
on X
~nd
D
15
any divisor, then
'-(D) - deg D + 1 - g +
~
(W-D).
8.THE OSCULATING HYPERPLANE OF A CURVE
LBt
X be
an
curve of genus
defined aver
-:c
ffiap. Then
Assume
d of
~
irreducible.
g defined
R. and' let f
non-singular,
over
.
projective.
K but viewed as
X +<cc PG(n,K)
algebraic
the set of
points
be a suitable ratianai
i5 viewed as the set of branches of X.
that 'C
"15
oot
contained
In
a
hyperplane.
i5 the number of points of intersection of
~
The
degree
with a generlc
hyperplane. Far aoy hyperplane H. if n p is the interseetion multipli
city cf H end ~ et P, then
15
a divisor of degree d
= LO po
Aisa
i2'=IH."IH a hyperplane)
i5 a linear system.
fl]5
contained
In
In this case, D,..,. D'
the
complete
linear
for aoy D.O'
system
ID[
In
çç . Hence
ID' ID'N DI,
,{here D 15 some element of 9 .
...... ,'.
A. ,\/,"J}
l
~
•
,~
t
e linear" sysLem defines an embedding f
by
L(D) = Ig€K(X)ldiv(g) + D> D) .
:
given
- 12 -
Given
A.
l inf:nf
system
degree
fl.S
the
Snme
not
neCÙSSA.ry.
it
q)
the
•
system containing q
complete
~ and possibly Unger dimensiono
is
has
Hence. although
simpler to consider complete linear systems.
Rnd this Wl':: do.
Let C6
of
degree
d
have
complete
ce .
be the sel of hyperplanes
i1nd let P be a f i xed· paint of
~i
Let
linear
system çp
A.ssociated
passlng through p with multiplicity at least i. Then
EFlCh fii
1S a projective space.
mension one in !ZIi' Such aD i
are J o " "
lf qi
#
fÀ +
i
1
,
then
~i+l has
codi-
is a (§l,P)-arder. So the (ç},P)-orders
.J o ' wherEl
o
Note that jl
=
= J
<
o
1 if and anIy if Il
Far example, let
~
IS
d.
non singular.
be a pIane cubico
Then
(J o ,J 1 ,J 2 )
=
(0,1,2) if P is neither singular oor
(0,1,3)
if P
(0,/.3)
if P 1S singular.
Note that, as the points of
15
r:c
aD
inflexion,
an inflexion,
are viewed as branches. each branch
has a unique tangent.
The Hasse derivative. satisfies the following properties:
i ;
i
i
E,.(i)t
)
DCi)O'.t
Ci) t
J l
J
( i i ) D(t i ) (fg)
l
=
.E
J=O
D(il f
t
•
DCi-ilg;
t
- 13 -
( i i i l DCi l D(j) =
t
The
t
hyp~rplane
unlque
P is the osculating
hyperplan~
r
X
D
det
For example. if
with
• •
f
multiplicity
Hp and has 8Quation
o
(lo l
interséction
x
• •
n
(lol
D
o
fn
=
•
•
'(ln-l)
D
fo
·(jn-l)
D
fn
O
the twisted cubie in PG(3.K),
~ 1S
= (l,t,t
2
3
.t ).
2
The osculating hyperplane at P(1.t,t ,t 3 ) is
x
det
l
•
3
3
t
t
O
l
2t
2
3t
O
O
l
= O ••
3t
that is.
2
3t xl + 3tx 2 - x 3
The
point
P on
~
is
a
=
O.
WeierstrHss point,
W-point
for-short.
- 14 -
g
Since
complete,
lS
the
Riernfl.nn"'Roch
theorem
gives
that.
11
d '2g- 2, then
( i ) n = d-g;
Ci i) dim !ili = d-g-i
Ci i i) J i =
Let
L.
=
l
n
L ,15 dual to
1
for
l
for
-<
l
d
<
l
-
d - 2g
l ;
2g.
meet ing 'fJ
hyperplanes
+
"t
P wi th
TI
j ,+1.
p >
Then
l
g)i and
. .. c Ln- l .
Also
Lo :::
the
{?},
set LI
1s
the
tangent
l1ne at
P,
f\nd
LO _ 1 1s
the osculating hyperplane al P.
The point P 1s a
g)-osculation
exists a hyperplane H such that
point
if jn>n.
that
15,
there
TI p > TI.
The integers ji are characterized by the following result.
THEOREM
•
•
8.1
(i )
.
lf
integer
r
such
that
then
"re
J o ·····'J 1 _1
I
smallest
.
1s
n(r)f(Q)
linearly
is
the
independent
the latter sel spans L.l - l '
(ii)lf
(r )
o
D
f(Q)"."D
(r )
s
f(Q)
o
are
<
linearly
rs
are
integers
independent.
such
then
that
J 1.< r ..
l