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HOMEWORK PROBLEMS FOR MATH 8320: 4871
PETE L. CLARK
This is the third set of homework problems for the 8320 course. As usual, unless
mention is made to the contrary,
k
is assumed to be an arbitrary ﬁeld. However,
we are content to develop the theory of hyperelliptic curves only in characteristic
diﬀerent from 2, and there is only one problem where the imperfection of the ground
ﬁeld is addressed in an important way.
Note: Perhaps unfortunately, this problem set contains few problems of medium
diﬃculty. Rather, most of the problems are either easy, hard or (so far as I know),
unsolved. In order to make your work more eﬃcient, I have labelled parts of prob
lems with a designation of (E), (M), (H) or (U), accordingly. Do understand that
this classiﬁcation is highly surjective.
Exercise 48(E): Show that the index of a variety is equal to the greatest com
mon divisor of all degrees of closed points
P
on
V
. (Recall that our deﬁnition of
the index was as the cardinality of the cokernel of the degree map from
Z
0
(
V
) to
Z
.)
Exercise 49: (a(E)) If
k
is not perfect, we can also deﬁne the
separable index
i
s
to be the gcd of all degrees of closed points with separable residue ﬁeld extensions.
What is an equivalent deﬁnition in terms of divisors? (Hint: you want to deﬁne a
subgroup of
Z
0
(
X
) generated by “separable” divisors.)
(b(E)) Show that
i
(
V
)

i
s
(
V
) for all
V
.
(c(U)) Find a nice variety
V
over some (imperfect) ﬁeld
k
such that
i
S
(
V
)
> i
(
V
).
Exercise 50(M): Show that the Weierstrass points on a hyperelliptic curve of genus
g
≥
2 are precisely the ramiﬁcation points of the covering
C
→
C/ι
=
P
1
. In
particular, there are exactly 2
g
+ 2 Weierstrass points.
Exercise 51(H): In this exercise you will ﬁll in the details of a statement and proof
of Hurwitz’s
L¨uckensatz
.
Exercise 52: Let
k
be an imperfect ﬁeld and
C
/k
a nice (in particular, geomet
rically regular) curve.
(a(E)) Must every automorphism of
C
be deﬁned over a separable extension of
k
?
(b(H)) What about if we assume that the genus is at least 2?
Exercise 53: Let
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This note was uploaded on 10/26/2011 for the course MATH 8320 taught by Professor Clark during the Fall '10 term at University of Georgia Athens.
 Fall '10
 Clark
 Math, Algebra

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