Math 431 ALGEBRAIC GEOMETRY
Homework 3 Solution Key
Ali Sinan Sert¨oz
7 January 2003
1)
Show that for any
g
≥
0, there is a compact Riemann surface of genus
g
. In particular show
that for any
g >
2 there exist a hyperelliptic and a nonhyperelliptic compact Riemann
surface of genus
g
. You may not be able to give the proofs in detail but you can quote some
technical results with full references which imply the existence of the required surfaces.
For
g
= 0 and
g
= 1 we have the projective line and the torus respectively.
To construct a hyperelliptic curve of genus
g
≥
2 we quote your text book page 141; for any
given 2
g
+ 2 mutually distinct points
a
1
, . . . , a
2
g
+2
in
C
we first consider the curve
C
=
(
y
2

2
g
+2
Y
i
=1
(
x

a
i
) = 0
)
∪
[0 : 0 : 1]
⊂
P
1
.
The normalization
C
0
of this curve is a hyperelliptic curve of genus
g
.
Now the remark on
page 144 of your textbook says that two such hyperelliptic curves are isomorphic if and only
if the associated points
a
1
, . . . , a
2
g
+2
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 Algebra, Geometry, Manifold, Algebraic geometry, Riemann surface, Hyperelliptic Curves, compact Riemann surface

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