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Past exam of the mathematics course of the University of Cambridge
/
2022
/
iii
/
Paper 152
/
1
/
a
/
i
/
Solution
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2022
iii
Paper 152
1
a
i
2026-09-28
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For
m
=
(
u
,
v
)
∈
M
≅
Z
2
, membership in the
dual cone
is equivalent to
u
≥
0
,
−
u
+
3
v
≥
0.
(1)
The
Hilbert basis of a rational cone
is
(
0
,
1
)
,
(
1
,
1
)
,
(
2
,
1
)
,
(
3
,
1
)
. Therefore the
coordinate ring of an affine toric variety
is
C
[
S
σ
]
=
C
[
t
2
,
t
1
t
2
,
t
1
2
t
2
,
t
1
3
t
2
]
⊆
C
[
t
1
±
1
,
t
2
±
1
]
.
(2)
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i
a
1
Paper 152
iii
2022
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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