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Irreducible components of two intersecting complex hyperbolas
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Mathematics
Area of mathematics
Geometry and topology
Algebraic geometry
Algebraic variety
Affine algebraic set
Created
2026-09-24
Updated
2026-09-24
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The algebraic
set
V
(
x
2
+
y
2
−
1
,
x
2
−
z
2
−
1
)
⊆
A
C
3
(1)
has the two
irreducible components
V
(
y
−
i
z
,
x
2
−
z
2
−
1
)
and
V
(
y
+
i
z
,
x
2
−
z
2
−
1
)
.
(2)
Each is irreducible because the linear change
u
=
x
−
z
,
v
=
x
+
z
identifies its
coordinate ring
with
C
[
u
,
v
]
/
(
uv
−
1
)
≅
C
[
u
,
u
−
1
]
,
(3)
an
integral domain
.
Ancestors
(7)
Affine algebraic set
Algebraic variety
Algebraic geometry
Geometry and topology
Area of mathematics
Mathematics
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Incoming links
(1)
Past exam of the mathematics course of the University of Cambridge
/
2021
/
ii
/
Paper 1
/
25I
/
Decomposition of the given vanishing locus
/
Solution
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