In algebraic geometry , the Hilbert function gives information about the number of hypersurfaces at a given degree. For sufficiently large arguments, it agrees with a polynomial called a Hilbert polynomial.
Hilbert function Let be a projective variety with a vanishing ideal
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⊂
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n
{\ displaystyle X \ subset P ^ {n}}
I.
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⊂
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[
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0
,
...
,
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]
{\ displaystyle I (X) \ subset K \ left [Z_ {0}, \ ldots, Z_ {n} \ right]}
For be
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1
{\ displaystyle d \ geq 1}
I.
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d
=
I.
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∩
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[
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d
{\ displaystyle I (X) _ {d} = I (X) \ cap K \ left [Z_ {0}, \ ldots, Z_ {n} \ right] _ {d}}
the homogeneous part of the degree . The coordinate ring is then a graduated ring
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{\ displaystyle d}
S.
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{\ displaystyle S (X)}
S.
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=
⨁
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≥
0
S.
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d
{\ displaystyle S (X) = \ bigoplus _ {d \ geq 0} S (X) _ {d}}
with .
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[
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...
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d
/
I.
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{\ displaystyle S (X) _ {d} = K \ left [Z_ {0}, \ ldots, Z_ {n} \ right] _ {d} / I (X) _ {d}}
The dimension of gives the number of independent, containing hypersurfaces of degree . The Hilbert function is defined by
I.
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)
d
{\ displaystyle I (X) _ {d}}
X
{\ displaystyle X}
d
{\ displaystyle d}
H
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{\ displaystyle h_ {X}}
H
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(
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=
dim
S.
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d
{\ displaystyle h_ {X} (d) = \ dim S (X) _ {d}}
so it gives the codimension of .
I.
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d
{\ displaystyle I (X) _ {d}}
Examples
Be . Then it's for everyone .
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=
{
[
1
:
0
:
0
]
,
[
0
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1
:
0
]
,
[
0
:
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:
1
]
}
⊂
P
2
{\ displaystyle X = \ left \ {\ left [1 \ colon 0 \ colon 0 \ right], \ left [0 \ colon 1 \ colon 0 \ right], \ left [0 \ colon 0 \ colon 1 \ right] \ right \} \ subset P ^ {2}}
H
X
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d
)
=
3
{\ displaystyle h_ {X} (d) = 3}
d
≥
1
{\ displaystyle d \ geq 1}
Be . Then and for everyone .
X
=
{
[
1
:
0
:
0
]
,
[
0
:
1
:
0
]
,
[
1
:
1
:
0
]
}
⊂
P
2
{\ displaystyle X = \ left \ {\ left [1 \ colon 0 \ colon 0 \ right], \ left [0 \ colon 1 \ colon 0 \ right], \ left [1 \ colon 1 \ colon 0 \ right] \ right \} \ subset P ^ {2}}
H
X
(
1
)
=
2
{\ displaystyle h_ {X} (1) = 2}
H
X
(
d
)
=
3
{\ displaystyle h_ {X} (d) = 3}
d
≥
2
{\ displaystyle d \ geq 2}
Let be a set made up of points. Then is for .
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⊂
P
n
{\ displaystyle X \ subset P ^ {n}}
m
{\ displaystyle m}
H
X
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d
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=
m
{\ displaystyle h_ {X} (d) = m}
d
≥
m
-
1
{\ displaystyle d \ geq m-1}
Let be a curve given by a homogeneous polynomial of degree . Then is for .
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⊂
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2
{\ displaystyle X \ subset P ^ {2}}
k
{\ displaystyle k}
H
X
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=
d
⋅
k
-
1
2
k
(
k
-
3
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{\ displaystyle h_ {X} (d) = d \ cdot k - {\ frac {1} {2}} k (k-3)}
d
≥
k
{\ displaystyle d \ geq k}
Hilbert polynomial Theorem: For every projective variety there is a polynomial of degree such that
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⊂
P
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{\ displaystyle X \ subset P ^ {n}}
H
X
∈
Q
[
d
]
{\ displaystyle H_ {X} \ in \ mathbb {Q} \ left [d \ right]}
dim
(
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)
{\ displaystyle \ dim (X)}
H
X
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=
H
X
(
d
)
{\ displaystyle H_ {X} (d) = h_ {X} (d)}
there are sufficiently large for all .
d
∈
Z
{\ displaystyle d \ in \ mathbb {Z}}
The polynomial is called the Hilbert polynomial of the variety .
H
X
{\ displaystyle H_ {X}}
X
{\ displaystyle X}
See also literature
D. Eisenbud : Commutative algebra. With a view toward algebraic geometry , Graduate Texts in Mathematics 150, Springer-Verlag New York, ISBN 0-387-94268-8
Web links
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
![{\ displaystyle I (X) \ subset K \ left [Z_ {0}, \ ldots, Z_ {n} \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d16d672978e565f94fc9993c793d1fdeabbc41f1)
![{\ displaystyle I (X) _ {d} = I (X) \ cap K \ left [Z_ {0}, \ ldots, Z_ {n} \ right] _ {d}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c96248f7a73e304af803f8d70a2816215a30ffdb)
![{\ displaystyle S (X) _ {d} = K \ left [Z_ {0}, \ ldots, Z_ {n} \ right] _ {d} / I (X) _ {d}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed7c8253bba3efe7e7be843f7967b7052213577)
![{\ displaystyle X = \ left \ {\ left [1 \ colon 0 \ colon 0 \ right], \ left [0 \ colon 1 \ colon 0 \ right], \ left [0 \ colon 0 \ colon 1 \ right] \ right \} \ subset P ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03fbdc9add4da9ec867028b408682d42aeeeec40)
![{\ displaystyle X = \ left \ {\ left [1 \ colon 0 \ colon 0 \ right], \ left [0 \ colon 1 \ colon 0 \ right], \ left [1 \ colon 1 \ colon 0 \ right] \ right \} \ subset P ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2d6944f9d3c258c42f08b096fd6cf9d3775478)
![{\ displaystyle H_ {X} \ in \ mathbb {Q} \ left [d \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1578ca79a38793ad7ceef5c415295db5349b4a5)
