aarjaneiro/planetmath huggingface dataset:
proof that Spec⁡(R) is quasi-compact

proof that

Spec

⁡

(

R

)

is quasi-compact

Note that most of the notation used here is defined in the entry prime spectrum.

The following is a proof that

Spec

⁡

(

R

)

is

quasi-compact

.

Proof.

Let

Λ

be an indexing set and

{

U

λ

}

λ

∈

Λ

be an

open cover

for

Spec

⁡

(

R

)

. For every

λ

∈

Λ

, let

I

λ

be an ideal of

R

with

U

λ

=

Spec

⁡

(

R

)

∖

V

⁢

(

I

λ

)

. Since

Spec

⁡

(

R

)

=

⋃

λ

∈

Λ

U

λ

=

⋃

λ

∈

Λ

(

Spec

⁡

(

R

)

∖

V

⁢

(

I

λ

)

)

=

Spec

⁡

(

R

)

∖

⋂

λ

∈

Λ

V

⁢

(

I

λ

)

=

Spec

⁡

(

R

)

∖

V

⁢

(

∑

λ

∈

Λ

I

λ

)

,

V

⁢

(

∑

λ

∈

Λ

I

λ

)

=

∅

. Thus, by this theorem (

http://planetmath.org/VIemptysetImpliesIR

),

∑

λ

∈

Λ

I

λ

=

R

. Since

1

R

∈

R

=

∑

λ

∈

Λ

I

λ

, there exists a finite subset

L

of

Λ

such that, for every

ℓ

∈

L

, there exists an

i

ℓ

∈

I

ℓ

with

1

R

=

∑

ℓ

∈

L

i

ℓ

.

Let

r

∈

R

. Then

r

=

r

⋅

1

R

=

r

⁢

∑

ℓ

∈

L

i

ℓ

=

∑

ℓ

∈

L

r

⋅

i

ℓ

∈

∑

ℓ

∈

L

I

ℓ

. Thus,

∑

ℓ

∈

L

I

ℓ

=

R

. Therefore,

V

⁢

(

∑

ℓ

∈

L

I

ℓ

)

=

∅

. Since

Spec

⁡

(

R

)

=

Spec

⁡

(

R

)

∖

V

⁢

(

∑

ℓ

∈

L

I

ℓ

)

=

Spec

⁡

(

R

)

∖

⋂

ℓ

∈

L

V

⁢

(

I

ℓ

)

=

⋃

ℓ

∈

L

(

Spec

⁡

(

R

)

∖

V

⁢

(

I

ℓ

)

)

=

⋃

ℓ

∈

L

U

ℓ

,

{

U

λ

}

λ

∈

Λ

restricts to a finite subcover. It follows that

Spec

⁡

(

R

)

is quasi-compact.
∎

Title

proof that

Spec

⁡

(

R

)

is quasi-compact

Canonical name

ProofThatoperatornameSpecRIsQuasicompact

Date of creation

2013-03-22 16:07:40

Last modified on

2013-03-22 16:07:40

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 14A15

Related topic

VIemptysetImpliesIR

Generated on Fri Feb 9 17:57:08 2018 by

LaTeXML
separated scheme

separated scheme

A scheme

X

is defined to be a

separated scheme

if the

morphism

d

:

X

→

X

×

Spec

⁡

ℤ

X

into the

fibre product

X

×

Spec

⁡

ℤ

X

which is induced by the

identity maps

i

:

X

⟶

X

in each

coordinate

is a

closed immersion

.

Note the

similarity

to the definition of a

Hausdorff topological space

. In the situation of

topological spaces

, a space

X

is

Hausdorff

if and only if the diagonal morphism

X

⟶

X

×

X

is a closed

embedding

of topological spaces. The definition of a separated scheme is very

similar

, except that the topological product is replaced with the scheme fibre product.

More generally, if

X

is a scheme over a base scheme

Y

, the scheme

X

is defined to be

separated

over

Y

if the

diagonal embedding

d

:

X

→

X

×

Y

X

is a closed immersion.

Title

separated scheme

Canonical name

SeparatedScheme

Date of creation

2013-03-22 12:50:25

Last modified on

2013-03-22 12:50:25

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 14A15

Defines

separated

Generated on Fri Feb 9 17:57:12 2018 by

LaTeXML
singular set

singular set

The

singular set

of a

variety

X

is the set of

singular points

. This is a

proper subvariety

. A

subvariety

Y

of

X

is contained in the singular set
if and only if its

local ring

𝒪

Y

is not

regular

(

http://planetmath.org/RegularLocalRing

).

Title

singular set

Canonical name

SingularSet

Date of creation

2013-03-22 13:20:17

Last modified on

2013-03-22 13:20:17

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14A15

Generated on Fri Feb 9 17:57:14 2018 by

LaTeXML
V⁢(I)=∅ implies I=R

V

⁢

(

I

)

=

∅

implies

I

=

R

Note that most of the notation used here is defined in the entry prime spectrum.

Theorem

.

If

R

is a

commutative ring

with

identity

and

I

is an ideal of

R

with

V

⁢

(

I

)

=

∅

, then

I

=

R

.

Proof.

Let

R

be a commutative ring with identity and

I

be an ideal of

R

with

I

≠

R

. Then, by this theorem (

http://planetmath.org/EveryRingHasAMaximalIdeal

), there exists a

maximal ideal

M

of

R

containing

I

. Since

M

is , then

M

is a proper

prime ideal

of

R

. Thus,

M

∈

V

⁢

(

I

)

. The theorem follows.
∎

Title

V

⁢

(

I

)

=

∅

implies

I

=

R

Canonical name

VIemptysetImpliesIR

Date of creation

2013-03-22 16:07:43

Last modified on

2013-03-22 16:07:43

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 14A15

Related topic

ProofThatOperatornameSpecRIsQuasiCompact

Generated on Fri Feb 9 17:57:16 2018 by

LaTeXML
bivector

bivector

A

bivector

is a two-dimensional analog to a one-dimensional vector. Whereas a vector is often utilized to represent a one-dimensional directed quantity (often visualized geometrically as a directed line-segment), a bivector is used to represent a two-dimensional directed quantity (often visualized as an oriented plane-segment).

Since a bivector is a two-dimensional entity, it can be built up from two

linearly independent

vectors,

𝐚

and

𝐛

by means of the

exterior product

.

𝐁

=

𝐚

∧

𝐛

The vectors

𝐚

and

𝐛

span the

subspace

represented by the bivector

𝐁

. Typically the

orientation

of the bivector is established by placing the two vectors tail-to-tail and sweeping from the first vector to the second. In this way, an oppositely oriented bivector may be obtained by reversing the order of the vectors in the exterior product.

𝐚

∧

𝐛

=

-

𝐛

∧

𝐚

Title

bivector

Canonical

name

Bivector

Date of creation

2013-03-22 14:51:48

Last modified on

2013-03-22 14:51:48

Owner

PhysBrain (974)

Last modified by

PhysBrain (974)

Numerical id

6

Author

PhysBrain (974)

Entry type

Definition

Classification

msc 14A20

Generated on Fri Feb 9 17:57:17 2018 by

LaTeXML
Segre map

Segre map

The

Segre map

is an embedding

s

:

ℙ

n

×

ℙ

m

→

ℙ

n

⁢

m

+

n

+

m

of the product of
two

projective spaces

into a larger projective space. It is important since it makes the
product of two

projective varieties

into a projective variety again.
Invariantly, it can described as follows. Let

V

,

W

be (finite dimensional)

vector spaces

; then

s

:

ℙ

⁢

V

×

ℙ

⁢

W

⟶

ℙ

⁢

(

V

⊗

W

)

[

x

]

,

[

y

]

⟼

[

x

⊗

y

]

In

homogeneous coordinates

, the pair of points

[

x

0

:

x

1

:

⋯

:

x

n

]

,

[

y

0

:

y

1

:

⋯

:

y

m

]

maps to

[

x

0

⁢

y

0

:

x

1

⁢

y

0

:

⋯

:

x

n

⁢

y

0

:

x

0

⁢

y

1

:

x

1

⁢

y

1

:

⋯

:

x

n

⁢

y

m

]

.

If we imagine the target space as the projectivized version of the space
of

(

n

+

1

)

×

(

m

+

1

)

matrices, then the image is exactly the set of matrices
which have rank 1; thus it is the common zero locus of the

equations

|

a

i

⁢

j

a

i

⁢

l

a

k

⁢

j

a

k

⁢

l

|

=

a

i

⁢

j

⁢

a

k

⁢

l

-

a

i

⁢

l

⁢

a

k

⁢

j

=

0

for all

0

≤

i

<

k

≤

n

,

0

≤

j

<

l

≤

m

.

Varieties

of this form (defined by vanishing
of minors in some space of matrices) are usually called

determinantal varieties

.

Title

Segre map

Canonical

name

SegreMap

Date of creation

2013-03-22 14:24:45

Last modified on

2013-03-22 14:24:45

Owner

halu (5781)

Last modified by

halu (5781)

Numerical id

4

Author

halu (5781)

Entry type

Definition

Classification

msc 14A25

Classification

msc 14M12

Synonym

Segre embedding

Generated on Fri Feb 9 17:57:19 2018 by

LaTeXML
ample

ample

An

invertible sheaf

𝔏

on a scheme

X

is called

ample

if for any

coherent sheaf

𝔉

,

𝔉

⊗

𝔏

n

is generated by global sections for sufficiently large

n

.

An invertible sheaf is ample if and only if

𝔏

m

is very ample for some

m

; this is very often taken as the

definition

of ample, which can be surprising.

Title

ample

Canonical

name

Ample

Date of creation

2013-03-22 13:52:47

Last modified on

2013-03-22 13:52:47

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

6

Author

archibal (4430)

Entry type

Definition

Classification

msc 14A99

Generated on Fri Feb 9 17:57:21 2018 by

LaTeXML
Cartier divisor

Cartier divisor

On a scheme

X

, a

Cartier divisor

is a global section of the sheaf

𝒦

*

/

𝒪

*

, where

𝒦

*

is the

multiplicative

sheaf of meromorphic functions

, and

𝒪

*

the multiplicative sheaf of invertible

regular functions

(the units of the

structure sheaf

).

More explicitly, a Cartier divisor is a choice of open cover

U

i

of

X

, and

meromorphic functions

f

i

∈

𝒦

*

⁢

(

U

i

)

, such that

f

i

/

f

j

∈

𝒪

*

⁢

(

U

i

∩

U

j

)

, along with two Cartier divisors being the same if the open cover of one is a refinement of the other, with the same

functions

attached to open sets, or if

f

i

is replaced by

g

⁢

f

i

with

g

∈

𝒪

*

.

Intuitively, the only carried by Cartier divisor is where it vanishes, and the order it does there. Thus, a Cartier divisor should give us a

Weil divisor

, and vice versa. On “nice” (for example,

nonsingular

over an algebraically closed field) schemes, it does.

Title

Cartier divisor

Canonical name

CartierDivisor

Date of creation

2013-03-22 13:52:29

Last modified on

2013-03-22 13:52:29

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14A99

Generated on Fri Feb 9 17:57:22 2018 by

LaTeXML
general position

general position

In

projective geometry

, a set of points is said to be in

general position

iff any

d

+

2

of them do not lie on a

d

-dimensional plane, i.e., 4 points are in general position iff no three of them are on the same line.

Dually a set of

d

-dimensional planes is said to be in general position iff no

d

+

2

of them meet in the same point, i.e., 4 lines are in general position iff no three of them meet in the same point.

Title

general position

Canonical name

GeneralPosition

Date of creation

2013-03-22 13:37:31

Last modified on

2013-03-22 13:37:31

Owner

jgade (861)

Last modified by

jgade (861)

Numerical id

8

Author

jgade (861)

Entry type

Definition

Classification

msc 14A99

Generated on Fri Feb 9 17:57:24 2018 by

LaTeXML
height of a prime ideal

height of a prime ideal

Let

R

be a

commutative ring

and

𝔭

a

prime ideal

of

R

. The

height

of

𝔭

is the

supremum

of all integers

n

such that there exists a chain

𝔭

0

⊂

⋯

⊂

𝔭

n

=

𝔭

of distinct prime ideals. The height of

𝔭

is denoted by

h

⁡

(

𝔭

)

.

h

⁡

(

𝔭

)

is also known as the rank of

𝔭

and the codimension of

𝔭

.

The

Krull dimension

of

R

is the supremum of the heights of all the prime ideals of

R

:

sup

⁡

{

h

⁡

(

𝔭

)

∣

𝔭

⁢

prime in

⁢

R

}

.

Title

height of a prime ideal

Canonical

name

HeightOfAPrimeIdeal

Date of creation

2013-03-22 12:49:25

Last modified on

2013-03-22 12:49:25

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 14A99

Synonym

height

Related topic

KrullDimension

Related topic

Cevian

Defines

rank of an ideal

Defines

codimension of an ideal

Generated on Fri Feb 9 17:57:25 2018 by

LaTeXML
invertible sheaf

invertible sheaf

A sheaf

𝔏

of

𝒪

X

modules on a ringed space

𝒪

X

is called

if there is another sheaf of

𝒪

X

-modules

𝔏

′

such that

𝔏

⊗

𝔏

′

≅

𝒪

X

. A sheaf is

invertible

if and only if it is

locally free

of rank 1, and its

inverse

is the sheaf

𝔏

∨

≅

ℋ

⁢

o

⁢

m

⁢

(

𝔏

,

𝒪

X

)

, by the map.

The set of invertible sheaves form an

abelian group

under tensor

multiplication

, called the

Picard group

of

X

.

Title

invertible sheaf

Canonical name

InvertibleSheaf

Date of creation

2013-03-22 13:52:34

Last modified on

2013-03-22 13:52:34

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 14A99

Generated on Fri Feb 9 17:57:27 2018 by

LaTeXML
locally free

locally free

A sheaf of

𝒪

X

-modules

ℱ

on a ringed space

X

is called

locally free

if for each point

p

∈

X

, there is an open neighborhood (

http://planetmath.org/

Neighborhood

)

U

of

x

such that

ℱ

|

U

is free (

http://planetmath.org/FreeModule

) as an

𝒪

X

|

U

-module, or equivalently,

ℱ

p

, the stalk of

ℱ

at

p

, is free as a

(

𝒪

X

)

p

-module. If

ℱ

p

is of finite rank (

http://planetmath.org/ModuleOfFiniteRank

)

n

, then

ℱ

is said to be of rank

n

.

Title

locally free

Canonical

name

LocallyFree

Date of creation

2013-03-22 13:52:31

Last modified on

2013-03-22 13:52:31

Owner

mps (409)

Last modified by

mps (409)

Numerical id

13

Author

mps (409)

Entry type

Definition

Classification

msc 14A99

Generated on Fri Feb 9 17:57:28 2018 by

LaTeXML
normal irreducible varieties are nonsingular in codimension 1

normal irreducible varieties are nonsingular in codimension 1

Theorem 1

.

Let

X

be a normal

irreducible variety

. The

singular set

S

⊂

X

has codimension 2 or more.

Proof.

Assume not. We may assume

X

is affine, since codimension is local. Now let

𝔲

be the ideal of

functions

vanishing on

S

. This is an ideal of height 1, so the

local ring

of

Y

,

𝒪

S

=

A

⁢

(

X

)

𝔲

, where

A

⁢

(

X

)

is the affine ring of

X

, is a 1-dimensional local ring, and

integrally closed

, since

X

is normal. Any integrally closed 1-dimensional local domain is a
DVR, and thus

regular

. But

S

is the singular set, so its local ring is not regular, a

contradiction

.
∎

Title

normal irreducible varieties are nonsingular in codimension 1

Canonical name

NormalIrreducibleVarietiesAreNonsingularInCodimension1

Date of creation

2013-03-22 13:20:20

Last modified on

2013-03-22 13:20:20

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

7

Author

archibal (4430)

Entry type

Theorem

Classification

msc 14A99

Generated on Fri Feb 9 17:57:30 2018 by

LaTeXML
Serre’s twisting theorem

Serre’s twisting theorem

Let

X

be a scheme, and

𝔏

an ample

invertible sheaf

on

X

. Then for any

coherent sheaf

ℱ

, and sufficiently large

n

,

H

i

⁢

(

ℱ

⊗

𝔏

n

)

=

0

, that is, the higher

sheaf cohomology

of

ℱ

⊗

𝔏

n

is trivial.

Title

Serre’s twisting theorem

Canonical name

SerresTwistingTheorem

Date of creation

2013-03-22 13:52:50

Last modified on

2013-03-22 13:52:50

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Theorem

Classification

msc 14A99

Generated on Fri Feb 9 17:57:32 2018 by

LaTeXML
sheaf of meromorphic functions

sheaf of meromorphic functions

Let

(

X

,

𝒪

X

)

be a ringed space. By definition, for every

U

we have a ring

𝒪

X

⁢

(

U

)

. If

S

U

is the set of elements that are not

zero divisors

, we can construct the

localization

K

U

=

S

U

-

1

⁢

𝒪

X

⁢

(

U

)

. If

𝒪

X

⁢

(

U

)

is actually an

integral domain

, then

K

will be its

field of fractions

. It is easy to verify that the restriction maps of the sheaf

𝒪

X

⁢

(

U

)

yield restriction maps on the rings

K

U

, so that we can define a

presheaf

U

↦

K

U

. Let

𝒦

X

be the

sheafification

of this presheaf. Then

𝒦

X

is called the

sheaf of meromorphic functions

.

If

X

is a connected

complex manifold

, then

X

has a sheaf

𝒪

X

of

holomorphic functions

making it into a ringed space. These rings are always integral domains, and their

quotients

are all the same, so

𝒦

X

is a constant sheaf; in fact it always takes the same value, a field

K

. We recognize

K

as precisely the field of

meromorphic functions

on

X

.

If

X

is a scheme, then

X

has an associated sheaf

𝒪

X

making it into a ringed space. If

X

is arbitrary, then

𝒦

X

will simply be a sheaf of rings. If, however,

X

is

integral

and quasicompact, then the situation is very similar to the situation of complex manifolds; the ring of

regular functions

on every Zariski open set is an integral domain, and all the restriction maps are injective. As a result, the sheaf of meromorphic functions is again a constant sheaf that always yields the same value, and this value is called the

function field

of

X

. This function field is an essential object of study in birational geometry.

If

X

is not reduced, its

structure sheaf

contains

nilpotents

. Thus

𝒦

is not a sheaf of fields, even locally. Such schemes arise when discussing infinitesimal deformations.

If

X

is reduced but not

irreducible

, then each

irreducible component

(if it is quasicompact) has a function field, and

𝒦

X

⁢

(

X

)

is in fact the

direct sum

of these function fields.

If

X

is a

differential manifold

, the

differentiable functions

on it form a sheaf of rings making

X

into a ringed space. Here the structure of

𝒦

X

is much more complicated; a

function

on

U

has an

inverse

if and only if its

support

on

U

has empty interior. So globally, this amounts to allowing functions to have poles provided the support of these poles has empty interior. This complicated structure makes the sheaf of meromorphic functions much less useful in the differentiable category than it is for schemes or complex manifolds.

Title

sheaf of meromorphic functions

Canonical name

SheafOfMeromorphicFunctions

Date of creation

2013-03-22 13:52:17

Last modified on

2013-03-22 13:52:17

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

6

Author

archibal (4430)

Entry type

Definition

Classification

msc 14A99

Defines

meromorphic

Generated on Fri Feb 9 17:57:33 2018 by

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very ample

very ample

An

invertible sheaf

𝔏

on a scheme

X

over a field

k

is called

very ample

if (1) at each point

x

∈

X

, there is a global section

s

∈

𝔏

⁢

(

X

)

not vanishing at

x

, and (2) for each pair of points

x

,

y

∈

X

, there is a global section

s

∈

𝔏

⁢

(

X

)

such that

s

vanishes at exactly one of

x

and

y

.

Equivalently,

𝔏

is very ample if there is an

embedding

f

:

X

→

ℙ

n

such that

f

*

⁢

𝒪

⁢

(

1

)

=

𝔏

, that is,

𝔏

is the

pullback

of the tautological bundle on

ℙ

n

.

If

k

is

algebraically closed

, Riemann-Roch (

http://planetmath.org/RiemannRochTheorem

) shows that on a curve

X

, any invertible sheaf of degree greater than or equal to twice the genus of

X

is very ample.

Title

very ample

Canonical name

VeryAmple

Date of creation

2013-03-22 13:52:44

Last modified on

2013-03-22 13:52:44

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 14A99

Generated on Fri Feb 9 17:57:35 2018 by

LaTeXML
algebraic equivalence of divisors

algebraic equivalence of divisors

Let

X

be a

surface

(a two-dimensional

algebraic variety

).

Definition 1

.

1.

An

algebraic

family of

effective divisors

on

X

parametrized by a

non-singular

curve

T

is defined to be an effective

Cartier divisor

𝒟

on

X

×

T

which is flat over

T

.

2.

If

ℱ

is an algebraic family of effective divisors on

X

, parametrized by a non-singular curve

T

, and

P

,

Q

∈

T

are any two

closed points

on

T

, then we say that the corresponding

divisors

in

ℱ

,

D

P

,

D

Q

, are prealgebraically

equivalent

.

3.

Two (Weil) divisors

D

,

D

′

on

X

are algebraically equivalent if there is a

finite sequence

D

=

D

0

,

D

1

,

…

,

D

n

=

D

′

with

D

i

and

D

i

+

1

prealgebraically equivalent for all

0

≤

i

<

n

.

Title

algebraic equivalence of divisors

Canonical name

AlgebraicEquivalenceOfDivisors

Date of creation

2013-03-22 15:34:10

Last modified on

2013-03-22 15:34:10

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 14C20

Generated on Fri Feb 9 17:57:36 2018 by

LaTeXML
divisor

divisor

A

divisor

D

on a projective nonsingular curve over an algebraically closed field is a formal sum of points

D

=

∑

n

p

⁢

p

where only finitely many of the

n

p

∈

ℤ

are nonzero.

The

degree

of a divisor

D

is

deg

⁢

(

D

)

=

∑

n

p

.

Title

divisor

Canonical

name

Divisor

Date of creation

2013-03-22 12:03:16

Last modified on

2013-03-22 12:03:16

Owner

nerdy2 (62)

Last modified by

nerdy2 (62)

Numerical id

7

Author

nerdy2 (62)

Entry type

Definition

Classification

msc 14C20

Synonym

Weil divisor

Defines

degree

Generated on Fri Feb 9 17:57:38 2018 by

LaTeXML
intersection divisor for a quartic

intersection divisor for a quartic

Let

C

be a

non-singular

curve in the plane, defined over an algebraically closed field

K

, and given by a polynomial

f

⁢

(

x

,

y

)

=

0

of degree

4

(i.e.

C

is a quartic). Let

L

be a (

rational

) line in the plane

K

2

. The

intersection

divisor

of

C

and

L

is of the form:

(

L

⋅

C

)

=

P

1

+

P

2

+

P

3

+

P

4

where

P

i

,

i

=

1

,

2

,

3

,

4

, are points in

C

⁢

(

K

)

. There are five possibilities:

1.

The

generic

position: all the points

P

i

are distinct.

2.

L

is tangent to

C

: there exist indices

1

≤

i

≠

j

≤

4

such that

P

i

=

P

j

. Without loss of generality we may assume

P

1

=

P

2

and

(

L

⋅

C

)

=

2

⁢

P

1

+

P

3

+

P

4

, and

P

3

≠

P

4

.

3.

L

is bitangent to

C

when

P

1

=

P

2

and

P

3

=

P

4

but

P

1

≠

P

3

. It may be shown that if

char

⁡

(

K

)

≠

2

then

C

has exactly

28

bitangent lines.

4.

L

intersects

C

at exactly two points, thus

P

1

=

P

2

=

P

3

≠

P

4

. The point

P

1

is called a

flex

.

5.

L

intersects

C

at exactly one point and

P

1

=

P

2

=

P

3

=

P

4

. This point is called a

hyperflex

. A quartic

C

may not have any hyperflex.

References

1

S. Flon, R. Oyono, C. Ritzenthaler,

Fast

addition

on non-hyperelliptic genus 3 curves

, can be found

http://eprint.iacr.org/2004/118.ps

here.

Title

intersection divisor for a quartic

Canonical

name

IntersectionDivisorForAQuartic

Date of creation

2013-03-22 15:44:59

Last modified on

2013-03-22 15:44:59

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 14C20

Defines

hyperflex

Defines

flex

Generated on Fri Feb 9 17:57:40 2018 by

LaTeXML
Néron-Severi group

Néron-Severi group

Definition 1

.

Let

S

be a

surface

(an

algebraic variety

of dimenion

2

) defined over an algebraically closed field

k

. The Néron-Severi group of

S

, denoted by

NS

⁡

(

S

)

, is the group of

Weil divisors

on

S

modulo

algebraic equivalence of divisors

(see parent entry).

Theorem

(Néron, Severi).

The Néron-Severi group of a surface is a

finitely generated

abelian group

.

Title

Néron-Severi group

Canonical name

NeronSeveriGroup

Date of creation

2013-03-22 15:34:13

Last modified on

2013-03-22 15:34:13

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 14C20

Generated on Fri Feb 9 17:57:41 2018 by

LaTeXML