Linear system of divisors explained

In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.

(X,l{O}_X)

.^[1]

Linear systems of dimension 1, 2, or 3 are called a pencil, a net, or a web, respectively.

A map determined by a linear system is sometimes called the Kodaira map.

Definitions

Given a general variety

, two divisors

D,E\inDiv(X)

are linearly equivalent if

E=D+(f)

, or in other words a non-zero element

of the function field

k(X)

. Here

(f)

denotes the divisor of zeroes and poles of the function

Note that if

has singular points, the notion of 'divisor' is inherently ambiguous (Cartier divisors, Weil divisors: see divisor (algebraic geometry)). The definition in that case is usually said with greater care (using invertible sheaves or holomorphic line bundles); see below.

A complete linear system on

is defined as the set of all effective divisors linearly equivalent to some given divisor

D\inDiv(X)

. It is denoted

|D|

. Let

l{L}

be the line bundle associated to

. In the case that

is a nonsingular projective variety, the set

|D|

is in natural bijection with

(\Gamma(X,l{L})\smallsetminus\{0\})/k^\ast,

^[2] by associating the element

E=D+(f)

|D|

to the set of non-zero multiples of

(this is well defined since two non-zero rational functions have the same divisor if and only if they are non-zero multiples of each other). A complete linear system

|D|

is therefore a projective space.

A linear system

ak{d}

is then a projective subspace of a complete linear system, so it corresponds to a vector subspace W of

\Gamma(X,l{L}).

The dimension of the linear system

ak{d}

is its dimension as a projective space. Hence

\dimak{d}=\dimW-1

Linear systems can also be introduced by means of the line bundle or invertible sheaf language. In those terms, divisors

(Cartier divisors, to be precise) correspond to line bundles, and linear equivalence of two divisors means that the corresponding line bundles are isomorphic.

Examples

Linear equivalence

Consider the line bundle

l{O}(2)

P³

whose sections

s\in\Gamma(P^3,l{O}(2))

define quadric surfaces. For the associated divisor

D_s=Z(s)

, it is linearly equivalent to any other divisor defined by the vanishing locus of some

t\in\Gamma(P^3,l{O}(2))

using the rational function

\left(t/s\right)

(Proposition 7.2). For example, the divisor

associated to the vanishing locus of

x²+y²+z²+w²

is linearly equivalent to the divisor

associated to the vanishing locus of

. Then, there is the equivalence of divisors

D=E+\left(

x²+y²+z²+w²
xy

\right)

Linear systems on curves

One of the important complete linear systems on an algebraic curve

of genus

is given by the complete linear system associated with the canonical divisor

, denoted

|K|=

	0(C,\omega
P(H
	C))

. This definition follows from proposition II.7.7 of Hartshorne since every effective divisor in the linear system comes from the zeros of some section of

\omega_C

Hyperelliptic curves

One application of linear systems is used in the classification of algebraic curves. A hyperelliptic curve is a curve

with a degre

morphism

f:C\toP¹

. For the case

g=2

all curves are hyperelliptic: the Riemann–Roch theorem then gives the degree of

K_C

2g-2=2

and

	0(K
h
	C)

, hence there is a degree

map to

P¹=

	0(C,\omega
P(H
	C))

g_r^d

	d
g
	r

is a linear system

ak{d}

on a curve

which is of degree

and dimension

. For example, hyperelliptic curves have a

	1
g
	2

since

|K_C|

defines one. In fact, hyperelliptic curves have a unique

	1
g
	2

from proposition 5.3. Another close set of examples are curves with a

	3
g
	1

which are called trigonal curves. In fact, any curve has a

	d
g
	1

for

d\geq(1/2)g+1

.^[3]

Linear systems of hypersurfaces in a projective space

Consider the line bundle

l{O}(d)

over

Pⁿ

. If we take global sections

V=\Gamma(l{O}(d))

, then we can take its projectivization

P(V)

. This is isomorphic to

P^N

where

N=\binom{n+d}{n}-1

Then, using any embedding

P^k\toP^N

we can construct a linear system of dimension

Linear system of conics

See main article: Linear system of conics.

Characteristic linear system of a family of curves

The characteristic linear system of a family of curves on an algebraic surface Y for a curve C in the family is a linear system formed by the curves in the family that are infinitely near C.^[4]

In modern terms, it is a subsystem of the linear system associated to the normal bundle to

C\hookrightarrowY

. Note a characteristic system need not to be complete; in fact, the question of completeness is something studied extensively by the Italian school without a satisfactory conclusion; nowadays, the Kodaira–Spencer theory can be used to answer the question of the completeness.

Other examples

The Cayley–Bacharach theorem is a property of a pencil of cubics, which states that the base locus satisfies an "8 implies 9" property: any cubic containing 8 of the points necessarily contains the 9th.

Linear systems in birational geometry

In general linear systems became a basic tool of birational geometry as practised by the Italian school of algebraic geometry. The technical demands became quite stringent; later developments clarified a number of issues. The computation of the relevant dimensions - the Riemann–Roch problem as it can be called - can be better phrased in terms of homological algebra. The effect of working on varieties with singular points is to show up a difference between Weil divisors (in the free abelian group generated by codimension-one subvarieties), and Cartier divisors coming from sections of invertible sheaves.

The Italian school liked to reduce the geometry on an algebraic surface to that of linear systems cut out by surfaces in three-space; Zariski wrote his celebrated book Algebraic Surfaces to try to pull together the methods, involving linear systems with fixed base points. There was a controversy, one of the final issues in the conflict between 'old' and 'new' points of view in algebraic geometry, over Henri Poincaré's characteristic linear system of an algebraic family of curves on an algebraic surface.

Base locus

The base locus of a linear system of divisors on a variety refers to the subvariety of points 'common' to all divisors in the linear system. Geometrically, this corresponds to the common intersection of the varieties. Linear systems may or may not have a base locus – for example, the pencil of affine lines

x=a

has no common intersection, but given two (nondegenerate) conics in the complex projective plane, they intersect in four points (counting with multiplicity) and thus the pencil they define has these points as base locus.

More precisely, suppose that

|D|

is a complete linear system of divisors on some variety

. Consider the intersection

\operatorname{Bl}(|D|):=

cap
	D_eff\in\|D\|

\operatorname{Supp}D_eff

where

\operatorname{Supp}

denotes the support of a divisor, and the intersection is taken over all effective divisors

D_eff

in the linear system. This is the base locus of

|D|

(as a set, at least: there may be more subtle scheme-theoretic considerations as to what the structure sheaf of

\operatorname{Bl}

should be).

One application of the notion of base locus is to nefness of a Cartier divisor class (i.e. complete linear system). Suppose

|D|

is such a class on a variety

, and

an irreducible curve on

. If

is not contained in the base locus of

|D|

, then there exists some divisor

\tildeD

in the class which does not contain

, and so intersects it properly. Basic facts from intersection theory then tell us that we must have

|D| ⋅ C\geq0

. The conclusion is that to check nefness of a divisor class, it suffices to compute the intersection number with curves contained in the base locus of the class. So, roughly speaking, the 'smaller' the base locus, the 'more likely' it is that the class is nef.

In the modern formulation of algebraic geometry, a complete linear system

|D|

of (Cartier) divisors on a variety

is viewed as a line bundle

l{O}(D)

. From this viewpoint, the base locus

\operatorname{Bl}(|D|)

is the set of common zeroes of all sections of

l{O}(D)

. A simple consequence is that the bundle is globally generated if and only if the base locus is empty.

The notion of the base locus still makes sense for a non-complete linear system as well: the base locus of it is still the intersection of the supports of all the effective divisors in the system.

Example

p:ak{X}\toP¹

given by two generic sections

f,g\in\Gamma(P^n,l{O}(d))

, so

ak{X}

given by the scheme

ak{X}=Proj\left(

k[s,t][x_0,\ldots,x_n]
(sf+tg)

\right)

This has an associated linear system of divisors since each polynomial,

s_0f+t_0g

for a fixed

[s_0:t_0]\inP¹

is a divisor in

Pⁿ

. Then, the base locus of this system of divisors is the scheme given by the vanishing locus of

f,g

, so

Bl(ak{X})=Proj\left(

k[s,t][x_0,\ldots,x_n]
(f,g)

\right)

A map determined by a linear system

Each linear system on an algebraic variety determines a morphism from the complement of the base locus to a projective space of dimension of the system, as follows. (In a sense, the converse is also true; see the section below)

Let L be a line bundle on an algebraic variety X and

V\subset\Gamma(X,L)

a finite-dimensional vector subspace. For the sake of clarity, we first consider the case when V is base-point-free; in other words, the natural map

V ⊗ _kl{O}_X\toL

is surjective (here, k = the base field). Or equivalently,

\operatorname{Sym}((V ⊗ _kl{O}_X) ⊗ _l{O_X}L^-1)\to

	infty
oplus
	n=0

l{O}_X

is surjective. Hence, writing

V_X=V x X

for the trivial vector bundle and passing the surjection to the relative Proj, there is a closed immersion:

i:X\hookrightarrow

	*
P(V
	X

⊗ L)\simeq

	*)
P(V
	X

=P(V^*) x X

where

\simeq

on the right is the invariance of the projective bundle under a twist by a line bundle. Following i by a projection, there results in the map:^[5]

f:X\toP(V^*).

When the base locus of V is not empty, the above discussion still goes through with

l{O}_X

in the direct sum replaced by an ideal sheaf defining the base locus and X replaced by the blow-up

\widetilde{X}

of it along the (scheme-theoretic) base locus B. Precisely, as above, there is a surjection

\operatorname{Sym}((V ⊗ _kl{O}_X) ⊗ _l{O_X}L^-1)\to

	infty
oplus
	n=0

l{I}ⁿ

where

l{I}

is the ideal sheaf of B and that gives rise to

i:\widetilde{X}\hookrightarrowP(V^*) x X.

Since

X-B\simeq

an open subset of

\widetilde{X}

, there results in the map:

f:X-B\toP(V^*).

Finally, when a basis of V is chosen, the above discussion becomes more down-to-earth (and that is the style used in Hartshorne, Algebraic Geometry).

Linear system determined by a map to a projective space

Each morphism from an algebraic variety to a projective space determines a base-point-free linear system on the variety; because of this, a base-point-free linear system and a map to a projective space are often used interchangeably.

For a closed immersion

f:Y\hookrightarrowX

of algebraic varieties there is a pullback of a linear system

ak{d}

, defined as

f^-1(ak{d})=\{f^-1(D)|D\inak{d}\}

(page 158).

O(1) on a projective variety

A projective variety

embedded in

P^r

has a natural linear system determining a map to projective space from

l{O}_X(1)=l{O}_X ⊗ _l{O


	P^r

} \mathcal_(1). This sends a point

x\inX

to its corresponding point

[x_{0: … :x}_r]\inP^r

References

Book: Phillip Griffiths

. P. Griffiths . Phillip Griffiths . J. Harris . Joe Harris (mathematician) . Principles of Algebraic Geometry . Wiley Classics Library . Wiley Interscience . 1994 . 0-471-05059-8 . 137 .

Hartshorne, R., Algebraic Geometry, Springer-Verlag, 1977; corrected 6th printing, 1993. .
Lazarsfeld, R., Positivity in Algebraic Geometry I, Springer-Verlag, 2004. .

Notes and References

[Alexander Grothendieck|Grothendieck, Alexandre]
Hartshorne, R. 'Algebraic Geometry', proposition II.7.2, page 151, proposition II.7.7, page 157, page 158, exercise IV.1.7, page 298, proposition IV.5.3, page 342
Kleiman. Steven L.. Laksov. Dan. 1974. Another proof of the existence of special divisors. Acta Mathematica. EN. 132. 163–176. 10.1007/BF02392112. 0001-5962. free.
Book: Arbarello . Enrico . Enrico Arbarello . Cornalba . Maurizio . Griffiths . Phillip . Phillip Griffiths . Geometry of algebraic curves . II, with a contribution by Joseph Daniel Harris . Grundlehren der Mathematischen Wissenschaften . 268 . Springer . Heidelberg . 2011 . 2807457 . 10.1007/978-1-4757-5323-3 . 3. 978-1-4419-2825-2 .
Book: Fulton, William. Intersection Theory. § 4.4. Linear Systems . 10.1007/978-1-4612-1700-8_5 . Springer. 1998.

Linear system of divisors explained

Definitions

Examples

Linear equivalence

Linear systems on curves

Hyperelliptic curves

grd

Linear systems of hypersurfaces in a projective space

Linear system of conics

Characteristic linear system of a family of curves

Other examples

Linear systems in birational geometry

Base locus

Example

A map determined by a linear system

Linear system determined by a map to a projective space

O(1) on a projective variety

See also

References

Notes and References

g_r^d