Linear system of divisors: Difference between revisions

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{{Short description|Concept in algebraic geometry}}

{{redirect-distinguish2|Kodaira map|[[Kodaira–Spencer map]] from cohomology theory}}

[[File:Apollonian circles.svg|thumb|A '''linear system of divisors''' algebraicizes the classic geometric notion of a [[family of curves]], as in the [[Apollonian circles]].]]

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.

These arose first in the form of a ''linear system'' of [[algebraic curve]]s in the [[projective plane]]. It assumed a more general form, through gradual generalisation, so that one could speak of '''linear equivalence''' of [[divisor (algebraic geometry)|divisor]]s ''D'' on a general [[Scheme (mathematics)|scheme]] or even a [[ringed space]] <math>(''X'', '' \mathcal{O''<sub>''X''}_X)</submath>).<ref>[[Alexander Grothendieck|Grothendieck, Alexandre]]; Dieudonné, Jean. ''EGA IV'', 21.3.</ref>

Linear systemsystems of dimension 1, 2, or 3 are called a '''[[Pencil (mathematics)|pencil]]''', a '''net''', or a '''web''', respectively.

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

==DefinitionDefinitions ==

Given the fundamental idea of a [[rational function]] on a general variety <math>X</math>, or in other words of a function <math>f</math> in the [[Function field of an algebraic variety|function field]] of <math>X</math>, <math>f \in k(X)</math>,two divisors <math>D,E \in \text{Div}(X)</math> are '''linearly equivalent divisors''' if

:<math>DE = ED + (f)\ </math>

wherefor some non-zero [[rational function]] <math>f</math> on <math>X</math>, or in other words a non-zero element <math>f</math> of the [[Function field of an algebraic variety|function field]] <math>k(X)</math>. Here <math>(f)</math> denotes the divisor of zeroes and poles of the function <math>f</math>.

Note that if <math>X</math> has [[Mathematical singularity|singular points]], the notion of 'divisor' is inherently ambiguous ([[Cartier divisor]]s, [[Weil divisor]]s: see [[divisor (algebraic geometry)]]). The definition in that case is usually said with greater care (using [[invertible sheaves]] or [[holomorphic line bundle]]s); see below.

A '''complete linear system''' on <math>X</math> is defined as the set of all effective divisors linearly equivalent to some given divisor <math>D \in \text{Div}(X)</math>. It is denoted <math>|D|</math>. Let <math>\mathcal{L}</math> be the line bundle associated to <math>D</math>. In the case that <math>X</math> is a nonsingular projective variety, the set <math>|D|</math> is in natural bijection with <math> (\Gamma(X,\mathcal{L}) \smallsetminus \{0\})/k^\ast, </math> <ref name=":0">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</ref> {{explain|dateby associating the element <math>E =July 2020}}D + (f)</math> of <math>|D|</math> to the set of non-zero multiples of <math>f</math> (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 <math>|D|</math> is therefore a projective space.

A '''linear system''' <math> \mathfrak{d} </math> is then a projective subspace of a complete linear system, so it corresponds to a vector subspace ''W'' of <math> \Gamma(X,\mathcal{L}). </math> The dimension of the linear system <math> \mathfrak{d} </math> is its dimension as a projective space. Hence <math> \dim \mathfrak{d} = \dim W - 1 </math>.

Since a Cartier divisor class is an isomorphism class of a line bundle, linearLinear systems can also be introduced by means of the [[line bundle]] or [[invertible sheaf]] language, without reference to divisors at all. In those terms, divisors <math>D</math> ([[Cartier divisor]]s, 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 <math>\mathcal{O}(2)</math> on <math>\mathbb{P}^3</math> whose sections <math>s \in \Gamma(\mathbb{P}^3,\mathcal{O}(2))</math> define [[quadric surfacessurface]]s. For the associated divisor <math>D_s = Z(s)</math>, it is linearly equivalent to any other divisor defined by the vanishing locus of some <math>t \in \Gamma(\mathbb{P}^3,\mathcal{O}(2)) </math> using the rational function <math>\left(t/s\right)</math><ref name=":0" /> (Proposition 7.2). For example, the divisor <math>D</math> associated to the vanishing locus of <math>x^2 + y^2 + z^2 + w^2</math> is linearly equivalent to the divisor <math>E</math> associated to the vanishing locus of <math>xy</math>. Then, there is the equivalence of divisors<blockquote><math>D = E + \left( \frac{x^2 + y^2 + z^2 + w^2}{xy} \right)</math></blockquote>

=== Linear systems on curves ===

One of the important complete linear systems on an algebraic curve <math>C</math> of [[Genus (mathematics)|genus]] <math>g</math> is given by the complete linear system associated with the canonical divisor <math>K</math>, denoted <math>|K| = \mathbb{P}(H^0(C,\omega_C))</math>. This definition follows from proposition II.7.7 of Hartshorne<ref name=":0" /> since every effective divisor in the linear system comes from the zeros of some section of <math>\omega_C</math>.

==== Hyperelliptic curves ====

One application of linear systems is used in the classification of algebraic curves. A [[hyperelliptic curve]] is a curve <math>C</math> with a [[Degree of a finite morphism|degree <math>2</math> morphism]] <math>f:C \to \mathbb{P}^1</math>.<ref name=":0" />. For the case <math>g=2</math> all curves are hyperelliptic: the [[Riemann–Roch theorem]] then gives the degree of <math>K_C</math> is <math>2g - 2 = 2</math> and <math>h^0(K_C) = 2</math>, hence there is a degree <math>2</math> map to <math>\mathbb{P}^1 = \mathbb{P}(H^0(C,\omega_C))</math>.

==== g_rd^dr ====

A <math>g^r_d</math> is a linear system <math> \mathfrak{d} </math> on a curve <math>C</math> which is of degree <math>d</math> and dimension <math>r</math>. For example, hyperelliptic curves have a <math>g^1_2</math> sincewhich is induced by the <math>|K_C|2:1</math>-map defines<math>C one\to \mathbb P^1</math>. In fact, hyperelliptic curves have a unique <math>g^1_2</math><ref name=":0" /> from proposition 5.3. Another close set of examples are curves with a <math>gg_1^1_33</math> which are called [[Trigonal curve|trigonal curves]]. In fact, any curve has a <math>g_dg^1d_1</math> for <math>d \geq (1/2)g + 1</math>.<ref>{{Cite journal|lastlast1=Kleiman|firstfirst1=Steven L.|last2=Laksov|first2=Dan|date=1974|title=Another proof of the existence of special divisors|url=https://projecteuclid.org/euclid.acta/1485889804|journal=Acta Mathematica|language=EN|volume=132|pages=163–176|doi=10.1007/BF02392112|issn=0001-5962|doi-access=free}}</ref>.

===Linear systems of hypersurfaces in <math>\mathbb{P}^n</math>a projective space===

Consider the line bundle <math>\mathcal{O}(d)</math> over <math>\mathbb{P}^n</math>. If we take global sections <math>V = \Gamma(\mathcal{O}(d))</math>, then we can take its projectivization <math>\mathbb{P}(V)</math>. This is isomorphic to <math>\mathbb{P}^N</math> where

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===Linear system of conics===

{{main|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''.<ref>{{cite book |last1=Arbarello |first1=Enrico |author1-link=Enrico Arbarello |last2=Cornalba |first2=Maurizio |last3=Griffiths |first3=Phillip |author3-link=Phillip Griffiths |title=Geometry of algebraic curves |volume=II, with a contribution by Joseph Daniel Harris |series=Grundlehren der Mathematischen Wissenschaften |issue=268 |publisher=Springer |___location=Heidelberg |year=2011 |mr=2807457 |doi=10.1007/978-1-4757-5323-3 |page=3|isbn=978-1-4419-2825-2 }}</ref>

In modern terms, it is a subsystem of the linear system associated to the [[normal bundle]] to <math>C \hookrightarrow Y</math>. 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 ===

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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 &mdash; the Riemann–Roch problem as it can be called &mdash; can be better phrased in terms of [[homological algebra]]. The effect of working on varieties with [[Mathematical singularity|singular points]] is to show up a difference between [[Weil divisor]]s (in the [[free abelian group]] generated by codimension-one subvarieties), and [[Cartier divisor]]s 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 ==

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Let ''L'' be a line bundle on an algebraic variety ''X'' and <math>V \subset \Gamma(X, L)</math> 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 <math>V \otimes_k \mathcal{O}_X \to L</math> is surjective (here, ''k'' = the base field). Or equivalently, <math>\operatorname{Sym}((V \otimes_k \mathcal{O}_X) \otimes_{\mathcal{O}_X} L^{-1}) \to \bigoplus_{n=0}^{\infty} \mathcal{O}_X</math> is surjective. Hence, writing <math>V_X = V \times X</math> for the trivial vector bundle and passing the surjection to the [[relative Proj]], there is a [[closed immersion]]:

:<math>i: X \hookrightarrow \mathbb{P}(V_X^* \otimes L) \simeq \mathbb{P}(V_X^*) = \mathbb{P}(V^*) \times X</math>

where <math>\simeq</math> 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:<ref>{{harvnbcite book|last = Fulton|locfirst = William|title = Intersection Theory|chapter = § 4.4. Linear Systems |doi = 10.1007/978-1-4612-1700-8_5 |publisher = Springer|year = 1998}}</ref>

:<math>f: X \to \mathbb{P}(V^*).</math>

When the base locus of ''V'' is not empty, the above discussion still goes through with <math>\mathcal{O}_X</math> in the direct sum replaced by an [[ideal sheaf]] defining the base locus and ''X'' replaced by the [[blowing up|blow-up]] <math>\widetilde{X}</math> of it along the (scheme-theoretic) base locus ''B''. Precisely, as above, there is a surjection <math>\operatorname{Sym}((V \otimes_k \mathcal{O}_X) \otimes_{\mathcal{O}_X} L^{-1}) \to \bigoplus_{n=0}^{\infty} \mathcal{I}^n</math> where <math>\mathcal{I}</math> is the ideal sheaf of ''B'' and that gives rise to

:<math>i: \widetilde{X} \hookrightarrow \mathbb{P}(V^*) \times X.</math>

Since <math>X - B \simeq</math> an open subset of <math>\widetilde{X}</math>, there results in the map:

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=== O(1) on a projective variety ===

A projective variety <math>X</math> embedded in <math>\mathbb{P}^r</math> has a canonicalnatural linear system determining a map to projective space from <math>\mathcal{O}_X(1) = \mathcal{O}_X \otimes_{\mathcal{O}_{\mathbb{P}^r}} \mathcal{O}_{\mathbb{P}^r}(1)</math>. This sends a point <math>x \in X</math> to its corresponding point <math>[x_0:\cdots:x_r] \in \mathbb{P}^r </math>.

== See also ==

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{{refbegin}}

* {{cite book | author=P. Griffiths | authorlink=Phillip Griffiths |author2=J. Harris |authorlink2=Joe Harris (mathematician) | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | page=137 }}

* [[Robin Hartshorne|Hartshorne, R.]], ''Algebraic Geometry'', [[Springer-Verlag]], 1977; corrected 6th printing, 1993. {{isbn|0-387-90244-9}}.

* [[Robert Lazarsfeld|Lazarsfeld, R.]], ''Positivity in Algebraic Geometry I'', Springer-Verlag, 2004. {{isbn|3-540-22533-1}}.

{{refend}}

*

[[Category:Geometry of divisors]]