Content deleted Content added
mNo edit summary |
|||
| (48 intermediate revisions by 25 users not shown) | |||
Line 1:
{{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>(
A map determined by a linear system is sometimes called the '''Kodaira map'''.
==
Given a general variety <math>X</math>, two divisors <math>D,E \in \text{Div}(X)</math> are '''linearly equivalent''' if
:<math>
Note that if
A '''complete linear system
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(
==Linear systems of hypersurfaces in <math>\mathbb{P}^n</math>==▼
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▼
:<math>N = \binom{n+d}{n} - 1</math>▼
Then, using any embedding <math>\mathbb{P}^k \to \mathbb{P}^N</math> we can construct a linear system of dimension <math>k</math>.▼
=== Linear equivalence ===
==Linear system of conics==▼
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 surface]]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<sub>d</sub><sup>r</sup> ====
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> which is induced by the <math>2:1</math>-map <math>C \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>g_1^3</math> which are called [[Trigonal curve|trigonal curves]]. In fact, any curve has a <math>g^d_1</math> for <math>d \geq (1/2)g + 1</math>.<ref>{{Cite journal|last1=Kleiman|first1=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>
▲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
▲:<math>N = \binom{n+d}{n} - 1</math>
▲Then, using any embedding <math>\mathbb{P}^k \to \mathbb{P}^N</math> we can construct a linear system of dimension <math>k</math>.
===Linear
===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 ===
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.
Line 73 ⟶ 61:
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 [[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
== Base locus ==
Line 91 ⟶ 79:
{{See also|Theorem of Bertini}}
=== Example ===
Consider the [[Lefschetz pencil]] <math>p:\mathfrak{X} \to \mathbb{P}^1</math> given by two generic sections <math>f,g \in \Gamma(\mathbb{P}^n,\mathcal{O}(d))</math>, so <math>\mathfrak{X}</math> given by the scheme<blockquote><math>\mathfrak{X} =\text{Proj}\left( \frac{k[s,t][x_0,\ldots,x_n]}{(sf + tg)} \right)</math></blockquote>This has an associated linear system of divisors since each polynomial, <math>s_0f + t_0g</math> for a fixed <math>[s_0:t_0] \in \mathbb{P}^1</math> is a divisor in <math>\mathbb{P}^n</math>. Then, the base locus of this system of divisors is the scheme given by the vanishing locus of <math>f,g</math>, so<blockquote><math>\text{Bl}(\mathfrak{X}) = \text{Proj}\left(
\frac{
k[s,t][x_0,\ldots,x_n]
}{
(f,g)
}
\right)</math></blockquote>
== A map determined by a linear system ==
<!-- the below is a coordinate-free approach; while useful and important in application, we should also give a less abstract construction as well. -->
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;
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>{{
:<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:
Line 107 ⟶ 104:
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).<!-- and we should give that version here as well. -->
== Linear system determined by a map to a projective space ==
{{expand section|date=August 2019}}
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 <math>f: Y \hookrightarrow X</math> of algebraic varieties there is a pullback of a linear system <math>\mathfrak{d}</math> on <math>X</math> to <math>Y</math>, defined as <math>f^{-1}(\mathfrak{d}) = \{ f^{-1}(D) | D \in \mathfrak{d} \}</math><ref name=":0" /> (page 158).
=== O(1) on a projective variety ===
A projective variety <math>X</math> embedded in <math>\mathbb{P}^r</math> has a natural 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 ==
* [[Brill–Noether theory]]
*[[Lefschetz pencil]]
*[[bundle of principal parts]]▼
==References==
{{reflist}}
{{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}}.▼
▲* Lazarsfeld, R., ''Positivity in Algebraic Geometry I'', Springer-Verlag, 2004. {{isbn|3-540-22533-1}}.
{{refend}}
▲== See also ==
▲*[[bundle of principal parts]]
[[Category:Geometry of divisors]]
| |||