Linear system of divisors: Difference between revisions

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==Definition ==

Given the fundamental idea of a [[rational function]] on a general variety ''V''<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>, divisors <math>D,E \in \text{Div}(X)</math> are ''V'linearly equivalent divisors','' if

divisors D and E are ''linearly equivalent'' if

:<math>D = E + (f)\ </math>

where <math>(''f'')</math> denotes the divisor of zeroes and poles of the function ''<math>f''</math>.

Note that if ''V''<math>X</math> has [[Mathematical singularity|singular points]], '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 ''V'' 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''}(''D'')</math> be the line bundle associated to ''<math>D''</math>. In the case that ''V''<math>X</math> is a nonsingular projective variety the set <math>|''D''|</math> is in natural bijection with <math> (\Gamma(V,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> and 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(V,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, linear 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 surfaces. 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 <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 finite 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_r^d ====

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> since <math>|K_C|</math> defines one. 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|last=Kleiman|first=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}}</ref>.

===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>.

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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.

In general, (under some assumptions<!-- needFor a refclosed -->), one can pullback a linear system as follows: letimmersion <math>f: XY \tohookrightarrow YX</math> be a morphism of algebraic varieties. Thenthere theis a pullback of a linear system <math>\mathfrak{d}</math> on ''<math>X</math> to <math>Y''</math>, isdefined 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 canonical 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]]

*[[bundle of principal parts]]

==References==

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{{refend}}

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== See also ==

*[[bundle of principal parts]]

[[Category:Geometry of divisors]]