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]].]]

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=== 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_rg^dr_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>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>

===Linear systems of hypersurfaces in a projective space===

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:<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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{{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}}