Linear system of divisors: Difference between revisions

Revision as of 19:02, 26 July 2024

Latest revision as of 01:10, 24 January 2025

(5 intermediate revisions by 3 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]].]]

Line 33 ⟶ 34:

==== 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|degredegree <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===

Line 97 ⟶ 98:

:<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: