Linear system of divisors: Difference between revisions

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==== 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|doi-access=free}}</ref>.

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

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{{refend}}

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[[Category:Geometry of divisors]]