Linear system of divisors: Difference between revisions

Revision as of 11:14, 9 October 2024

Revision as of 16:20, 14 November 2024

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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 [[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_d^r ====