Editing Hyperelliptic curve (section)

==Occurrence and applications==

All curves of genus 2 are hyperelliptic, but for genus ≥ 3 the generic curve is not hyperelliptic. This is seen heuristically by a [[moduli space]] dimension check. Counting constants, with ''n'' = 2''g'' + 2, the collection of ''n'' points subject to the action of the automorphisms of the projective line has (2''g'' + 2) &minus; 3 degrees of freedom, which is less than 3''g'' &minus; 3, the number of moduli of a curve of genus ''g'', unless ''g'' is 2. Much more is known about the ''hyperelliptic locus'' in the moduli space of curves or [[abelian varieties]],{{clarify|What does the reference to abelian varieties mean?|date=December 2012}}  though it is harder to exhibit ''general'' non-hyperelliptic curves with simple models.<ref>{{cite journal
 | last = Poor | first = Cris
 | doi = 10.1090/S0002-9939-96-03312-6
 | issue = 7
 | journal = Proceedings of the American Mathematical Society
 | mr = 1327038
 | pages = 1987–1991
 | title = Schottky's form and the hyperelliptic locus
 | volume = 124
 | year = 1996| doi-access = free
 }}</ref> One geometric characterization of hyperelliptic curves is via [[Weierstrass point]]s. More detailed geometry of non-hyperelliptic curves is read from the theory of [[canonical curve]]s, the [[canonical bundle#Canonical maps|canonical mapping]] being 2-to-1 on hyperelliptic curves but 1-to-1 otherwise for ''g'' > 2. [[Trigonal curve]]s are those that correspond to taking a cube root, rather than a square root, of a polynomial.

The definition by quadratic extensions of the rational function field works for fields in general except in characteristic 2; in all cases the geometric definition as a ramified double cover of the projective line is available, if the extension is assumed to be separable.

Hyperelliptic curves can be used in [[hyperelliptic curve cryptography]] for [[cryptosystem]]s based on the [[discrete logarithm problem]].

Hyperelliptic curves also appear composing entire connected components of certain strata of the moduli space of Abelian differentials.<ref>{{cite journal |arxiv=math.GT/0201292 | doi=10.1007/s00222-003-0303-x | volume=153 | title=Connected components of the moduli spaces of Abelian differentials with prescribed singularities | year=2003 | journal=Inventiones Mathematicae | pages=631–678 | last1 = Kontsevich | first1 = Maxim | last2 = Zorich | first2 = Anton| issue=3 | bibcode=2003InMat.153..631K | s2cid=14716447 }}</ref>

Hyperellipticity of genus-2 curves was used to prove [[Mikhail Leonidovich Gromov|Gromov]]'s [[filling area conjecture]] in the case of fillings of genus =1.

===Classification===

Hyperelliptic curves of given genus ''g'' have a moduli space, closely related to the ring of [[invariants of a binary form]] of degree 2''g''+2.{{specify|date=August 2019}}