Hyperelliptic curve: Difference between revisions
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If <math>p</math> is a rational point on a hyperelliptic curve, then for all <math>k</math> we have <math>h^0((2(k+1))p)\geq h^0(2kp)+1</math>. Hence we must have <math>h^0((2g-2)p)\geq g</math>. However, by Riemann-Roch this implies that the divisor <math>(2g-2)p</math> is [[rationally equivalent]] to the canonical class <math>K_C</math>. Hence the canonical class of <math>C</math> is <math>g-1</math> times the hyperelliptic class of <math>C</math>, and the canonical image of <math>C</math> is a rational curve of degree <math>g-1</math>. | If <math>p</math> is a rational point on a hyperelliptic curve, then for all <math>k</math> we have <math>h^0((2(k+1))p)\geq h^0(2kp)+1</math>. Hence we must have <math>h^0((2g-2)p)\geq g</math>. However, by Riemann-Roch this implies that the divisor <math>(2g-2)p</math> is [[rationally equivalent]] to the canonical class <math>K_C</math>. Hence the canonical class of <math>C</math> is <math>g-1</math> times the hyperelliptic class of <math>C</math>, and the canonical image of <math>C</math> is a rational curve of degree <math>g-1</math>. | ||
== | == Moduli of hyperelliptic curves == | ||
=== | === Binary forms === | ||
=== | === Stable hyperelliptic curves === | ||
[[Category:Mathematics Workgroup]] | [[Category:Mathematics Workgroup]] | ||
[[Category:CZ Live]] | [[Category:CZ Live]] |
Revision as of 03:44, 22 February 2007
In algebraic geometry a hyperelliptic curve is an algebraic curve which admits a double cover . If such a double cover exists it is unique, and it is called the "hyperelliptic double cover". The involution induced on the curve by the interchanging between the two "sheets" of the double cover is called the "hyperelliptic involution". The divisor class of a fiber of the hyperelliptic double cover is called the "hyperelliptic class".
Weierstrass points
By the Riemann-Hurwitz formula the hyperelliptic double cover has exactly branch points. For each branch point we have . Hence these points are all Weierstrass points. Moreover, we see that for each of these points , and thus the Weierstrass weight of each of these points is at least . However, by the second part of the Weierstrass gap theorem, the total weight of Weierstrass points is , and thus the Weierstrass points of are exactly the branch points of the hyperelliptic double cover.
Curves of genus 2
If the genus of is 2, then the degree of the canonical class is 2, and . Hence the canonical map is a double cover.
The canonical embedding
If is a rational point on a hyperelliptic curve, then for all we have . Hence we must have . However, by Riemann-Roch this implies that the divisor is rationally equivalent to the canonical class . Hence the canonical class of is times the hyperelliptic class of , and the canonical image of is a rational curve of degree .