Hyperelliptic curve: Difference between revisions
imported>William Hart |
imported>David Lehavi (started discussion on moduli) |
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In [[algebraic geometry]] a hyperelliptic curve is an algebraic curve <math>C</math> which admits a double cover <math>f:C\to\mathbb{P}^1</math>. If such a double cover exists it is unique, and it is called the "hyperelliptic double cover". The involution induced on the curve <math>C</math> 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". | In [[algebraic geometry]] a hyperelliptic curve is an algebraic curve <math>C</math> fo genus geate then 1, which admits a double cover <math>f:C\to\mathbb{P}^1</math>. If such a double cover exists it is unique, and it is called the "hyperelliptic double cover". The involution induced on the curve <math>C</math> 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 === | === Weierstrass points === | ||
By the [[Riemann-Hurwitz formula]] the hyperelliptic double cover has exactly <math>2g+2</math> branch points. For each branch point <math>p</math> we have <math>h^0(2p)= 2</math>. Hence these points are all Weierstrass points. Moreover, we see that for each of these points <math>h^0((2(k+1))p)\geq h^0(2kp)+1</math>, and thus the [[Weierstrass weight]] of each of these points is at least <math>\sum_{k=1}^g (2k-k)=g(g-1)/2</math>. However, by the second part of the [[Weierstrass gap theorem]], the total weight of Weierstrass points is <math>g(g^2-1)</math>, and thus the Weierstrass points of <math>C</math> are exactly the branch points of the hyperelliptic double cover. | By the [[Riemann-Hurwitz formula]] the hyperelliptic double cover has exactly <math>2g+2</math> branch points. For each branch point <math>p</math> we have <math>h^0(2p)= 2</math>. Hence these points are all Weierstrass points. Moreover, we see that for each of these points <math>h^0((2(k+1))p)\geq h^0(2kp)+1</math>, and thus the [[Weierstrass weight]] of each of these points is at least <math>\sum_{k=1}^g (2k-k)=g(g-1)/2</math>. However, by the second part of the [[Weierstrass gap theorem]], the total weight of Weierstrass points is <math>g(g^2-1)</math>, and thus the Weierstrass points of <math>C</math> are exactly the branch points of the hyperelliptic double cover. | ||
Given a set <math>B</math> of <math>2g+2</math> distinct points on <math>\mathbb{P}^1</math>, there is a uniqe double cover of <math>C\to\mathbb{P}^1</math> whose [[branch divisor]] is the set <math>B</math>. From an algebro-geometric point of view this on can construct the curve <math>C</math> by taking the <math>Proj</math> of the sheaf whose sections <math>g</math> over an open subset <math>U\subset\mathbb{P}^1</math> satisfy <math>g^2\in O_U(B)</math>. | |||
=== The plane model === | |||
=== Curves of genus 2 === | === Curves of genus 2 === | ||
If the genus of <math>C</math> is 2, then the degree of the [[canonical class]] <math>K_C</math> is 2, and <math>h^0(K_C)=2</math>. Hence the [[canonical map]] is a double cover. | If the genus of <math>C</math> is 2, then the degree of the [[canonical class]] <math>K_C</math> is 2, and <math>h^0(K_C)=2</math>. Hence the [[canonical map]] is a double cover. | ||
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== Moduli of hyperelliptic curves == | == Moduli of hyperelliptic curves == | ||
Since for any set <math>B</math> of <math>2g+2</math> on <math>\mathbb{P}^1</math> there is a unique double cover <math>C\to\mathbb{P}^1</math> branch divisor <math>B</math>, the [[course moduli space]] of hyperelliptic curves of genus <math>g</math> is isomorphic to the moduli of <math>2g+2</math> points on <math>\mathbb{P}^1</math>, up to projective transformations. However, as there are more then three points in <math>B</math>, there is a finite non-empty subset of <math>Aut(\mathbb{P}^1)=PGL_2</math> that send three of the points in <math>B</math> to <math>0,1,\infty</math>. Thus, the moduli of <math>2g+2</math> distinct points on <math>\mathbb{P}^1</math> up to projective transformations is a finite quotient of the space of distincit <math>2g-1</math> on <math>\mathbb{P}^1\setminus\{0,1,\infty\}</math>. Specifically this space is an [[affine space]] of dimension <math>2g-1</math>. | |||
=== Binary forms === | === Binary forms === | ||
Revision as of 11:18, 24 February 2007
In algebraic geometry a hyperelliptic curve is an algebraic curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} fo genus geate then 1, which admits a double cover Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:C\to\mathbb{P}^1} . If such a double cover exists it is unique, and it is called the "hyperelliptic double cover". The involution induced on the curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2g+2} branch points. For each branch point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(2p)= 2} . Hence these points are all Weierstrass points. Moreover, we see that for each of these points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0((2(k+1))p)\geq h^0(2kp)+1} , and thus the Weierstrass weight of each of these points is at least Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=1}^g (2k-k)=g(g-1)/2} . However, by the second part of the Weierstrass gap theorem, the total weight of Weierstrass points is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(g^2-1)} , and thus the Weierstrass points of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} are exactly the branch points of the hyperelliptic double cover.
Given a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2g+2} distinct points on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{P}^1} , there is a uniqe double cover of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\to\mathbb{P}^1} whose branch divisor is the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} . From an algebro-geometric point of view this on can construct the curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} by taking the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Proj} of the sheaf whose sections Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} over an open subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U\subset\mathbb{P}^1} satisfy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g^2\in O_U(B)} .
The plane model
Curves of genus 2
If the genus of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is 2, then the degree of the canonical class Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_C} is 2, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(K_C)=2} . Hence the canonical map is a double cover.
The canonical embedding
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is a rational point on a hyperelliptic curve, then for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0((2(k+1))p)\geq h^0(2kp)+1} . Hence we must have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0((2g-2)p)\geq g} . However, by Riemann-Roch this implies that the divisor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2g-2)p} is rationally equivalent to the canonical class Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_C} . Hence the canonical class of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g-1} times the hyperelliptic class of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} , and the canonical image of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is a rational curve of degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g-1} .
Moduli of hyperelliptic curves
Since for any set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2g+2} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{P}^1} there is a unique double cover Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\to\mathbb{P}^1} branch divisor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} , the course moduli space of hyperelliptic curves of genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is isomorphic to the moduli of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2g+2} points on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{P}^1} , up to projective transformations. However, as there are more then three points in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} , there is a finite non-empty subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Aut(\mathbb{P}^1)=PGL_2} that send three of the points in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0,1,\infty} . Thus, the moduli of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2g+2} distinct points on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{P}^1} up to projective transformations is a finite quotient of the space of distincit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2g-1} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{P}^1\setminus\{0,1,\infty\}} . Specifically this space is an affine space of dimension .
