Hyperelliptic curve: Difference between revisions
imported>David Lehavi (added discussion on level 2 structure) |
imported>David Lehavi (added pictures of plane model, and degenration of bitangents) |
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=== The plane model === | === The plane model === | ||
[[Image:hyperelliptic_plane.png|200px|thumb|A plane model for a hyperelliptic curve of genus 3. Two Weierstrass points have non-real coordinates, and one is at infinity]] | |||
=== 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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theta characteritic on a curve of genus <math>g</math>. Hence our description exhausts all the theta characteritics. | theta characteritic on a curve of genus <math>g</math>. Hence our description exhausts all the theta characteritics. | ||
[[Image:family_bitangents.png|200px|thumb|a family of bitangents to cannonical curves of genus 3 degenerate to a line connecting images of two Weierstrass points of a cannonical image of a hyperelliptic curve of genus 3]] | |||
* In genus 2, there are 6 Weiertrass points. Each of them is an odd theta characteritics. There are <math>binom(6,3)=20</math> three-tuples of distinct Weierstrass points, and hence there are <math>20/2=10</math> odd theta characteritic, as expected. It is interesting to understand the geometrical meaning of pairs of Weierstrass points: if <math>p_1,p_2</math> are two distinct Weierstrass points on <math>C</math> then <math>p_1+p_2-H_C</math> is a 2-torsion point on the [[Jacobian]] of <math>C</math>, in this way we can express all the <math>2^4-1=15=binom(6,2)</math> non trivial 2-torsion points on this Jacobian. Moreover, if <math>p_3+p_4-H_C</math> is another (and different) two torsion point, then the <math>Weil pairing</math> between the two 2-torsion points is given by <math>\{p_1,p_2\}\cap\{p_3,p_4\}</math>. | * In genus 2, there are 6 Weiertrass points. Each of them is an odd theta characteritics. There are <math>binom(6,3)=20</math> three-tuples of distinct Weierstrass points, and hence there are <math>20/2=10</math> odd theta characteritic, as expected. It is interesting to understand the geometrical meaning of pairs of Weierstrass points: if <math>p_1,p_2</math> are two distinct Weierstrass points on <math>C</math> then <math>p_1+p_2-H_C</math> is a 2-torsion point on the [[Jacobian]] of <math>C</math>, in this way we can express all the <math>2^4-1=15=binom(6,2)</math> non trivial 2-torsion points on this Jacobian. Moreover, if <math>p_3+p_4-H_C</math> is another (and different) two torsion point, then the <math>Weil pairing</math> between the two 2-torsion points is given by <math>\{p_1,p_2\}\cap\{p_3,p_4\}</math>. | ||
* in genus 3, there are 8 Weierstrass points. The even theta characteritics are given by the empty set, and by paritions of these 8 points to two distinct sets - hence we get <math>1+binom(8,4)/2=36</math> even theta characteritics. the odd theta characteritics are given by the <math>binom(8,2)=28</math> pairs of Weierstrass points. The cannonical map in thithis case maps the curve <math>C</math> to a double cover of plane conic. Consider a family of cannonicaly embedded genus 3 curves with parmeter <math>t</math> given by <math>Nulls(Q^2+tF)</math>, where we identify the double conic <math>Q^2</math> with the cannonical image of <math>C</math>, where the Weierstarss points the intersection points of <math>Nulls(Q)\cap Nulls(F)</math>. Then it can be shown that the "limit" of the bitangents of the curves in the family (which are the odd theta characteritics) are the lines connecting pairs of images of Weierstrass points on <math>C</math>. | * in genus 3, there are 8 Weierstrass points. The even theta characteritics are given by the empty set, and by paritions of these 8 points to two distinct sets - hence we get <math>1+binom(8,4)/2=36</math> even theta characteritics. the odd theta characteritics are given by the <math>binom(8,2)=28</math> pairs of Weierstrass points. The cannonical map in thithis case maps the curve <math>C</math> to a double cover of plane conic. Consider a family of cannonicaly embedded genus 3 curves with parmeter <math>t</math> given by <math>Nulls(Q^2+tF)</math>, where we identify the double conic <math>Q^2</math> with the cannonical image of <math>C</math>, where the Weierstarss points the intersection points of <math>Nulls(Q)\cap Nulls(F)</math>. Then it can be shown that the "limit" of the bitangents of the curves in the family (which are the odd theta characteritics) are the lines connecting pairs of images of Weierstrass points on <math>C</math>. | ||
Revision as of 13:34, 26 February 2007
In algebraic geometry a hyperelliptic curve is an algebraic curve 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 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.
Given a set of distinct points on , there is a uniqe double cover of whose branch divisor is the set . 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 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 . Hence the canonical map is a double cover.
The canonical system
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 . 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 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} .
Level 2 structure
If is a set of at most 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} 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} such that 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 \# S-(g+1)} is even, then is a theta characteritic 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} ; i.e. 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 2D_S\sim K_C} in the Picard group 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} . Moreover, it can be shown that , and if there are two such sets 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 S\neq S'} , then either 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 D_S\neq D_S'} or is the set of all Weirstrass 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 C} .
![{\displaystyle D_{S}:=[S]+{\frac {\#S-(g+1)}{2}}H_{C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5dceb38f21bd3b14cb4af2876a6f218e45184da)
If we count each 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 S} together with its complementary set in the set of Weierstrass points (and then divide by 2) then the combinatorial description above tells us that any parition of the set of Weierstrass points to two sets such that the difference between the cardinalities is divisible by 4 induces a theta characteritic. Hence we constructed 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 \frac{1}{2}\sum_{4|2g+2-2k}binom(2g+2,k)} . This combinatorial sum 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 2^{2g}} which is the number of theta characteritic on a curve of genus . Hence our description exhausts all the theta characteritics.
- In genus 2, there are 6 Weiertrass points. Each of them is an odd theta characteritics. There are 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 binom(6,3)=20} three-tuples of distinct Weierstrass points, and hence there are 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 20/2=10} odd theta characteritic, as expected. It is interesting to understand the geometrical meaning of pairs of Weierstrass points: 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_1,p_2} are two distinct Weierstrass 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 C} then 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_1+p_2-H_C} is a 2-torsion point on the Jacobian 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} , in this way we can express all the non trivial 2-torsion points on this Jacobian. Moreover, 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_3+p_4-H_C} is another (and different) two torsion point, then 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 Weil pairing} between the two 2-torsion points is given by 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_1,p_2\}\cap\{p_3,p_4\}} .
- in genus 3, there are 8 Weierstrass points. The even theta characteritics are given by the empty set, and by paritions of these 8 points to two distinct sets - hence we get 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 1+binom(8,4)/2=36} even theta characteritics. the odd theta characteritics are given by 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 binom(8,2)=28} pairs of Weierstrass points. The cannonical map in thithis case maps 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} to a double cover of plane conic. Consider a family of cannonicaly embedded genus 3 curves with parmeter 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 t} given by 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 Nulls(Q^2+tF)} , where we identify the double conic 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 Q^2} with the cannonical 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} , where the Weierstarss points the intersection 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 Nulls(Q)\cap Nulls(F)} . Then it can be shown that the "limit" of the bitangents of the curves in the family (which are the odd theta characteritics) are the lines connecting pairs of images of Weierstrass 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 C} .
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 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} .