Hyperelliptic curve: Difference between revisions
imported>David Lehavi (added pictures of plane model, and degenration of bitangents) |
imported>David Lehavi (added compactifications of the moduli) |
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In [[algebraic geometry]] a hyperelliptic curve is an algebraic curve <math>C</math> | In [[algebraic geometry]] a hyperelliptic curve is an [[algebraic curve]] <math>C</math> of [[genus]] greater 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 | Given a set <math>B</math> of <math>2g+2</math> distinct points on <math>\mathbb{P}^1</math>, there is a unique 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 === | === The plane model === | ||
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If <math>S</math> is a set of at most <math>g+1</math> Weierstrass points of <math>C</math> such that <math>\# S-(g+1)</math> is even, then <math>D_S:=[S]+\frac{\#S-(g+1)}{2}H_C</math> is a [[theta characteritic]] of <math>C</math>; i.e. <math>2D_S\sim K_C</math> in the Picard group of <math>C</math>. Moreover, it can be shown that <math>h^0(D_S)=1+\frac{\#S-(g+1)}{2}</math>, and if there are two such sets <math>S\neq S'</math>, then either <math>D_S\neq D_S'</math> or <math>S'\cup S</math> is the set of all Weirstrass points on <math>C</math>. | If <math>S</math> is a set of at most <math>g+1</math> Weierstrass points of <math>C</math> such that <math>\# S-(g+1)</math> is even, then <math>D_S:=[S]+\frac{\#S-(g+1)}{2}H_C</math> is a [[theta characteritic]] of <math>C</math>; i.e. <math>2D_S\sim K_C</math> in the Picard group of <math>C</math>. Moreover, it can be shown that <math>h^0(D_S)=1+\frac{\#S-(g+1)}{2}</math>, and if there are two such sets <math>S\neq S'</math>, then either <math>D_S\neq D_S'</math> or <math>S'\cup S</math> is the set of all Weirstrass points on <math>C</math>. | ||
If we count each set <math>S</math> 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 | If we count each set <math>S</math> 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 characteristic. Hence we constructed <math>\frac{1}{2}\sum_{4|2g+2-2k}binom(2g+2,k)</math>. This combinatorial sum is <math>2^{2g}</math> which is the number of | ||
theta | theta characteristic on a curve of genus <math>g</math>. Hence our description exhausts all the theta characteristics. | ||
[[Image:family_bitangents.png|200px|thumb|a family of bitangents to | [[Image:family_bitangents.png|200px|thumb|a family of bitangents to canonical curves of genus 3 degenerate to a line connecting images of two Weierstrass points of a canonical image of a hyperelliptic curve of genus 3]] | ||
* In genus 2, there are 6 | * In genus 2, there are 6 Weierstrass points. Each of them is an odd theta characteristics. 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 characteristic, 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 | * in genus 3, there are 8 Weierstrass points. The even theta characteristics are given by the empty set, and by partitions of these 8 points to two distinct sets - hence we get <math>1+binom(8,4)/2=36</math> even theta characteristics. the odd theta characteristics are given by the <math>binom(8,2)=28</math> pairs of Weierstrass points. The canonical map in this case maps the curve <math>C</math> to a double cover of plane conic. Consider a family of canonically embedded genus 3 curves with parameter <math>t</math> given by <math>Nulls(Q^2+tF)</math>, where we identify the double conic <math>Q^2</math> with the canonical image of <math>C</math>, where the Weierstrass 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>. | ||
== 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 | 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 [[irreducible]] [[affine scheme]] of dimension <math>2g-1</math>. | ||
=== | === Compactifications of the moduli === | ||
= | There are two standard "natural" compactifications of the moduli of <math>n</math> distinct points on <math>\mathbb{P}^1</math>. One is the [[binary forms]] compactification <math>\overline{\mathcal{B}}_n</math>: the [[GIT quotient]] of homogenous polynomials of degree <math>n</math> in two variables by the group <math>PGL_2</math> acting on the linear span of the variables. The second compactification is the [[Deligne-Mumford compactification]] of the [[moduli of pointed curves]] <math>\overline{\mathcal{M}}_{0,n}</math>. For the case <math>n=2g+2</math> Avritzer and Lange constructed a surjective morphism <math>\overline{\mathcal{M}}_{0,2g+2}\to\overline{\mathcal{B}}_{2g+2}</math>, thus relating these compactifications of the space of hyperelliptic case. Finally, the closure of the locus of hyperelliptic curves in the moduli [[stack]] <math>\overline{\mathcal{M}}_g</math> of curves of genus <math>g</math>, is a degree <math>1/2</math> cover of <math>\overline{\mathcal{M}}_{0,2g+2}</math>. | ||
[[Category:Mathematics Workgroup]] | [[Category:Mathematics Workgroup]] | ||
[[Category:CZ Live]] | [[Category:CZ Live]] |
Revision as of 21:39, 26 February 2007
In algebraic geometry a hyperelliptic curve is an algebraic curve of genus greater then 1, 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.
Given a set of distinct points on , there is a unique double cover of whose branch divisor is the set . From an algebro-geometric point of view this on can construct the curve by taking the of the sheaf whose sections over an open subset satisfy .
The plane model
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 system
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 .
Level 2 structure
If is a set of at most Weierstrass points of such that is even, then is a theta characteritic of ; i.e. in the Picard group of . Moreover, it can be shown that , and if there are two such sets , then either or is the set of all Weirstrass points on .
If we count each set 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 characteristic. Hence we constructed . This combinatorial sum is which is the number of theta characteristic on a curve of genus . Hence our description exhausts all the theta characteristics.
- In genus 2, there are 6 Weierstrass points. Each of them is an odd theta characteristics. There are three-tuples of distinct Weierstrass points, and hence there are odd theta characteristic, as expected. It is interesting to understand the geometrical meaning of pairs of Weierstrass points: if are two distinct Weierstrass points on then is a 2-torsion point on the Jacobian of , in this way we can express all the non trivial 2-torsion points on this Jacobian. Moreover, if is another (and different) two torsion point, then the between the two 2-torsion points is given by .
- in genus 3, there are 8 Weierstrass points. The even theta characteristics are given by the empty set, and by partitions of these 8 points to two distinct sets - hence we get even theta characteristics. the odd theta characteristics are given by the pairs of Weierstrass points. The canonical map in this case maps the curve to a double cover of plane conic. Consider a family of canonically embedded genus 3 curves with parameter given by , where we identify the double conic with the canonical image of , where the Weierstrass points the intersection points of . 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 .
Moduli of hyperelliptic curves
Since for any set of on there is a unique double cover branch divisor , the course moduli space of hyperelliptic curves of genus is isomorphic to the moduli of points on , up to projective transformations. However, as there are more then three points in , there is a finite non-empty subset of that send three of the points in to . Thus, the moduli of distinct points on up to projective transformations is a finite quotient of the space of distincit on . Specifically this space is an irreducible affine scheme of dimension .
Compactifications of the moduli
There are two standard "natural" compactifications of the moduli of distinct points on . One is the binary forms compactification : the GIT quotient of homogenous polynomials of degree in two variables by the group acting on the linear span of the variables. The second compactification is the Deligne-Mumford compactification of the moduli of pointed curves . For the case Avritzer and Lange constructed a surjective morphism , thus relating these compactifications of the space of hyperelliptic case. Finally, the closure of the locus of hyperelliptic curves in the moduli stack of curves of genus , is a degree cover of .