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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 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".
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=== Weierstrass points ===
In [[algebraic geometry]] and in the theory of [[Riemann surfaces]] a '''hyperelliptic curve''' is an [[algebraic curve]] <math>C</math> of [[genus]] greater than 1, which admits a double cover <math>\scriptstyle f:C\to\mathbb{P}^1</math>. These curves are among the simplest algebraic curves: they are all [[birationally equivalent]] to curves of the form <math>y^2-f(x)</math> in the affine plane, where <math>f(x)</math> is a polynomial in <math>x</math>, and the degree of <math>f(x)</math> is either twice the genus of the curve plus 2, or twice the genus of the curve plus one.
 
If a double cover <math>f:C\to\mathbb{P}^1</math> exists, then it is the unique double cover <math>C\to\mathbb{P}^1</math>, and it is called the "hyperelliptic double cover". The involution induced on the curve <math>C</math> by 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 <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 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>.
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 one 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 ==
[[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.]]
[[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.]]
If the <math>2g+2</math> Bracnh points of a hyperelliptic double cover <math>C\to\mathbb{P}^1</math> are <math>w_1,\ldots w_{2g+2}</math> then C is [[birational]] to the plane curve <math>Nulls(y^2=(x-w_1)(x-w_2)\cdots(x-w_{2g+1})(x-w_{2g+2}))\subset\mathbb{A}^2</math>, where if one of the branch points is infinity, we omit the corresponding term in the product. The closure of this curve in the projective plane has a singularity on the line <math>\mathbb{P}^2\setminus\mathbb{A}^2</math>.
If the <math>2g+2</math> branch points of a hyperelliptic double cover <math>C\to\mathbb{P}^1</math> are <math>w_1,\ldots w_{2g+2}</math> then C is [[birational]] to the plane curve <math>\mbox{Nulls}(y^2=(x-w_1)(x-w_2)\cdots(x-w_{2g+1})(x-w_{2g+2}))\subset\mathbb{A}^2</math>, where if one of the branch points is infinity, we omit the corresponding term in the product. The closure of this curve in the projective plane has a singularity on the line <math>\mathbb{P}^2\setminus\mathbb{A}^2</math>.


=== 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. The [[Jacobian variety]] of such a curve is an [[Abelian surface]].
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. The [[Jacobian variety]] of such a curve is an [[Abelian surface]].


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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>.
=== Level 2 structure ===
=== Level 2 structure ===
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 characteristic]] of <math>C</math>; i.e. <math>2D_S\sim K_C</math> is 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 Weierstrass 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 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
Every theta characteristics may be obtained in this way: Indeed 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 partition of the set of Weierstrass points into two sets such that the difference between the cardinalities is divisible by 4, induces a theta characteristic. In this way we count a total of <math>\frac{1}{2}\sum_{4|2g+2-2k}\mbox{binom}(2g+2,k)</math> distinct theta characteristics . This combinatorial sum is <math>2^{2g}</math> which is the number of theta characteristics on a curve of genus <math>g</math>. Hence our description exhausts all the theta characteristics.
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 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]]
[[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 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 2, there are 6 Weierstrass points. Each of them is an odd theta characteristic. There are <math>\mbox{binom}(6,3)=20</math> three-tuples of distinct Weierstrass points, and hence there are <math>20/2=10</math> odd theta characteristics, 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=\mbox{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)  2-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 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>.
* 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 into two distinct sets - hence we get <math>1+\mbox{binom}(8,4)/2=36</math> even theta characteristics. The odd theta characteristics are given by the <math>\mbox{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 a plane conic. Consider a family of canonically embedded genus 3 curves with parameter <math>t</math> given by <math>\mbox{Nulls}(Q^2+tF)</math>, where we identify the double conic <math>Q^2</math> with the canonical image of <math>C</math> and where the Weierstrass points are the intersection points of <math>\mbox{Nulls}(Q)\cap \mbox{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 characteristics) 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 [[irreducible]] [[affine scheme]] of dimension <math>2g-1</math>.
Since for any set <math>B</math> of <math>2g+2</math> points on <math>\mathbb{P}^1</math> there is a unique double cover <math>C\to\mathbb{P}^1</math> with branch divisor <math>B</math>, the [[coarse 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 than three points in <math>B</math>, there is a finite non-empty subset of <math>\mbox{Aut}(\mathbb{P}^1)=\mbox{PGL}_2</math> that sends three of the points in <math>B</math> to <math>0,1,\infty</math>. Thus, the moduli of space which parametrizes <math>2g+2</math> distinct points on <math>\mathbb{P}^1</math> up to projective transformations, is a finite quotient of the space of distinct <math>2g-1</math> points 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 ===
=== 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>.
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 homogeneous 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 in the 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>.
 
=== References ===
* ACGH
* Avritzer and Lange
* Dolgachev "Invariant theory"
* Harris and Morison "Moduli of curves"


[[Category:Mathematics Workgroup]]
== References ==
[[Category:CZ Live]]
*  E. Arbarello, M. Cornalba, P. Griffiths, J. Harris: ''Geometry of Algebraic Curves, I''. Grund. series, vol. 267, Springer.  ISBN 0387909974 (theta characteristics on hyperelliptic curves)
* D. Avritzer, H. Lange ''The moduli spaces of hyperelliptic curves and binary forms'' Math Z. Volume 242, Number 4, 2002
* I. Dolgachev ''Invariant theory'' ISBN 0521525489 (binary forms compactification)
* J. Harris and D. Morison ''Moduli of curves'' ISBN 0387984291 (degeneration of bitangents)[[Category:Suggestion Bot Tag]]

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In algebraic geometry and in the theory of Riemann surfaces a hyperelliptic curve is an algebraic curve of genus greater than 1, which admits a double cover . These curves are among the simplest algebraic curves: they are all birationally equivalent to curves of the form in the affine plane, where is a polynomial in , and the degree of is either twice the genus of the curve plus 2, or twice the genus of the curve plus one.

If a double cover exists, then it is the unique double cover , and it is called the "hyperelliptic double cover". The involution induced on the curve by 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 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 unique double cover of 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 one 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

A plane model for a hyperelliptic curve of genus 3. Two Weierstrass points have non-real coordinates, and one is at infinity.

If the branch points of a hyperelliptic 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} 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 w_1,\ldots w_{2g+2}} then C is birational to the plane 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 \mbox{Nulls}(y^2=(x-w_1)(x-w_2)\cdots(x-w_{2g+1})(x-w_{2g+2}))\subset\mathbb{A}^2} , where if one of the branch points is infinity, we omit the corresponding term in the product. The closure of this curve in the projective plane has a singularity on the line 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}^2\setminus\mathbb{A}^2} .

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 Jacobian variety of such a curve is an Abelian surface.

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 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 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 , 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 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} is a set of at most 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 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:=[S]+\frac{\#S-(g+1)}{2}H_C} is a theta characteristic 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} is 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 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(D_S)=1+\frac{\#S-(g+1)}{2}} , 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 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'\cup S} is the set of all Weierstrass points on .

Every theta characteristics may be obtained in this way: Indeed 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 partition of the set of Weierstrass points into two sets such that the difference between the cardinalities is divisible by 4, induces a theta characteristic. In this way we count a total 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 \frac{1}{2}\sum_{4|2g+2-2k}\mbox{binom}(2g+2,k)} distinct theta characteristics . This combinatorial sum is which is the number of theta characteristics on a curve 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} . Hence our description exhausts all the theta characteristics.

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 Weierstrass points. Each of them is an odd theta characteristic. 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 \mbox{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 characteristics, 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 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 2^4-1=15=\mbox{binom}(6,2)} 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) 2-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 characteristics are given by the empty set and by partitions of these 8 points into 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+\mbox{binom}(8,4)/2=36} even theta characteristics. The odd theta characteristics 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 \mbox{binom}(8,2)=28} pairs of Weierstrass points. The canonical map in this 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 a plane conic. Consider a family of canonically embedded genus 3 curves with parameter 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 \mbox{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 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} and where the Weierstrass points are 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 \mbox{Nulls}(Q)\cap \mbox{Nulls}(F)} . Then it can be shown that the "limit" of the bitangents of the curves in the family (which are the odd theta characteristics) 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} 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 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} with 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 coarse 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 than 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 \mbox{Aut}(\mathbb{P}^1)=\mbox{PGL}_2} that sends 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 space which parametrizes 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 distinct 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} 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\setminus\{0,1,\infty\}} . Specifically this space is an irreducible affine scheme 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} .

Compactifications of the moduli

There are two standard "natural" compactifications of 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 n} 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} . One is the binary forms compactification 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 \overline{\mathcal{B}}_n} : the GIT quotient of homogeneous polynomials 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 n} in two variables by the group 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 PGL_2} acting on the linear span of the variables. The second compactification is the Deligne-Mumford compactification of the moduli of pointed curves 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 \overline{\mathcal{M}}_{0,n}} . For the case 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 n=2g+2} Avritzer and Lange constructed a surjective morphism 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 \overline{\mathcal{M}}_{0,2g+2}\to\overline{\mathcal{B}}_{2g+2}} , thus relating these compactifications in the hyperelliptic case. Finally, the closure of the locus of hyperelliptic curves in the moduli stack 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 \overline{\mathcal{M}}_g} of 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 a 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 1/2} 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 \overline{\mathcal{M}}_{0,2g+2}} .

References

  • E. Arbarello, M. Cornalba, P. Griffiths, J. Harris: Geometry of Algebraic Curves, I. Grund. series, vol. 267, Springer. ISBN 0387909974 (theta characteristics on hyperelliptic curves)
  • D. Avritzer, H. Lange The moduli spaces of hyperelliptic curves and binary forms Math Z. Volume 242, Number 4, 2002
  • I. Dolgachev Invariant theory ISBN 0521525489 (binary forms compactification)
  • J. Harris and D. Morison Moduli of curves ISBN 0387984291 (degeneration of bitangents)
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