Weber's formula for the bitangents of a smooth plane quartic
aa r X i v : . [ m a t h . AG ] D ec Weber ’s formula for the bitangents of asmooth plane quartic
Alessio Fiorentino ∗ Abstract
In a section of his 1876 treatise
Theorie der Abelschen Functionen vom Geschlecht
Thetanullwerte ). Thepresent note is devoted to a modern presentation of Weber’s formula. Inthe end a connection with the universal bitangent matrix is also displayed.
1. Introduction
The problem of characterizing the complex principally polarized abelian vari-eties of dimension g which are Jacobian varieties of smooth projective curvesof genus g , is a long-standing research subject that dates back to Riemann andSchottky. In fact, the question can be simply answered whenever g ≤
3, as inthis case every indecomposable principally polarized abelian variety is knownto be the Jacobian variety of an irreducible smooth projective curve (uniquelydetermined up to isomorphisms). A naturally related question is how to ex-plicitly recover the curve from a given principally polarized abelian variety. Asolution to this problem for non-hyperelliptic curves of genus 3 has been foundin [Gu11], where an equation for the curve is recovered by resorting to the Rie-mann model of the curve associated with Steiner complexes of bitangents. Oneof the first mathematicians who succeeded in establishing a link between thegeometry of the curve and the algebraic structure defined by its period matrixwas Heinrich Martin Weber. In his work [We76] he actually provided both aformula to recover the bitangents of the the curve from its period matrix and areverse formula to recover the fourth powers of theta constants (
Thetanullwerte ),valued at the point corresponding to the period matrix of the curve, from itsbitangents. A detailed explanation of the latter formula, along with a modernproof of it, can be found in [NR15]; instead, this note aims to prove Weber’sformula for the bitangents of the curve from a modern point of view. ∗ This work was supported by the Centre Henri Lebesgue (Programme PIA- ANR-11-LABX-0020-01) . Acknowledgments The author wishes to thank Christophe Ritzenthaler for bringing Weber’s workto his attention as well as for all the enlightening discussions. He is also gratefulto Riccardo Salvati Manni for the several conversations on the subject.
3. Quadratic forms on symplectic vector spaces over F F and is motivated by theneed for a coordinate-free presentation of Weber’s formula; a more detailedexplanation of the subject can be found in Dolgachev’s book [Do12] and inGross and Harris’s paper [GH04].Let g ≥ V a vector space of dimension 2 g over F providedwith a symplectic form ω . A quadratic form q on the symplectic vector space( V , ω ) is a map q : V F such that: q ( λ v ) = λ q ( v ) ∀ λ ∈ F , ∀ v ∈ Vq ( v + w ) = q ( v ) + q ( w ) + ω ( v , w ) ∀ v , w ∈ V ;There are 2 g distinct quadratic forms on ( V , ω ). By definition q , q ‘ ∈ Q ( V ) if andonly if q ‘ − q = α where α is a linear form on V ; therefore, for any q , q ‘ ∈ Q ( V )there exists a unique v ∈ V such that: q ‘( w ) = q ( w ) + ω ( v , w ) ∀ w ∈ V ; (1)Thanks to (1), a free and transitive action of V on Q ( V ) is well defined by setting v + q : = q ′ , hence the set Q ( V ) is an a ffi ne space over V , which means it can beidentified with V whenever a quadratic form is fixed as origin. Furthermore,the disjoint union V ˙ ∪ Q ( V ) can be thought as a vector space of dimension 2 g + F . Once a symplectic basis e , . . . , e g , f , . . . , f g is chosen for V , a quadraticform q is naturally defined as origin for the a ffi ne space Q ( V ): q ( w ) : = λ · µ = λ µ + · · · λ g µ g ∀ w = ( λ, µ ) = g X i = λ i e i + g X i = µ i f i (2)Then, by (1), each q ∈ Q ( V ) can be identified with the unique column vector v = [ m ′ m ′′ ] such that: q ( w ) = λ · µ + λ · m ′ + m ′′ · µ ∀ w = ( λ, µ ) (3)Besides, since the subgroup of GL ( V ) that preserves the symplectic form ω isisomorphic to SP (2 g , F ), an action of SP (2 g , F ) on the a ffi ne space Q ( V ) is welldefined by setting γ · q ( v ) : = q ( γ − v ) for any v ∈ V . The orbits of Q ( V ) underthis action are described in terms of the Arf invariant of a quadratic form: a ( q ) : = g X i = q ( e i ) q ( f i ) ∀ q ∈ Q ( V ); Such a linear form is well defined, as any element of F has exactly one square root, whichactually coincides with the element itself. Q ( V )is seen to decompose into two orbits: the set Q ( V ) + of even quadratic forms,namely those whose Arf invariant is equal to 0 (the cardinality of this orbitis equal to 2 g − (2 g + Q ( V ) − of odd quadratic forms, namelythose whose Arf invariant is equal to 1 (the cardinality of this orbit is equal to2 g − (2 g − q defined in (2) is even and that a ( q ) = m ′ · m ′′ for any quadratic form q whosecoordinates with respect to q are [ m ′ m ′′ ], as in (3). Remarkable orbits of non-ordered collections of quadratic forms are also characterized in terms of the Arfinvariant; in particular, a triple q , q , q ∈ Q ( V ) is called syzygetic (resp. azygetic )if a ( q ) + a ( q ) + a ( q ) + a ( q + q + q ) = = q , . . . , q n ∈ Q ( V ) with n ≥ { q i , q j , q k } ⊂ { q , . . . , q n } is syzygetic (resp. azygetic). An Aronholdsystem is a collection of 2 g + q , . . . , q g + ∈ Q ( V ) which is abasis for the vector space V ˙ ∪ Q ( V ) and such that for any q = P g + i = λ i q i ∈ Q ( V )the following expression holds: a ( q ) = g + X i = λ i − + ( g ≡ , g ≡ , SP (2 g , F ) on them is transitive.
4. Theta characteristics and quadratic forms
This section is intended to recall the link between the above mentioned algebraicsettings and the geometry of the projective curves. Classical references for thissubject are [ACGH85] and [GH78]. We will also follow the exposition outlinedin [Gu02] and [NR15].Let C be a smooth complex non-hyperelliptic curve of genus g canonicallyembedded in P g − by means of a basis ω , . . . , ω g of the cohomology space H ( C , Ω ), and let S g denote the Siegel upper-half space of degree g , namelythe tube domain of complex symmetric g × g matrices with positive definiteimaginary part. Once a symplectic basis δ , . . . , δ g , δ ′ , . . . , δ ′ g of the homologyspace H ( C , Z ) is chosen, the g × g period matrix of the curve ( R δ j ω i , R δ ′ j ω i )defines a lattice in C g and consequently a complex torus whose isomorphismclass has a representative of the form J C : = C g / ( Z g + τ Z g ) with τ ∈ S g . Thiscomplex torus is known as the Jacobian variety of the curve C and is a principallypolarized abelian variety, whose set of 2-torsion points J C [2] can be clearlyidentified with the set of the representatives ( h + τ · k ) with h , k ∈ Z g . Hence, J C [2] admits a vector space structure over Z and is furthermore endowedwith a symplectic form given by the Weyl pairing. The a ffi ne space Q ( J C [2])of the quadratic forms on J C [2] can be therefore identified with Z g × Z g , oncea quadratic form is fixed as origin. This a ffi ne space is strictly related to thegeometry of the curve, since its points can be identified with the so-called thetacharacteristics. A theta characteristic on C is a divisor D such that 2 D ∼ K C where K C is the canonical divisor of the curve. Once a point P ∈ C is fixed, thewell known Abel-Jacobi map φ P : Div ( C ) → J C is defined on the group Div ( C )of the equivalence classes of divisors; since p = φ P ( D ′ − D ) ∈ J C [2] whenever3he divisors D and D ′ are linearly equivalent to distinct theta characteristics, afree transitive action of the vector space J C [2] is well defined on the set of thetacharacteristics as well, by setting D + p : = D ′ . Then, for any theta characteristic D a quadratic form in Q ( J C [2]) is uniquely defined by setting: q D ( p ) : = [dim L ( D + p ) + dim L ( D )] mod2where L ( D ) and L ( D + p ) are the Riemann-Roch spaces respectively associatedwith the divisors D and D + p . This gives a bijection between the set of thetacharacteristics and Q ( J C [2]). The theta characteristics can be therefore identifiedwith the vectors in Z g × Z g as long as a theta characteristic D is fixed; a canon-ical choice for such a D is suggested by Riemann’s theorem on the geometryof the theta divisor, as we will briefly recall.A Riemann theta function of level 2 with characteristic m = ( m ′ m ′′ ), where m ′ , m ′′ ∈ Z g is a holomorphic function θ m : S g × C g C defined by the series: θ m ( τ, z ) ≔ X n ∈ Z g e (cid:20) t (cid:18) n + m ′ (cid:19) · τ · (cid:18) n + m ′ (cid:19) + (cid:18) n + m ′ (cid:19) · (cid:18) z + m ′′ (cid:19)(cid:21) where e ( z ) : = exp( π i z ) and the symbol · stands for the usual inner product. Asa consequence of the reduction formula : θ m + n ( τ, z ) = ( − m ′ · n ′′ θ m ( τ, z ) ∀ m = ( m ′ m ′′ ) , ∀ n = ( n ′ n ′′ ) (4)these functions are uniquely determined up to a sign by the so-called reducedcharacteristics [ m ] : = [ m ′ m ′′ ] with m ′ , m ′′ ∈ Z g . The theta constant (Thetanullwert) with characteristic m is the function defined by setting θ m ( τ ) : = θ m ( τ, addition formula (cf. [Ig72] for a general formulation in terms of real characteristics): θ m ( τ, u + v ) θ m ( τ, u − v ) θ m ( τ ) θ m ( τ ) == / g X [ a ] ∈ Z g / Z g e ( m ′ · a ′′ ) θ n + a ( τ, u ) θ n + a ( τ, u ) θ n + a ( τ, v ) θ n + a ( τ, v ) (5)where the sum runs over a set of representatives for Z g / Z g and { n , n , n , n } and { m , m , m , m } are any two collections of four characteristics that satisfythe following identity:( n , n , n , n ) =
12 ( m , m , m , m ) · − − − − − − If τ ∈ S g identifies the Jacobian variety J C of the curve, the theta divisor Θ iswell defined on J C as the pull-back of the divisor { z ∈ C g | θ ( τ ) = } , and theChern class of the holomorphic line bundle associated with such a divisor givesa principal polarization on J C . The following classical theorem holds: Theorem 1 ( Riemann’s theorem).
There exists a theta characteristic D on the curveC such that: W g − = Θ + D where W g − : = { D ∈ Div ( C ) | deg ( D ) = g − , dim L ( D ) > } . Furthermore, dim L ( D ) is even, and mult p ( Θ ) = dim L ( D + p ) for any p ∈ J C [2]4f such a theta characteristic D is fixed, a quadratic form q in Q ( J C [2]) is fixed aswell; then, any theta characteristic on the curve is of the form D + v with v = [ m ′ m ′′ ]and m ′ , m ′′ ∈ Z g and the corresponding quadratic form is q + v . A Riemanntheta function θ [ m ] with reduced characteristic [ m ] = [ m ′ m ′′ ] can be thereforeregarded as a function θ [ q ] associated with the quadratic form q = q + [ m ′ m ′′ ].The function z → θ [ q ]( τ, z ) is even (resp. odd) whenever q is even (resp. odd),hence the theta constant θ [ q ] is non-trivial if and only if q is even; furthermore,for any q = q + [ m ′ m ′′ ] and for any ( k , h ) ∈ Z g × Z g the following transformationlaw holds (cf. [RF74]): θ [ q ] (cid:18) τ, z + h + τ · k (cid:19) = e (cid:18) − k · ( m ′′ + h ) − k · z − t k · τ · k (cid:19) θ [ q + [ kh ]]( τ, z )Thanks to this formula the pull-back of the zero locus of any Riemann thetafunction θ [ q ] also defines a divisor Θ [ q ] on J C and mult ( Θ [ q + v ]) = mult v ( Θ )for any v ∈ J C [2]. Riemann’s theorem thus implies that the e ff ective thetadivisors are those associated with odd quadratic forms. Therefore, for any oddquadratic form q ∈ Q ( J C [2]) − the associated theta characteristic is of the type D q = P + · · · + P g − and is actually the divisor that is cut on the canonical curveby a hyperplane tangent at the image points of P , . . . , P g − in P g − under thecanonical map; the direction of such a hyperplane in P g − is then given by thegradient of the corresponding Riemann theta function valued at z = z θ [ q ]( τ ) : = ∂θ [ q ] ∂ z ( τ, , . . . , ∂θ [ q ] ∂ z g ( τ, ! which is non-trivial if and only if q is odd. The Jacobian determinant of g Riemann theta functions valued at z = D [ q , . . . , q g ]( τ ) : = (grad z θ [ q ] ∧ · · · ∧ grad z θ [ q g ])( τ ) (6)The algebraic link between theta constants and Jacobian determinants is dis-played by Igusa’s conjectural formula (cf. [Ig83]), which has been proved upto the case g =
5. Bitangents of a plane quartic: Weber’s formula
For the rest of the paper we will be only concerned with the g = C of genus 3 is a smooth plane quartic,whose 28 bitangents are in bijection with the 28 odd quadratic forms on J C [2].By virtue of the geometrical link recalled in the previous section, there existhomogeneus coordinates ( Z : Z : Z ) in P such that the equations of the 28bitangents are: X i = ∂θ [ q ] ∂ Z i ( τ, Z i = , ∀ q ∈ Q ( J C [2]) − (7)In this case, an Aronhold system is a collection of seven odd quadratic forms q , . . . , q such that each sub-triple { q i , q j , q k } ⊂ { q , . . . , q } is azygetic, which5eans q i + q j + q k is even; there exist exactly 288 distinct Aronhold systemswhen g =
3. Once an Aronhold system is fixed, the remaining 21 odd quadraticforms can be simply described in terms of it as follows: q ij : = q S + q i + q j ∀ i , j (8)where q S : = P i = q i is an even quadratic form. The other 35 even quadraticforms di ff erent form q S are easily seen to be described in terms of the Aronholdsystem as follows: q ijk : = q i + q j + q k ∀ i , j , k distinct (9)An Aronhold system of bitangents for the plane quartic is then a collectionof seven bitangents associated with an Aronhold system of quadratic forms;this geometrically translates into the condition that for any collection of threebitangents out of the seven, the six corresponding points of tangency on thequartic do not lie in the same conic. The datum of an Aronhold system isenough to recover an equation for the plane quartic along with equations forthe remaining 21 bitangents; this is basically done by means of the Steiner com-plexes of bitangents determined by the sub-collections of six bitangents in theAronhold system. We will only recall here the main features of the method ofreconstruction with a particular focus on the Riemann model of the curve (cf.[We76] for details and [Do12] for a modern exposition of the subject).The following statement holds: Proposition 1.
Let q be a non-null quadratic form on J C [2] and let { q , q ′ } , { q , q ′ } and { q , q ′ } be three pairs of odd quadratic forms on J C [2] such that q i + q ′ i = q for anyi = , , . Then, for any two of these pairs there exists a conic that passes through theeight points of tangency; in particular, an equation for the quartic is given by: f ξ f ξ − ( f ξ + f ξ + f ξ ) = or, in Weber’s notation: p f ξ + p f ξ + p f ξ = where { f i , ξ i } is a suitable pair of linear forms associated with the bitangents corre-sponding to the pair { q i , q ′ i } . As any subtriple q i , q j , q k of an Arnohold system is an azygetic triple, it canbe completed to three pairs { q i , q ′ i } , { q j , q ′ j } and { q k , q ′ k } such as in the statementof Proposition 1. Thus, any three bitangents in an Aronhold system cannotintersect at a same point, because such a point would be a singular point of thecurve by (10), while the curve is smooth; this proves the following: Corollary 1.
Up to a projective transformation, an Aronhold system of bitangents forthe quartic is given by the following equations in P : β : X = β : a X + a X + a X = β : X = β : a X + a X + a X = β : X = β : a X + a X + a X = β : X + X + X = for suitable ( a i : a i : a i ) ∈ P . Proposition 2 ( Riemann’s model).
Let β , . . . , β an Aronhold system of bitangentsfor the curve as in (11) and q , . . . , q the corresponding quadratic forms. The threepairs { q , q } , { q , q } and { q , q } (cf. (8)) are such as in the statement of Proposition1, and an equation for the curve is given by: X ξ X ξ = ( X ξ + X ξ + X ξ ) where ξ ij are linear forms associated with the bitangents corresponding to q ij anddetermined by the linear system: ( ξ + ξ + ξ + X + X + X = ξ a i + ξ a i + ξ a i + k i ( a i X + a i X + a i X ) = i = , , with k , k , k ∈ C ∗ unique solution of the linear system: λ a λ a λ a λ a λ a λ a λ a λ a λ a k k k = − − − (13) where λ , λ , λ ∈ C ∗ are such that: a a a a a a a a a λ λ λ = − − − Note that the unicity of such a construction is a consequence of the resultsproved by Caporaso and Sernesi (cf. [CS03]) and by Lehavi (cf. [Le05]).As the curve C is fixed, for the sake of simplicity we shall omit the symbol of thevariable τ in the expressions of theta functions and theta constants throughoutthe rest of this section. Furthermore, by a slight abuse of notation we shalldenote by ( q ) = ( q ′ , q ′′ ) the non-reduced characteristic that corresponds to thecoordinates of the quadratic form q with respect to a fixed quadratic form q and by ( P i q i ) the non-reduced characteristic P i ( q i ). To prove Weber’s formulawe need the following Proposition first. Proposition 3.
Let { q , q , q , q } any azygetic -tuple of odd quadratic forms, and let { q , q , q } one of the two distinct triples which complete the -tuple to an Aronholdsystem { q , q , q , q , q , q , q } . Then:D [ q , q , q ] D [ q , q , q ] = − e (( q + q + q ) ′ · ( q + q ) ′′ ) θ ( q + q + q ) θ ( q + q + q ) θ ( q + q + q ) θ ( q + q + q ) θ ( q + q + q ) θ ( q + q + q ) where D [ q i , q j , q k ] are the Jacobian determinants of the corresponding Riemann thetafunctions with reduced characteristics valued at z = , as in (6).Proof. If we set u = n = ( q + q ), n = ( q + q ), n = ( q + q ) and n =
0, we get for any z ∈ C g :0 = X q ∈ Q ( J C [2]) χ ( q ) θ ( q + q + q ) θ ( q + q + q ) θ ( q + q + q )( z ) θ ( q )( z )7here χ ( q ) : = e (( q + q + q ) ′ · q ′′ ). The right side of the identity is the sumof two terms S − and S + , obtained by letting q run respectively over Q ( J C [2]) − and over Q ( J C [2]) + . Thanks to the labelling introduced in (8) for the elementsof Q ( J C [2]) − one easily derives S − = S (4) − + S (6) − , where: S (4) − = X i = χ ( q i ) θ ( q + q + q i ) θ ( q + q + q i ) θ ( q + q + q i )( z ) θ [ q i ]( z ) S (6) − = X j , k ∈{ , , , } s.t. j < k χ ( q jk ) θ ( q + q + q jk ) θ ( q + q + q jk ) θ ( q + q + q jk )( z ) θ [ q jk ]( z )As for S + , the labelling introduced in (9) for the elements of Q ( J C [2]) + shows that S + = S (4) + + S (6) + where S (4) + is the term given by summing on the four quadraticforms q i with i ∈ { , , , } , while S (6) + is the term given by summing on thesix quadratic forms q jk with j , k ∈ { , , , } and j < k . A straightforwardcomputation with the reduction formula shows that S (6) + and S (6) − cancel out,whereas: S (4) + = X i = e ( a ( q ) + a ( q )) χ ( q i ) θ ( q + q + q i ) θ ( q + q + q i ) θ ( q + q + q i )( z ) θ [ q i ]( z ) = S (4) − Therefore, one finally obtains the identity: X k = χ ( q k ) θ ( q + q + q k ) θ ( q + q + q k ) θ ( q + q + q k )( z ) θ [ q k ]( z ) = ∀ z ∈ C g By taking the derivative with respect to each z j for j = , , z = X k = χ ( q k ) θ ( q + q + q k ) θ ( q + q + q k ) θ ( q + q + q k ) ∂θ [ q k ] ∂ z j z = = j = , , (cid:3) We can now state the main theorem of this note:
Theorem 2 ( Weber’s formula).
Let τ ∈ S the period matrix of a smooth planequartic C. If q , · · · q is an Aronhold system of quadratic forms on the -torsion pointsof the Jacobian variety C / ( Z + τ Z ) , then for the coe ffi cients in (11) one has:a ij = η i e ( q ′ j · ( q + q + i ) ′′ ) θ ( q + q r + q j ) θ ( q + q s + q j ) θ ( q + i + q r + q j ) θ ( q + i + q s + q j ) i , j = , , where r and s are such that { + i , r , s } = { , , } and η i is a non-zero scalar factor thatonly depends on the index i, which is due to the fact that the equations for β , β and β in (11) are defined up to a scalar. Remark 1.
The reduction formula (4) can be used in Weber’s formula to express thecoe ffi cients of the bitangents in terms of reduced characteristics. In this case one has:a ij = ρ ij · η i e ( q ′ j · ( q + q + i ) ′′ ) θ [ q + q r + q j ] θ [ q + q s + q j ] θ [ q + i + q r + q j ] θ [ q + i + q s + q j ] i , j = , , here, for any i and j, ρ ij is the product of the reduction signs (cf. (4)) of the four thetaconstants appearing in the expression of a ij .Proof of Weber’s formula. Let f i = f i ( X , X , X ) be linear forms associated withthe bitangent β i for any i = , . . .
7; the equations (11) yield the following linearsystem for the f i : f = f + f + f ; f = a f + a f + a f f = a f + a f + a f f = a f + a f + a f (14)By (7), there also exists a projective transformation ϕ : P P such that: f i ( X , X , X ) = h i X j = ∂θ [ q i ] ∂ Z j z = ϕ j ( X , X , X ) ∀ i = , . . . , ffi cients h i ∈ C ∗ . Thus, each equation in (14) yelds linearsystems in the variables h i and a ij : h ∂θ [ q ] ∂ Z j z = = X i = h i ∂θ [ q i ] ∂ Z j z = j = , , h + i ∂θ [ q + i ] ∂ Z j z = = X l = a il h l ∂θ [ q l ] ∂ Z j z = j = , , i = , , h = D [ q , q , q ] D [ q , q , q ] h ; h = D [ q , q , q ] D [ q , q , q ] h ; h = D [ q , q , q ] D [ q , q , q ] h ;By replacing these solutions into (16), one gets the coe ffi cients for β + i for i = , , a i = µ i D [ q + i , q , q ] D [ q , q , q ] ; a i = µ i D [ q , q + i , q ] D [ q , q , q ] ; a i = µ i D [ q , q , q + i ] D [ q , q , q ] ; (17)where µ i : = h + i / h ∈ C ∗ . Therefore, the bitangents β + i are uniquely determinedas points in P by duality. By repeating the same procedure as before withequations (12), one obtains for suitable coe ffi cients h , h , h ∈ C ∗ : h ∂θ [ q ] ∂ Z j z = = h ∂θ [ q ] ∂ Z j z = + h ∂θ [ q ] ∂ Z j ( z = + h ∂θ [ q ] ∂ Z j ( z = k i h + i ∂θ [ q + i ] ∂ Z j z = = h a i ∂θ [ q ] ∂ Z j z = + h a i ∂θ [ q ] ∂ Z j ( z = + h a i ∂θ [ q ] ∂ Z j z = with i , j = , ,
3. By solving these linear systems, one has likewise:1 a i = k i µ i D [ q + i , q , q ] D [ q , q , q ] ; 1 a i = k i µ i D [ q , q + i , q ] D [ q , q , q ] ; 1 a i = k i µ i D [ q , q , q + i ] D [ q , q , q ] ;9hus, by applying Proposition 3 to the azygetic 4-tuples of the two Aronholdsystems: { q , q , q , q , q , q , q } { q , q , q , q , q , q , q } one gets an explicit expression for the square power of the constant factor interms of theta constants: µ i = k i D [ q , q , q ] D [ q , q , q ] D [ q + i , q , q ] D [ q + i , q , q ] == k i e (( q + q + i ) ′ · ( q + q + q + q ) ′′ ) θ ( q + i + q r + q s ) θ ( q + q r + q s ) i = , , r and s are such that { + i , r , s } = { , , } . Therefore, by replacing thisexpression into (17) one finally has: a ij = − ǫ i e π i ( q + q + i ) ′ · ( q + q + q + q ) ′′ e (( q j + q r + q s ) ′ · ( q + q + i ) ′′ ) θ ( q + q r + q j ) θ ( q + q s + q j ) θ ( q + i + q r + q j ) θ ( q + i + q s + q j )where ǫ i is a fixed root of 1 / k i for any i = , ,
3, and − e (( q r + q s ) ′ · ( q + q + i ) ′′ ) isa sign that can be absorbed into the definition of the root, as it only depends onthe index i . This proves the statement. (cid:3) The following corollary follows as a straightforward consequence:
Corollary 2.
By setting in Weber’s formula: η i : = ǫ i e π i ( q + q + i ) ′ · ( q + q + q + q ) ′′ i = , , where ǫ i is a chosen sign that only depends on i, the corresponding choice of representa-tives for the points ( a i : a i : a i ) in P for i = , , is such that ( k , k , k ) = (1 , , is the unique solution of (13).Proof. By the proof of Weber’s formula one has η i = σ i e π i ( q + q + i ) ′ · ( q + q + q + q ) ′′ where σ i is a non-zero factor that reduces to a sign for each i = , , k = k = k = (cid:3) As an example, we can fix a system of coordinates for the quadratic formsas in (3) and consider the following Aronhold system in terms of reducedcharacteristics: n = " ; n = " ; n = " ; n = " ; n = " ; n = " ; n = " ;Then, we can apply Weber’s formula with the choice made in Corollary 2 andcompute the reduction signs (see Remark 1): ρ = + ρ = + ρ = + ρ = + ρ = + ρ = + ρ = + ρ = − ρ = − = ǫ i θ θ θ θ ; a = ǫ i θ θ θ θ ; a = − ǫ θ θ θ θ ; a = ǫ i θ θ θ θ ; a = ǫ i θ θ θ θ ; a = ǫ θ θ θ θ ; a = ǫ i θ θ θ θ ; a = ǫ i θ θ θ θ ; a = ǫ θ θ θ θ ; Remark 2.
The formula in (17) for the Aronhold system (11) is in accordance withthe modular description of the universal matrix of bitangents obtained in [DFSM14].If a system of coordinates for the quadratic forms is fixed as in (3), an Aronhold systemq , . . . , q such that q = P i = q i is given by:n = " ; n = " ; n = " ; n = " ; n = " ; n = " ; n = " ; Then the first row of the universal bitangent matrix (cf. [DFSM14]) gives the followingmodular expressions for the corresponding bitangents: β ′ : D [ n + n , n + n , n + n ] P j = ∂θ n ∂ Z j , z = Z j = β ′ : D [ n + n , n + n , n + n ] P j = ∂θ n ∂ Z j z = Z j = β ′ : D [ n , n , n ] P j = ∂θ n ∂ Z j z = Z j = β ′ i : D [ n , n , n ] P j = ∂θ ni ∂ Z j z = Z j = i = , , , where, as above, D [ n i , n j , n k ] : = grad z θ n i ∧ grad z θ n j ∧ grad z θ n k . A straightforwardcomputation shows that the ordered collection of bitangents β , . . . β given by Weber’sformula is sent to the ordered collection β ′ , . . . β ′ by the projective transformation φ : P → P , defined by the matrix:A φ : = D [ n , n , n ] ∂θ n ∂ Z z = D [ n , n , n ] ∂θ n ∂ Z z = D [ n , n , n ] ∂θ n ∂ Z z = D [ n , n , n ] ∂θ n ∂ Z z = D [ n , n , n ] ∂θ n ∂ Z z = D [ n , n , n ] ∂θ n ∂ Z z = D [ n , n , n ] ∂θ n ∂ Z z = D [ n , n , n ] ∂θ n ∂ Z z = D [ n , n , n ] ∂θ n ∂ Z z = References [ACGH85] E. Arbarello, M. Cornalba, P. A. Gri ffi ths, J. Harris, Geometry of alge-braic curves , Die Grundlehren der matematischen Wissenschaften in Einzel-darstellungen 267, Springer-Verlag (1985);11CS03] L. Caporaso, E. Sernesi,
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