Discriminants of cubic curves and determinantal representations
aa r X i v : . [ m a t h . N T ] S e p DISCRIMINANTS OF CUBIC CURVES AND DETERMINANTALREPRESENTATIONS
MANH HUNG TRAN
Abstract.
The discriminant of a smooth plane cubic curve over the complex num-bers can be written as a product of theta functions. This provides an importantconnection between algebraic and analytic objects. In this paper, we perform a newapproach to obtain this classical result by using determinantal representations. Moreprecisely, one can represent a non-singular cubic form as the determinant of a matrixwhose elements are linear forms. Theta functions naturally appear in this represen-tation and thus in the discriminant of the cubic.
Contents
1. Introduction 12. Determinantal representations of Weierstrass cubics 33. Determinantal representations of complex plane curves 54. Discriminants of plane cubic curves 95. Plane quartics and Klein’s formula 14Acknowledgement 14References 151.
Introduction
The discriminant of a plane cubic curve is a polynomial of degree 12 in coefficientsof the cubic with 2040 monomials (see [8, p. 4]). But over C , we have short expressionsin terms of theta constants. Consider the classical case where our smooth projectivecubic curve C is defined by the affine Weierstrass equation: y = 4 x − g x − g . Using the Weierstrass parametrization, there exists a unique lattice Λ = ω Z + ω Z with some complex numbers ω , ω such that Im( ω /ω ) > C ( C ) ∼ = C / Λ. Let τ := ω /ω and apply the discriminant formula ∆ = 2 ( g − g ) from [1, p. 367-368],we have that ∆ = 2 (cid:18) πω (cid:19) ( θ (0 , τ ) θ (0 , τ ) θ (0 , τ )) . (1) Here θ , θ and θ are the three even Jacobi theta functions. The details of thetafunctions will be described in Section 2.We want to study the above discriminant formula with a new approach using deter-minantal representations. For a homogeneous polynomial φ , we construct a matrix U whose elements are linear forms such that we can write φ = λ det( U ) for some constant λ = 0. The study of φ has thus been moved to the study of the matrix U . In general,only plane curves and quadratic, cubic surfaces admit a determinantal representationas confirmed in [6]. The reader can have a look at [3] for a general discussion of thistopic.Starting with Weierstrass cubics, we find theta functions in their determinantal rep-resentations as well as in the discriminants. Let a = θ (0 , τ ) , b = θ (0 , τ ) , c = θ (0 , τ ) , (2)we will prove the following Theorem 1.1.
Let C φ be a smooth curve given by the Weierstrass form φ ( x, y, z ) = y z − x + g xz + g z , where g and g belong to a field K . Then φ admits determinantal representations x + tz y + dz (3 t − g ) z x − tz y − dzz − x − tz , with t, d ∈ K being arbitrary such that d = 4 t − g t − g . When K = C , there is anatural choice for t, d which produces a determinantal representation for φ in terms oftheta constants as follows x − π ω ( a + b ) z y − ( πω ) c z x + π ω ( a + b ) z yz − x + π ω ( a + b ) z . Here the even theta constants a, b, c were defined as in (2) . The first part of this theorem uses the method in [14, Section 2] where the authorestablished similar representations for other type of Weierstrass equations of the form y z = x ( x + ϑ z )( x + ϑ z ) with some constants ϑ , ϑ ∈ K . The discriminant formula(1) is then a consequence of the second part of this theorem using resultant as in Section2. Our goal is to study this phenomena for general smooth cubic curves using deter-minantal representations. One can actually provide determinantal representations forany non-rational complex plane curve by using a result in [2] as we will see later in ISCRIMINANTS OF CUBIC CURVES AND DETERMINANTAL REPRESENTATIONS 3
Section 3. This deduces in particular the formula of the discriminant of plane cubicsby using resultant. Since a cubic curve C φ over C always has a flex point, it can betransformed to a Weierstrass form after a linear coordinate change M (see [5, Sec-tion 4.4]). The resulting Weierstrass form is isomorphic to C / Λ for a unique lattice Λcoming the Weierstrass parametrization. Write Λ = ω Z + ω Z for some ω , ω ∈ C satisfying Im( ω /ω ) > τ = ω /ω , we will prove the following result. Theorem 1.2.
Let C φ be a smooth plane cubic curve over C defined by a cubic form φ and ∆ φ be the discriminant of φ , we have ∆ φ = 2 det( M ) (cid:18) πω (cid:19) ( abc ) , (3) where a, b, c were defined as in (2) . This result is known but the above approach with determinantal representations isnew.The discriminant formulae (1) and (3) are remarkable since they provide a connectionbetween algebraic (discriminants) and analytic (theta functions) objects. There is asimilar formula in the case of quartic curves studied by Klein [10, p. 72]. We will providean overview to this formula and a known result on determinantal representations ofplane quartics in the last section.2.
Determinantal representations of Weierstrass cubics
We will in this section study discriminants of smooth curves in Weierstrass form andprovide a proof to Theorem 1.1. Consider a smooth curve C φ given by φ ( x, y, z ) = y z − x + g xz + g z , (4)where g and g are elements in a field K . We want to find the 3 × L, M, N such that det( xL + yM + zN ) = φ ( x, y, z ) . The following determinantal representations of φ : x + tz y + dz (3 t − g ) z x − tz y − dzz − x − tz (5)is obtained from [14, Section 2], where t, d ∈ K be such that d = 4 t − g t − g . It canbe checked that the determinant of (5) is equal to φ .Now we move to the theory of theta functions to study the case when K = C . Thefollowing discussion bases on Wang and Guo [15]. In this case, there exists a uniquelattice Λ coming from the Weierstrass parametrization such that C φ ( C ) ∼ = C / Λ. Here
ISCRIMINANTS OF CUBIC CURVES AND DETERMINANTAL REPRESENTATIONS 4
Λ = ω Z + ω Z for some ω , ω ∈ C with τ = ω /ω ∈ H . The two coefficients g and g of the curve given by φ can be determined by (see [15, p. 509]) g = 23 (cid:18) πω (cid:19) ( a + b + c ) ,g = 427 (cid:18) πω (cid:19) ( a + b )( b + c )( c − a ) , where a = θ (0 , τ ) = e πiτ θ ( τ, τ ), b = θ (0 , τ ) = θ (0 , τ ) and c = θ (0 , τ ) = θ ( , τ ) withthe even Jacobi theta functions: θ ( z, τ ) = θ ( z, τ ) := ∞ X n = −∞ exp( πin τ + 2 πinz ) ,θ ( z, τ ) = exp( πiτ / πiz ) θ ( z + τ / , τ ) ,θ ( z, τ ) = θ ( z + τ / , τ ) . The above a, b, c are called even theta constants.Since ( t, d ) is a point on the affine curve associated to C φ defined by { z = 0 } , itis determined by theta constants via Weierstrass P -function and so are all the coeffi-cients in the linear matrix (5). To be precise, we consider the Weierstrass P -functionassociated to the lattice Λ defined for all s / ∈ Λ as P ( s ) = P ( s ; ω , ω ) := 1 s + X ( m,n ) ∈ Z \ (0 , (cid:18) s + mω + nω ) − mω + nω ) (cid:19) . As in [15, p. 469], it satisfies the differential equation P ′ ( s ) = 4 P ( s ) − g P ( s ) − g . The point ( t, d ) on the curve can be parametrized as t = P ( s ) and d = P ′ ( s ) for some s / ∈ Λ. It is known that the discriminant of the cubic (4) is given by the formula∆ φ = 2 ( g − g ) = 2 (cid:18) πω (cid:19) ( abc ) . (6)We will give another proof for the formula (6) using resultant and the determinan-tal representation (5). From [8, p. 434], the discriminant of a homogeneous cubicpolynomial φ ( x, y, z ) can be computed by resultant defined there as∆ φ = − Res( φ x , φ y , φ z ) / . (7)The reader can have a look at [8, Chapter 13] for a general discussion about resultants.We choose the minus sign here so that the sign of the discriminant is compatible toother sections of the paper. To simplify the computation, we choose a special valuefor the Weierstrass function P ( s ), namely, the 2-torsion point s = ω /
2. In this case P ( ω /
2) = − π ω ( a + b ) and P ′ ( ω /
2) = 0 by [15, p. 470,509]. Then d = 0 and ISCRIMINANTS OF CUBIC CURVES AND DETERMINANTAL REPRESENTATIONS 5 t = − π ω ( a + b ). Besides, using the Jacobi’s identity a + c = b (see [15, p. 504]),the matrix (5) can be written in the form x − π ω ( a + b ) z y − ( πω ) c z x + π ω ( a + b ) z yz − x + π ω ( a + b ) z . (8)Theorem 1.1 has thus been proved. From the representation φ = det( U ), where U isgiven by (8), we get that φ x = − x + 23 (cid:18) πω (cid:19) ( a + b + c ) z ,φ y = 2 yz, and φ z = y + 43 (cid:18) πω (cid:19) ( a + b + c ) xz + 49 (cid:18) πω (cid:19) ( a + b )( b + c )( c − a ) z . The discriminant ∆ φ of the cubic φ is then obtained via (7)∆ φ = 2 (cid:18) πω (cid:19) ( abc ) . In fact, we can directly use (7) to the curve (4). However, this approach of determi-nantal representations might be applied to more general cases. More details will beexplained in Section 4.3.
Determinantal representations of complex plane curves
In Section 2, we have already seen that one can compute the discriminant of smoothcurves over C in Weierstrass form by using determinantal representations. Our goalis to generalize this result to any smooth cubic curve. To do this, one can computedeterminantal representations of plane curves of arbitrary degrees based on Theorem5.1 in [2]. We will in this section prove Theorem 3.3. Let us first introduce somenotations.Let X be a compact Riemann surface, let L is a line bundle of half differentials on X (a theta characteristic), i.e., L ⊗ is the canonical bundle ω X on X and let χ be aflat line bundle over X such that h ( χ ⊗ L ) = 0. We associate to χ the Cauchy kernel K ( χ ; · , · ) as defined in Section 2 of [2]. Let λ , λ be two scalar meromorphic functionson X , which generate the whole field of meromorphic functions. Assume that all polesof λ , λ are simple and labeled as P , ..., P d ∈ X . One can write the Laurent expansionof λ k at P i (1 ≤ i ≤ d, k = 1 ,
2) with some fixed local coordinate t i = t i ( P ) centered at P = P i λ k ( P ) = − c ik t i − d ik + O ( | t i | ) . Then we define the d × d matrices L, M, N by ISCRIMINANTS OF CUBIC CURVES AND DETERMINANTAL REPRESENTATIONS 6 L = diag ≤ i ≤ d ( c i ) , M = diag ≤ i ≤ d ( − c i ) , N = ( n ij ) i,j , where n ij = d i c i − d i c i , i = j ;( c i c j − c j c i ) K ( χ ; P i , P j ) dt j ( P j ) , i = j .The result mentioned in [2] is the following Proposition 3.1.
The map π : X → C given by π ( P ) = ( λ ( P ) , λ ( P )) maps X \{ P , ..., P d } onto the affine part C of an algebraic curve C ⊂ P and extends to aproper birational map π : X → C of X in P . The defining irreducible homogeneouspolynomial φ ( x, y, z ) of C is such that (up to multiplying by some constant) φ ( x, y, z ) = det( xL + yM + zN ) . Here the affine part C of C is defined by { z = 0 } . The authors in [2] prove a more general version of the above proposition where theyconsider χ to be any flat vector bundle. We restrict here to the case of line bundlesince it is enough for our purpose.Suppose in this case that χ is defined by a unitary representation of the fundamentalgroup of X given by χ ( α i ) = exp( − πia i ) and χ ( β i ) = exp(2 πib i ) , i = 1 , ..., g, where a i , b i ∈ R , g is the genus of X and α , ..., α g , β , ..., β g form a symplectic basis of H ( X, Z ). Let ( η , ..., η g ) be a basis of holomorphic 1-forms on X , we form from thesebases the period matrix (Ω | Ω ) which is the g × g -matrix whose entries are(Ω ) ij = Z α j η i and (Ω ) ij = Z β j η i , for i, j = 1 , ..., g. We choose the canonical basis ( η , ..., η g ) of holomorphic 1-forms such that R α i η j = δ ij ,then the corresponding period matrix will be of the form ( I g | Ω). The matrix Ω liesin the Siegel upper half space H g and it is called the Riemann period matrix of X withrespect to the homology basis α , ..., α g , β , ..., β g . We fix such a symplectic homologybasis and the resulting period matrix Ω. Let J ( X ) = C g / ( Z g + Ω Z g ) be the Jacobian of X and ϕ : X → J ( X ) be the Abel-Jacobi map with any fixed base point. An explicitformula for the Cauchy kernel is given in [2, Theorem 4.1] as follows K ( χ ; P, Q ) = θ [ δ ]( ϕ ( Q ) − ϕ ( P )) θ [ δ ](0) E ( Q, P ) , (9)where θ [ δ ] is the associated theta function with characteristic δ = b + Ω a = ϕ ( χ )( a = ( a j ) j and b = ( b j ) j ) and E ( · , · ) is the prime form on X × X . Recall from [7, ISCRIMINANTS OF CUBIC CURVES AND DETERMINANTAL REPRESENTATIONS 7
Chapter II] that the prime form E is a bi-half-differential with simple zeros along thediagonal of X × X .Here the theta characteristic L is chosen such that ϕ ( L ) = −K , where K is thevector of Riemann constants. As a consequence of the Riemann singularity theorem, θ ( b + Ω a ) = 0 if and only if h ( χ ⊗ L ) = 0. Hence θ [ δ ](0) = 0 and the formula (9)makes sense.From Proposition 3.1, one can explicitly provide determinantal representations forcomplex plane curves using theta functions and the Abel-Jacobi map. The reader canhave a look at [9, Section 4], [13, Theorem 6] or [4, Theorem 2.2] for reference. Notethat results in the reference above only apply to the family of hyperbolic curves witha normalization. However, it can be written in the following general form. Proposition 3.2.
Let C φ ⊂ P be a non-rational irreducible complex plane curvedefined by φ = 0 , where φ ( x, y, z ) is an irreducible homogeneous polynomial of degree d . Suppose the d intersection points of C φ with the line { y = 0 } are distinct non-singular points P , ..., P d with coordinates P i = (1 , , β i ) , β i = 0 . Then φ ( x, y, z ) = λ det( xM + yN + zI ) , where λ = φ (0 , , , M = diag( − β , ..., − β d ) and N = ( n ij ) i,j with n ii = − β i φ y (1 , , β i ) φ x (1 , , β i ) and for i = jn ij = β i − β j θ [ δ ](0) . θ [ δ ]( ϕ ( P j ) − ϕ ( P i )) E ( P j , P i ) . p d ( − y/x )( P i ) p d ( − y/x )( P j ) . Here δ is an even theta characteristic such that θ [ δ ](0) = 0 , ϕ : X → J ( X ) is theAbel-Jacobi map from the desingularizing Riemann surface X of C φ to its Jacobianand E ( ., . ) is the prime form on X × X . We want to generalize Proposition 3.2 in such a way that the line { y = 0 } is replacedby a general line passing through distinct points of C φ .Let l be a line defined by αx + βy + γz = 0 so that its affine part l defined by αx + βy + γ = 0 intersects the affine part C φ of C φ at d distinct non-singular points P i , i = 1 , ..., d . Since α and β can not be both zero, we can suppose w.l.o.g that β = 0(the case α = 0 can be treated similarly). In this case, we can suppose further that β = −
1. Therefore, the line l can be rewritten as y = αx + γz . Assume that theintersections points P i of l and C φ have non-zero x -coordinates so that we can write P i = (1 /β i , α/β i + γ ) with β i = β j if i = j . Thus the intersection points of l and C φ are P i = (1 , α + γβ i , β i ). We now prove the following ISCRIMINANTS OF CUBIC CURVES AND DETERMINANTAL REPRESENTATIONS 8
Theorem 3.3.
Let C φ ⊂ P be a non-rational irreducible complex plane curve de-fined by φ = 0 , where φ ( x, y, z ) is an irreducible homogeneous polynomial of degree d . Suppose the d intersection points of C φ with the line { y = αx + γz } are distinctnon-singular points P , ..., P d with coordinates P i = (1 , α + γβ i , β i ) , β i = 0 . Then up tomultiplying by some constant φ ( x, y, z ) = det(( M − αN ) x + N y + ( I − γN ) z ) , where M = diag( − β , ..., − β d ) and N = ( n ij ) i,j with n ii = − β i φ y ( P i )( φ x + αφ y )( P i ) and for i = jn ij = θ [ δ ]( ϕ ( P j ) − ϕ ( P i )) θ [ δ ](0) E ( P j , P i ) β i − β j p β i ( αdx − dy )( P i ) p β j ( αdx − dy )( P j ) . Here δ is an even theta characteristic such that θ [ δ ](0) = 0 , ϕ : X → J ( X ) is theAbel-Jacobi map from the desingularizing Riemann surface X of C φ to its Jacobianand E ( ., . ) is the prime form on X × X .Proof. Apply Proposition 3.1 with the pair of meromorphic functions on the desingu-larizing Riemann surface X of C φ : λ = 1 y − αx − γ , λ = xy − αx − γ and the local coordinates t = αx − y + γx at the poles P i (zeros of αx − y + γ ). The nextstep is to write down Laurent expansions of λ , λ at P i . We have λ = − /t ⇒ c i = 1 , d i = 0 ∀ i. At P i we have λ = − t ( x ). Since1 x = β i + d ( x ) d ( αx − y + γx ) ( P i ) t + O ( | t | ) = β i + β i dxd ( y − αx ) ( P i ) t + O ( | t | ) , one deduces that c i = β i , d i = β i dxd ( y − αx ) ( P i ) . We then obtain from Proposition 3.1 that (up to some constant) φ = det(( M − αN ) x + N y + ( I − γN ) z ) , where M = diag( − β , ..., − β d ) and N = ( n ij ) with n ii = β i dxd ( y − αx ) ( P i ) ISCRIMINANTS OF CUBIC CURVES AND DETERMINANTAL REPRESENTATIONS 9 and for i = j n ij = θ [ δ ]( ϕ ( P j ) − ϕ ( P i )) θ [ δ ](0) E ( P j , P i ) β i − β j d ( αx − y + γx )( P j ) . Here δ is an even theta characteristic with θ [ δ ](0) = 0. Note that the affine part C φ of C φ is defined by { ( λ ( P ) , λ ( P )) | P ∈ X \ { P , ..., P d }} . Furthermore, if we replace N by the matrix N ′ which has the same diagonal elementsas N but different off-diagonal elements n ′ ij = θ [ δ ]( ϕ ( P j ) − ϕ ( P i )) θ [ δ ](0) E ( P j , P i ) β i − β j q d ( αx − y + γx )( P i ) q d ( αx − y + γx )( P j ) , then the determinantal representation does not change. Indeed, let U = ( M − αN ) x + N y + ( I − γN ) z and U ′ = ( M − αN ′ ) x + N ′ y + ( I − γN ′ ) z, if we multiply the i th -column of U and the i th -row of U ′ (for i = 1 , ..., d ) with the term q d ( αx − y + γx )( P i ) then both of them will become the same matrix U ∗ . Consequently,det( U ) = det( U ′ ) = det( U ∗ ) Q di =1 q d ( αx − y + γx )( P i ) . Observe that d ( αx − y + γx )( P i ) = β i ( αdx − dy )( P i ) and dxd ( y − αx ) ( P i ) = − φ y ( P i )( φ x + αφ y )( P i )by implicit function theorem with the fact that ( φ x + αφ y )( P i ) = 0. Indeed, sincethe polynomial f ( x ) := φ ( x, αx + γ,
1) has distinct roots (1 /β i ) we conclude that f ′ (1 /β i ) = 0 and hence ( φ x + αφ y )( P i ) = 0. This completes the proof of Theorem3.3. (cid:3) Proposition 3.2 is then established by reducing to the case α = γ = 0. Theorem 3.3will be applied in the next section to get a formula for the discriminant of plane cubiccurves. Remark 3.4.
One can also reformulate the analogous statement to the Theorem 3.3if the line y = αx + γz is replaced by x = αy + γz . Discriminants of plane cubic curves
We now study the main object of interest in which we consider a smooth plane curve C φ over C defined by the cubic form φ . The affine part of the curve C φ is parametrized ISCRIMINANTS OF CUBIC CURVES AND DETERMINANTAL REPRESENTATIONS 10 as { ( x, y,
1) = ( R ( P ( s ) , P ′ ( s )) , R ( P ( s ) , P ′ ( s )) , } , where P ( s ; ω , ω ) is the Weierstrass P -function associated to some ω , ω ∈ C satisfy-ing Im( ω /ω ) > τ of genus one case instead of Ω forthe period matrix. Moreover, we use the general Jacobian C / ( ω Z + ω Z ) in place ofthe normalized one C / ( Z + τ Z ) for τ = ω /ω in order to use the properties of thefunction P . By this change, an extra factor 1 /ω appears in the below elements n ij ( i = j ) in comparing with Theorem 3.3. This idea was mentioned in [4, Theorem 2.4].In addition, we use the notation θ δ ( δ = 1 , , ,
4) for theta functions as in Section 2instead of θ [ δ ].The prime form E ( P, Q ) in genus one case is better understood so that a consequenceof Theorem 3.3 is obtained as follows
Corollary 4.1.
Let C φ ⊂ P be a smooth plane cubic curve defined by φ = 0 , where φ ( x, y, z ) is a non-singular homogeneous cubic polynomial. Suppose the line y = αx + γz intersects C φ at 3 distinct points P , P , P with coordinates P i = (1 , α + γβ i , β i ) , β i = 0 .Then up to multiplying by some constant φ ( x, y, z ) = det(( M − αN ) x + N y + ( I − γN ) z ) , (10) where M = diag( − β , − β , − β ) and N = ( n ij ) i,j with n ii = − β i φ y ( P i )( φ x + αφ y )( P i ) and for i = jn ij = θ ′ (0) θ δ (( Q j − Q i ) /ω ) ω θ δ (0) θ (( Q j − Q i ) /ω ) β i − β j p β i ( αR ′ − R ′ )( Q i ) p β j ( αR ′ − R ′ )( Q j ) Here δ is any even theta characteristic, i.e., δ = 2 , or and Q i = ϕ ( P i ) . Note thatwe also have an analogous statement of this corollary by Remark 3.4. The field of meromorphic functions on a general genus one curve is generated by P , P ′ associated to some periods ω , ω . Thus R and R are rational functions on P , P ′ . Ingeneral, R and R have complicated expressions. But we have better interpretationsin the case of plane cubic curves. In this case, C φ always has a flex point and hence canbe transformed to a Weierstrass form after a linear coordinate change (see [5, Section4.4]). Thus, we are able to present rational functions R , R as: R ( s ) = λ P ( s ) + λ P ′ ( s ) + λ ,R ( s ) = λ P ( s ) + λ P ′ ( s ) + λ . (11) ISCRIMINANTS OF CUBIC CURVES AND DETERMINANTAL REPRESENTATIONS 11
The constants λ ij ∈ C satisfy λ λ = λ λ and depend on the coefficients of φ . Herewe fix any flex point and the corresponding periods ω , ω coming from the Weierstrassparametrization of the Weierstrass form.To shorten the determinantal representation, one should look at 2-torsion pointsto simplify θ and P . More precisely, we consider the line l which intersects C φ atthe points P i such that the corresponding points Q i on the torus C / ( ω Z + ω Z ) are ω / , ( ω + ω ) / ω / x -coordinates of P i are allnon-zero. We will treat the case l to have the form y = αx + γz and then make use ofCorollary 4.1. The other case can be treated similarly using Remark 3.4. The choice of2-torsion points gives us the convenience to work with some computations below. Let a = θ (0 , τ ) , b = θ (0 , τ ) , c = θ (0 , τ ), where τ = ω /ω , we will prove the following Proposition 4.2.
Let C φ ⊂ P be a smooth plane curve defined by φ = 0 , where φ ( x, y, z ) is a non-singular homogeneous cubic polynomial. Suppose the line y = αx + γz intersects C φ at 3 distinct points P , P , P with coordinates P i = (1 , α + γβ i , β i ) , β i = 0 so that the corresponding points Q i = ϕ ( P i ) of P i on the torus C / ( ω Z + ω Z ) are ω / , ( ω + ω ) / and ω / respectively. Denote by k = αλ − λ , then we have thefollowing expressions (up to some constant) for the discriminant ∆ φ of φ ∆ φ = λ ω k π ( abc ) ( β − β ) ( β − β ) ( β − β ) and ∆ φ = 16 (cid:18) λ πβ β β kω (cid:19) ( abc ) . Proof.
By [15, p. 470, 509], we have P ′ ( Q i ) = 0 for all i and P ( Q ) = π ω ( b + c ) , P ( Q ) = π ω ( a − c ) , P ( Q ) = − π ω ( a + b ) . Besides, P ′′ ( s ) = 6( P ( s )) − g / g = ( πω ) ( a + b + c ) as in [15, p. 469]. Thus P ′′ ( Q ) = 2 π b c ω , P ′′ ( Q ) = − π a c ω , P ′′ ( Q ) = 2 π a b ω . We also have for each i − φ y ( P i )( φ x + αφ y )( P i ) = dxd ( y − αx ) ( P i ) = R ′ ( R ′ − αR ′ ) ( Q i ) = λ λ − αλ . Now, choose δ = 3 to simplify the matrix N in Corollary 4.1. Let k = − λ /k andnote that θ ′ (0) = πabc as in [15, p. 507], we have n ii = k β i and n = n = 0 as θ ((1 + τ ) /
2) = 0. Moreover, n = n = π ( abc ) θ (cid:0) τ (cid:1) ( β − β ) ω k b θ (cid:0) τ (cid:1) β β P ′′ ( Q ) P ′′ ( Q ) = ω ( β − β ) k π β β b c , ISCRIMINANTS OF CUBIC CURVES AND DETERMINANTAL REPRESENTATIONS 12 n = n = π ( abc ) θ (cid:0) (cid:1) ( β − β ) ω k b θ (cid:0) (cid:1) β β P ′′ ( Q ) P ′′ ( Q ) = − ω ( β − β ) k π β β a b . Here we use the fact that (see [15, p. 502]) θ (cid:16) τ (cid:17) = q − a, θ (cid:16) τ (cid:17) = iq − c, θ (cid:18) (cid:19) = c, θ (cid:18) (cid:19) = a with q = e πiτ . One has 1 /β i = λ P ( Q i ) + λ from (11) and the fact R ( Q i ) = 1 /β i .Therefore, β − β β β = λ ( P ( Q ) − P ( Q )) = − λ π c ω ,β − β β β = λ ( P ( Q ) − P ( Q )) = − λ π b ω , (12) β − β β β = λ ( P ( Q ) − P ( Q )) = − λ π a ω . It can be seen from (12) that λ = 0. Similarly we have λ = αλ from the identities R ( Q i ) = α/β i + γ . Breaking out the determinant (10), one get the following expressionfor φ (up to some constant λ ) − β β β x + 3 β β β k x y + ( β n + β n − β β β k ) xy + k ( β β β k − β n − β n ) y + ( β β + β β + β β ) x z − k ( β β + β β + β β ) xyz +( k ( β β + β β + β β ) − n − n ) y z − ( β + β + β ) xz + k ( β + β + β ) yz + z . (13)Consequently, Res( φ x /λ, φ y /λ, φ z /λ ) =( − β − β ) ( β − β ) n n ( β − β ) ( β n − β n − β n + β n ) . The term β n − β n − β n + β n is equal to( β − β )( β − β ) ω k π b (cid:18) β − β c β β + β − β a β β (cid:19) = − λ ω ( β − β )( β − β )4 k π a c , where the later equality comes from (12). Furthermore,( n n ) = ω ( β − β ) ( β − β ) k π ( abc ) ( β β ) ( β β ) = λ ω ( β − β ) ( β − β ) k π a b c . The later equality again comes from (12). Hence∆ φ = −
127 Res( φ x , φ y , φ z ) = λ λ ω k π ( abc ) ( β − β ) ( β − β ) ( β − β ) . An alternative form of the discriminant can be established by using (12):∆ φ = 16 (cid:18) λλ πβ β β kω (cid:19) ( abc ) . (14) ISCRIMINANTS OF CUBIC CURVES AND DETERMINANTAL REPRESENTATIONS 13
This completes the proof of Proposition 4.2. (cid:3)
We now simplify the formula (14) by looking at the relationships between λ, λ , k and β β β . It can be seen from (13) that λβ β β = − φ (1 , , x = λ X + λ Y + λ ,y = λ X + λ Y + λ then the affine curve φ ( x, y,
1) = 0 will be transformed to a Weierstrass equation − Y + 4 X − g X − g = 0. In addition, the inverse transformation X = l x + l y + l ,Y = l x + l y + l would transform the Weierstrass equation − Y + 4 X − g X − g = 0 to:4 l x + 12 l l x y + 12 l l xy + 4 l y + (12 l l − l ) x +(24 l l l − l l ) xy + (12 l l − l ) y + (12 l l − l l − l g ) x + (15)(12 l l − l l − l g ) y + 4 l − l − l g − g . One can check that l = λ /D, l = − λ /D, l = − λ /D and l = λ /D with D = λ λ − λ λ . Compare the coefficients of x and x y in (13) and (15), we have ( l = − λβ β β , l l = 3 λβ β β k . Or ( λ = − λβ β β D , λ λ = − λβ β β k D . The second identity shows that λ = − λ / ( λβ β β D ). Hence λλ β β β = − Theorem 4.3.
Let C φ be a smooth plane cubic curve as in Proposition 4.2. Then thediscriminant ∆ φ of φ satisfies ∆ φ = 2 ( λ λ − λ λ ) (cid:18) πω (cid:19) ( abc ) . Let us look at the example when φ is given in the Weierstrass form − y +4 x − g x − g .In this case, λ = λ = 1 and λ = λ = 0. We thus recover the classical formula∆ φ = 2 ( πω ) ( abc ) .From Remark 3.4, one can also treat the other case where the line l passes through2-torsion points of C φ . Furthermore, the set of cubics φ in the above theorem forms ISCRIMINANTS OF CUBIC CURVES AND DETERMINANTAL REPRESENTATIONS 14 an open dense subset of the space of all ternary cubics and we have thus obtainedTheorem 1.2. 5.
Plane quartics and Klein’s formula
In this section, we provide an overview to a beautiful formula of Klein on planequartics, which are non-hyperelliptic curves of genus three. More concretely, let C F be a smooth plane curve over C given by a quartic F , let α , α , α , β , β , β be asymplectic basis of H ( C F , Z ) and let η , η , η be the classical basis of holomorphic1-forms of Ω C ( C F ) defined in [11, p. 329]. We construct from these the period matrix[Ω Ω ] whose entries are(Ω ) ij = Z α i η j and (Ω ) ij = Z β i η j , for i, j = 1 , , . De note by τ = Ω − Ω , the discriminant ∆ F of F satisfies the following formula (see[10, p. 72], [11, Theorem 2.2.3]):∆ F = 2 π (det Ω ) Y δ even θ δ (0 , τ ) . (16)Here θ δ is the Riemann theta function with characteristic δ = ( δ , δ ), where δ , δ ∈ { , } , defined for any z ∈ C as: θ δ ( z, τ ) = X n ∈ Z exp 2 πi (cid:18)
12 ( n + δ ) t τ ( n + δ ) + ( n + δ ) t ( z + δ ) (cid:19) . The product in (16) runs over all 36 even theta characteristics of genus three. Thecharacteristic δ is called even if the corresponding theta function θ δ is an even functionin z . This formula should be compared with (1) in the cubic case. One can ask if it ispossible to use determinantal representations to establish the formula (16) or not? Forthis, the authors in [12, Corollary 5.3] have obtained determinantal representations forplane quartics described by theta constants. More concretely, they use plane quartics’bitangents to construct the representations. Thus it might be interesting to explorethe problem in this case. Acknowledgement
I would like to thank my advisor Dennis Eriksson for introducing me to the topic aswell as providing me with many important ideas, corrections and comments.
ISCRIMINANTS OF CUBIC CURVES AND DETERMINANTAL REPRESENTATIONS 15
References [1] M. Artin, F. Rodriguez-Villegas, J. Tate,
On the Jacobians of plane cubics , Adv. Math. ,2005, 366-382.[2] J. A. Ball, V. Vinnikov,
Zero-pole interpolation for matrix meromorphic functions on a compactRiemann surface and a matrix Fay trisecant identity , Amer. J. Math. , 1999, 841-888.[3] A. Beauville,
Determinantal hypersurfaces , Michigan Math. Journal , 2000, 39-64.[4] M. T. Chien, H. Nakazato, Computing the determinantal representations of hyperbolic forms ,Czechoslovak Math. Journal,
66 (141) , 2016, 633-651.[5] J. Cremona,
G1CRPC: Rational points on curves , Section 4.4, 29-31, .[6] L. E. Dickson,
Determination of all homogeneous polynomials expressible as determinants withlinear elements , Trans. Amer. Math. Soc., , 1921, 167-179.[7] J. D. Fay, Theta functions on Riemann surfaces , Lecture Notes in Mathematics, , Springer-Verlag, Berlin-New york, 1973.[8] I. M. Gelfand, M. M. Kapranov, A. V. Zelevinsky
Discriminants, resultants, and multidimensionaldeterminants , Mathematics: Theory and Applications. Birkhauser Boston Inc., 1994.[9] J. W. Helton, V. Vinnikov,
Linear matrix inequality representation of sets , Comm. Pure Appl.Math.,
60 (5) , 2007, 654-674.[10] F. Klein,
Zur theorie der abelschen funktionen , Math. Annalen, , 1889-90, 1-83.[11] G. Lachaud, C. Ritzenthaler, A. Zykin, Jacobians among abelian threefolds: a formula of Kleinand a question of Serre , Math. Res. Lett.,
17 (2) , 2010, 323-333.[12] F. D. Piazza, A. Fiorentino, R. Salvati Manni,
Plane quartics: the universal matrix of bitangents ,Israel J. Math., , 2017, 111-138.[13] D. Plaumann, B. Sturmfels, C. Vinzant,
Computing linear matrix representations of Helton-Vinnikov curves , Operation Theory: Advances and Applications , 2012, 259-277.[14] V. Vinnikov,
Complete description of determinantal representations of smooth irreducible curves ,Linear Algebra Appl., , 1989, 103-140.[15] Z. X. Wang, D. R. Guo,
Special Functions , World Scientific Publishing, Teaneck, NJ, 1989.
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