Effective reconstruction of generic genus 4 curves from their theta hyperplanes
EEFFECTIVE RECONSTRUCTION OF GENERIC GENUS 4CURVES FROM THEIR THETA HYPERPLANES
DAVID LEHAVI
Abstract.
Effective reconstruction formulas of a curve from its thetahyperplanes are known classically in genus 2 (where the theta hyper-planes are Weierstrass points), and 3 (where, for a generic curve, thetheta hyperplanes are bitangents to a plane quartic). However, forhigher genera, no formula or algorithm are known. In this paper wegive an explicit (and simple) algorithm for computing a generic genus 4curve from it’s theta hyperplanes. Introduction
The quest for methods of reconstructing a curve from its theta hyper-planes goes back to the 19th and early 20th century geometers Aronholdand Coble: in the non hyperelliptic genus 3 case, theta hyperplanes aresimply bitangents, and both Aronhold and Coble provided formulas for re-constructing curves from certain ordered subsets of the 28 bitangents of thecurve (see [A], [Co] chapter IV, and [Dol] sections 6.1.2 and 6.2.2).Recent years witnessed some revived interest in generalizations of thisproblem from several directions: first relaxing the need for ordered thetahyperplanes (see [CS1], [L]), and then generalizations to higher genus curves(see [CS2]) and abelian varieties (see [GS-M1], [GS-M2]). However for g > C of genus4; since C is generic we assume that all its odd theta characteristics are 1dimensional - i.e. if θ is an odd theta characteristic of C then dim H ( θ, C ) =1. Hence, for each odd theta characteristic θ there exists a unique hyperplane l θ – called a theta hyperplane – in the dual canonical system of C such thatwhen C is identified with it’s canonical image, the points in the intersectionproduct C · l θ are all double, and the points in C · l θ sum up to θ . Thishyperplane is the projectivization of the plane T θ Θ C ⊂ T θ J C under theidentification of T θ J C = T J C = H ( K C ) ∗ . Recall (see e.g. [Dol] 5.4.2),that if α is a non-trivial 2 torsion point on the Jacobian J C , then the
Steiner
Date : November 12, 2018.1991
Mathematics Subject Classification. a r X i v : . [ m a t h . AG ] J un DAVID LEHAVI system Σ C,α of the pair (
C, α ) is defined to be the set { θ : 2 θ = K C and dim H ( θ, C ) = dim H ( θ + α, C ) ≡ } . The number of theta characteristics in a Steiner system of a genus g curveis 2 g − · (2 g − − · − (2 −
1) = 120 odd thetacharacteristics of the Jacobian. For each pair θ, θ + α , and correspondingtheta hyperplanes l θ , l θ + α , we let q { θ,θ + α } ∈ |O | K C | ∗ (2) | be the image of { l θ , l θ + α } ∈ S | K C | under the embedding: S | K C | → |O | K C | ∗ (2) | .We set V C := H ( O | K C | ∗ (2)), and denote the map H ( O | K C + α | ∗ (2)) → H (2 K C ) by i , the projection V C → H (2 K C ) by p , and the pre-image p − iH ( O | K C + α | ∗ (2)) ⊂ V C by V C,α . Finally, we denote the projectivizationof a linear space by P . The first component of our reconstruction algorithmis given by the following: Theorem 1.
Let
C, α, V C , and V C,α be as above, (specifically, recall that C isgeneric), and let Q C be the unique quadric surface containing the canonicalimage of C , then (1) for all α ∈ J C [2] (cid:114) { } we havespan ( { q { θ,θ + α } } θ ∈ Σ C,α ) = P V C,α ;(2) moreover, ∩ α ∈ JC [2] (cid:114) { } P V C,α = { [ Q C ] } , where [ Q C ] denotes the mod-uli point representing Q C in the space P V C . Note that the theorem may be rephrased in the following “genus free”terms: the intersection ∩ α ∈ JC [2] (cid:114) { } span ( { q { θ,θ + α } } θ ∈ Σ C,α ) is the locus ofquadrics over | K C | ∗ containing C . The beauty in this statement arises fromits relation to the Enriques-Babbage Theorem (see e.g. [ACGH] chapter VI § cut out by quadrics .The proof of the theorem eventually reduces to the analysis of bi-ellipticcurves: the first part is proved by considering a (degenerated) double coverof the nodal cubic; whereas the second part is proved by considering a curvewith a big automorphism group.Turning our attention back to the genus 4 case, we now aim to locate anirreducible cubic surface containing the canonical image of the curve C . Tothis end we will make heavy use of the following classical Theorem: Theorem (Wirtinger - see [W], [Co] chapter V, or [Ca]) . Let C be a genericgenus complex curve, and let α be a non trivial -torsion point on theJacobian of C . Then the image of C in the Prym canonical system | K C + α | ∗ is a sextic with nodes, which are the intersection points of four lines ingeneral position: l , l , l , l (not to be confused with the hyperplanes l θ ; wewill never use the two notations in the same context). Moreover, let S bethe blow-up of | K C + α | ∗ at the six intersection points l i ∩ l j , let H be thepullback to S of a generic hyperplane in | K C + α | ∗ and let E be the exceptional ECONSTRUCTION OF GENUS 4 CURVES FROM THETA HYPERPLANES 3 divisor in S , then the complete linear system |O S (3 H − E ) | is canonicallyisomorphic to | K C | . Finally, the image of S in the dual of this linear system– which we will denote below by W C,α – is a Cayley cubic (the unique spacecubic with four nodes), where the four nodes are the blow downs of the stricttransforms in S of the four lines l , . . . , l , and where by its definition W C,α contains the canonical image of C . Armed with this theorem we can state the following:
Theorem 2.
The four hyperplanes through each of the four triplets of nodesof W C,α are – set theoretically – the four intersection points in P V C of thesix dimensional projective subspace P (( V ∗ C / (( V C /V C,α ) ∗ ∧ ( V C /V C,α ) ∗ )) ∗ ) , and the 2nd Veronese image of | K C | in P V C . Moreover, each of these fourpoints has multiplicity in the intersection. Given Theorems 1 and 2, the canonical image of the curve C is readilyreconstructed as the intersection of Q C and W C,α – which is the only cubicwith nodes at the four intersection points of triplets of the four hyperplanesdetermined in Theorem 2.
Acknowledgment.
Sam Grushevsky and an anonymous reviewer read early versions of this workand gave the author extremely valuable feedback.2.
Proof of theorem 1
We start by analyzing special Steiner systems of bi-elliptic curves. Belowwe denote the ramification (resp. branch) locus of a map − by R − (resp. B − ). Theorem 3 (Coble, [Co] chapter IV) . Let π : C be → E be a bi-elliptic doublecover where the genus of C be is . We first note that C be is not hyperelliptic.Let π be the involution induced by π on the dual canonical system | K C be | ∗ (which is a projective linear involution). Then the fixed spaces of π are ahyperplane H π , and a point which we call the focal point . Identifying C be with its canonical image, the intersection H π · C be is the ramification divisor R π , which is comprised of six distinct points; the image of C in H π underthe projection from the focal point is E embedded as a plane cubic; H π isnaturally identified with a g on E , denoted by | L | ∗ which satisfied L ∼ B π .Conversely, the curve C be can be reconstructed from such data of E , B π (comprised of six distinct points), and L in the following way: Construct P as a cone over | L | ∗ , and in it reconstructs C be as the intersection of twosurfaces: • the cone ˜ E over the image of E in the linear system | L | ∗ , and, • a quadric surface ˜ Q in P ramified over | L | ∗ at the unique conicsatisfying intersecting E at B π . DAVID LEHAVI
Proof.
We first prove, arguing by contradiction, that C be is not hyperelliptic.Suppose it is, and denote the hyperelliptic system on it by | H C be | . Since thehyperelliptic involution commutes with all other involutions, we would havetwo double covers f : | H C be | ∗ → P and g : E → P such that C be = | H C be | ∗ × P E . However, we then have the following inequality in Div E : B π ≤ f ∗ B g , which is impossible since deg B g = 2 ⇒ deg f ∗ B g = 4, while deg B π = 6.Note that since π is a double cover, B π is a sum of six distinct points.We proceed to analyze the action of π : As K E is trivial, the image of R π in | K C be | ∗ is cut out by some hyperplane H π . Moreover, since C be does notadmit a g , at most 3 of the 6 points of R π are collinear. Since 6 points on H π , no 4 of which are collinear, are fixed by π , the entire plane H π is alsofixed by π .Recall that a projective linear involution which admits a fixed hyperplanealso admit a fixed point out of this hyperplane. We call this point the focalpoint and project the dual canonical system | K C be | ∗ from this point to H π .As π is linear, all the lines through the focal point (and some point on H π )are π invariant. Hence, the intersections of the canonical image of C be withthese lines are exactly the fibers of π , and the degree 2 map C be → H π factors through a map E → H π ; moreover, as C be is a degree 6 embedding,the induced map E → H π is a degree 3 embedding. Setting L ∈ Div E tobe the (pullback to E under an the embedding of the) intersection of theimage of E in H π and some line in H π , we may identify H π with the dualcomplete linear system | L | ∗ . Moreover, since the images of the B π in | L | ∗ isthe intersection of the image of E in H π , and the quadric surface Q C be , theysit on a conic in H π ; thus, we have 2 L ∼ B π in Pic E . Expressing | K C be | ∗ as a cone over | L | ∗ , we see that C be is the intersection of the cone over theimage of E in | L | ∗ through the focal point, and the quadric surface ramifiedover | L | ∗ at the unique conic passing through the images of B π there. (cid:3) Henceforth, we will identify E with it’s image in | L | ∗ , and C be with it’scanonical image. We now turn to the identification of some of the thetahyperplanes of C be : Proposition-Definition 4.
Assuming C be does not admit a theta null,there are exactly 24 theta hyperplanes of C be invariant under the bi-ellipticinvolution. They are given as follows: Denote the 6 distinct points of B π by b , . . . , b ; for each b i let x i , x i , x i , x i be the four points in E satisfying2 x ij + b i ∼ L . For each i, j we denote by l ij the line which satisfies l ij · E =2 x ij + b i . Let H ij be the pullback of l ij to | K C be | ∗ under the identificationof | K C be | ∗ as a cone over | L | ∗ , then H ij · C be = π − (2 x ij + b i ) . This proposition is an explicit form of proposition 2 in [B], where thegenera of the curves involved are 4 and 1.
ECONSTRUCTION OF GENUS 4 CURVES FROM THETA HYPERPLANES 5
Proof.
Identifying C be with it’s canonical image we have H ij · C be = 2 π − x ij + π − b i ; hence H ij represents an effective theta characteristic. Since we as-sume that C be does not admit a theta null, H ij is a theta hyperplane.Conversely, assume that H is a theta hyperplane invariant under the bi-elliptic involution, and let l be the projection of H to the linear system | L | ∗ ,then the following properties hold: • If some b i satisfies E · l > b i , then H · C be > π − ( b i ), which is theramification point lying over b i – with multiplicity 2. • If some point y (cid:54) = b . . . , b satisfies l · E > ay for some positive a ,then both points in π − ( y ) are in the intersection product H · C be ,each with intersection multiplicity a .Thus, if the intersection product l · E contains B branching points and n other points with positive intersection multiplicities a k , then2( B + a + · · · + a n ) = C be · H = 6 , where all the a i s are even . The case where n = 0 is the case where Q C be ∩ H π contains the line l , whichimplies that this intersection is the union of two lines, which implies that Q C be is singular, which implies that C be has a theta null. Whence, we haveonly one possible solution: B = n = 1 , a = 2. (cid:3) Proposition 5.
Let β be a non trivial torsion point in J E , then for each i = 1 , . . . , and j = 1 , , , there is some j (cid:48) such that x ij − x ij (cid:48) = β . Thisis a pairing on the x ij s, which induces a pairing on the H ij s. Moreover,denoting by π ∗ the map Pic( E ) → Pic( C be ) induced by π , we have π ∗ x ij − π ∗ x ij (cid:48) = π ∗ β ∈ J C be [2] (cid:114) { } .Proof. For each i, j the shifts of x i,j by the four points of J E [2] give thefour x i,j † , where j † = 1 , , ,
4. Hence, β induces a natural partition to pairs x ij , x ij (cid:48) , so that x ij − x ij (cid:48) = β for all i, j . The second part follows since π ∗ is an embedding. (cid:3) In the proof of Corollary 8 below, as well as in the proof of Theorem 1we will apply a restricted form the following:
Proposition 6 (Degeneracy loci of maps between vector bundles) . Thedegeneracy loci of a map between two vector bundles over a base scheme areclosed subschemes of the base.
This proposition, as well as the proof, are classical. The propositionfollows from a choice of a local basis to the bundles – which is possible sincethe statement of the proposition is local, from induction on the degeneracyrank, and from the fact that the determinant is trivial on a closed subscheme.As we just indicated, we will not apply the full strength of Proposition 6,but rather the following corollary:
DAVID LEHAVI
Corollary 7.
Let V /X be a vector bundle over a base X , and let V , . . . V n be sub-bundles of V . Then the function dim (cid:104)V | x , . . . , V n | x (cid:105) is lower semi-continuous on X , and the function dim( ∩ ni =1 V i | x ) is upper semi-continuouson X .Proof. By induction it suffices to prove this claim for two sub-bundles. ByProposition 6 the degeneracy locus of the map V ⊕ V → V is a closedsubscheme of the base. Since the degeneracy loci of both V → V and V → V are empty by definition (as they are sub-bundles), we are done. (cid:3) Lemma 8.
Let
E, L, β, b i , . . . b , and x ij be as above and generic, then thetwelve reducible conics l ij ∪ l ij (cid:48) in | L | ∗ span the dimensional space |O | L | ∗ (2) | . Note that Lemma 8 has nothing to do with C be ; it is a statement aboutplane cubics per se. Proof.
We apply the lower semi-continuity of the dimension of span fromCorollary 7. We consider the degenerated case where E is a nodal cubic,and β is degenerated to the trivial 2-torsion point. As a model for the nodalcubic we will use the plane cubic E given by the nulls of x + y − xyz . Theisomorphism between C ∗ and E is given by φ : t (cid:55)→ ( − t : t : 1 − t ) . The generic “combinatorial” scenario, where we have 24 points x ij coming inquartets indexed by the j coordinates, where the differences x ij − x ij are thefour 2-torsion points and where x ij − x ij (cid:48) = β , degenerates in the E case tothe following scenario: We have 12 “doubled” points x ij which come in pairsindexed by the j coordinate, where x i , x i = φ ( ± t i ) for some t i , and whereeach x ij is paired with itself . As in the generic case the six points b i haveto sit on a conic; however, as for this degenerated case we have b i = φ ( t i ),this constrain now translates to the easier constrain: (cid:81) i =1 t i = 1. Finally,instead of 24 l ij s we now have 12, where each one is “trivially paired” withitself. These 12 l ij s are given by T E ( φ ( t )) =(3 x − yz : 3 y − xz : − xy ) | x = − t,y = t ,z =1 − t =(3 t − t (1 − t ) : 3 t + t (1 − t ) : − t ) ∼ (2 t + t : 1 + 2 t : − t ) , for t = ± t , . . . , ± t . We will show that if t , . . . , t are all distinct, then thespan of the “doubled” T E ( ± t i ) for i = 1 , . . . , |O | P | ∗ (2) | .Squaring the projective linear form T E ( t ) we get (using the lexicographicorder on degree 2 monomials):(4 t + 4 t + t : 2 t + 5 t + 2 t : − t − t : 1 + 4 t + 4 t : − t − t : t ) . Let v t :=(4 t + 4 t + t , t + 5 t + 2 t , − t − t , t + 4 t , − t − t , t )=(4 t + t , t , − t , t , − t , t ) + t (4 t : 2 , − t + 2 t , t , − t , , ECONSTRUCTION OF GENUS 4 CURVES FROM THETA HYPERPLANES 7 then our aim is to show that the evaluations of v t at ± t , ± t span the 6dimensional affine space H ( O | L | ∗ (2)). Taking t (cid:54) = 0 , ∞ , the linear span of v t i , v − t i is equal to the linear span of ( v t i + v − t i ) , ( v t i − v − t i ). Denoting s = t , the vectors ( v t i + v − t i ) , ( v t i − v − t i ) are given by(4 s + s , s , − s , s , − s, s ) , t (2 s , , − s + s , s, − s , s + s , s , − s , s , − s, s ) at five distinct values of s .Performing the column operationscol (cid:55)→ col + 4col , col (cid:55)→ col + col , col (cid:55)→ col − on this matrix we get the matrix whose rows are the evaluations of the vector( s , , − s , , − s, s ). Except for the second (trivial) column, and up topermutations and sign changes of the columns, this matrix is a Vandermondematrix; thus it is of degree 5 for any 5 distinct values of s . Since the rowsof the last matrix span the kernel of the operator v (cid:55)→ v (0 , , , , , t ,retracing the column operations we performed on the original matrix, we seethat the span of the rows is the kernel of the operator v (cid:55)→ v (0 , , , , , − t .Finally, as t (2 s , , − s + s , s, − s , , , , , , − t = t (cid:54) = 0 generically , we see that for five distinct non zero values of t , the projective space spannedby the squares of the 10 of the l ij s is 5 dimensional. (cid:3) Corollary 9.
Let C be , E, b i , L, β, x ij be generic as above. Then the map |O | L | ∗ (2) | → P V C be [ q ] (cid:55)→ [ the cone over q through the focal point ] is an embedding into P V C be ,π ∗ β , whose image is spanned by the twelve q { θ,θ + π ∗ β } ’scorresponding to the partition, described in Proposition-Definition 4, of the hyperplanes H ij to twelve pairs.Proof. The map is an embedding since | K C be | ∗ is a cone over | L | ∗ . ByLemma 8, |O | L | ∗ (2) | is spanned by the twelve reducible conics l ij ∪ l ij (cid:48) ; henceby Proposition-Definition 4, the image is spanned by the twelve q { θ,θ + π ∗ β } sdescribed above. Finally, the twelve q { θ,θ + π ∗ β } ’s lie in P V C be ,π ∗ β by theirdefinition; hence, so does their span. (cid:3) Remark 10 (An alternative view of Corollary 9) . As Q C be contains thecurve C be , projecting P V C be ,π ∗ β from [ Q C be ] we get P V C be ,π ∗ β . Composingthis projection on the embedding from Corollary 9, we get an isomorphismbetween |O | L | ∗ (2) | and P V C be ,π ∗ β . In fact, more is true: in [LR] (Theorem2.9 in the journal version, or 2.15 in the arxiv version) it is proved, by acareful analysis of Coble’s construction, that there is a natural isomorphismbetween |O | L + β | ∗ | and |O | K Cbe + π ∗ β | ∗ | . DAVID LEHAVI
Proof of part 1 of Theorem 1.
In Corollary 9 we showed that for C be , E, β as in the corollary there are exactly twelve quadrics q θ,θ + π ∗ β passing throughthe focal point, and that these quadrics span a five dimensional subspace of P V C be ,π ∗ β . As the Steiner system Σ C be ,π ∗ β admits 56 −
24 = 32 theta hyper-planes which do not pass through the focal point of the involution, there are16 quadrics q θ,θ + π ∗ β which do not sit on the 5 dimensional space spannedby the 12 quadrics from 4. Hence, dim span( { q { θ,θ + π ∗ β } } θ ∈ Σ Cbe,π ∗ β ) ≥
6. ByCorollary 7 we see that dim span( { q { θ,θ + α } } θ ∈ Σ C,α ) ≥ { q { θ,θ + α } } θ ∈ Σ C,α ) ⊂ P V C,α , which is 6 dimensional, we see that for a generic curve C , dim P V C,α =6. (cid:3)
To prove the second part of theorem 1 we analyze the mutual structureof several Steiner systems on one curve. As we have already analyzed (someof) the structure of a specific Steiner system on a bi-elliptic curve, we willintroduce and analyze below a special genus 4 curve with many bi-ellipticinvolutions; this would immediately give us many Steiner systems on thiscurve, of the form already analyzed above.
Proposition-Definition 11 (Kuribayahi and Kuribayashi, see [KK] Propo-sition 2.4(f)(1)) . There is exactly one genus 4 curve, denoted here by C × ,whose automorphism group is isomorphic to ( Z / (cid:110) D . Endowing thedual canonical system of C × with the coordinates x , . . . x , and present-ing the automorphism group as automorphism of the canonical system, thegroup is generated by the automorphisms: τ = ( x : x : x : x ) (cid:55)→ ( ωx : ω x : x : x ) , where ω = 1 ,τ = ( x : x : x : x ) (cid:55)→ ( − x : − x : x : x ) , and τ = ( x : x : x : x ) (cid:55)→ ( x : x : x : x ) . Proof.
See [KK]. Verifying that the automorphism group contains a groupisomorphic to ( Z / (cid:110) D , which is all we need for our purpose here, isimmediate using the explicitly model in Corollary 12 below. (cid:3) Corollary 12 (Swinarski, see [S]) . The canonical image of the curve C × ,in the coordinates of 11, is the intersection of the following cubic and quadric: x − x + x − x , x x + x x . Proof.
Direct verification. (cid:3)
Proposition 13.
The quotient of C × under τ is elliptic. Denoting thequotient by τ by π τ and using our notations from Theorem 3, the focalpoint is (1 : 1 : 0 : 0) and H π τ = { ( a : − a : b : c ) | a, b, c ∈ C } . The imageof the elliptic quotient by τ in H π τ is given by the null set of (2 a ) + b − c − a ) bc .Proof. The claim about the invariant sub-spaces follows immediately by di-rect verification. By Corollary 12, C × sits on the null set of the cubic ECONSTRUCTION OF GENUS 4 CURVES FROM THETA HYPERPLANES 9 surface x − x + x − x − x − x )( x x + x x ) = ( x − x ) + x − x − x − x ) x x . The plane cubic curve is the intersection of this cone and the invarianthyperplane; i.e. in the plane coordinates given above it is the curve(2 a ) + b − c − a ) bc. (cid:3) Proof of part 2 of Theorem 1.
By definition, for any
C, α we have[ Q C ] ∈ ∩ α ∈ JC (cid:114) { } P V C,α . As V C,α are sub-bundles of V C over the moduli of C, α , Corollary 7 tells usthat in order to proof part 2 of Theorem 1 we need only prove it for thecurve C × .If σ is an involution of C × , we will denote by α σ one of the three 2-torsion points in J C × [2] (cid:114) { } invariant under the involution: these are thepullback of the three non trivial 2-torsion points of the quotient by σ , whichis an elliptic curve (“morally”, we don’t care which one of the three we pickis that as we are dealing with second symmetric products of linear systems).By Corollary 9, (and using the notation L as in the corollary), the space V C × ,α τ is spanned by the pullback of S H ( L ) to S H ( K C × ) = V C × under the presentation of | K C × | ∗ as a cone over | L | ∗ , and a quadric formin V C × ,α τ which is not trivial on the focal point; as the quadric Q C × does not pass through the focal point, and is in V C × ,α τ , we may choosethis quadric form to be x x + x x .In the notations used in 11, L is spanned by x − x , x , x ; hence: V C × ,α τ = (cid:104) x x + x x , ( x − x ) , ( x − x ) x , ( x − x ) x , x , x x , x (cid:105) = (cid:104) x + x , ( x − x ) x , ( x − x ) x , x x , x , x x , x (cid:105) . By symmetry, V C × ,α τ τ = (cid:104) x + ω x , ( x − ωx ) x , ( x − ωx ) x , x x , x , x x , x (cid:105) . In order to compute the intersection of these spaces, we observe that V C × may be broken to the direct sum of the following spaces:( (cid:5) ) (cid:104) x , x (cid:105) ⊕ (cid:104) x x , x x (cid:105) ⊕ (cid:104) x x , x x (cid:105) ⊕ (cid:104) x x , x , x x , x (cid:105) . Considering the spanning elements (indeed - bases) we chose for the spaces V C × ,α τ and V C × ,α τ τ , it is clear that both spaces are direct sums of theirintersections with the components in equation ( (cid:5) ), that both intersectionswith the last component in equation ( (cid:5) ) form the entire last component,and finally that the intersections of these spaces with each of the otherthree components in equation ( (cid:5) ) are one dimensional and different fromeach other. Hence we have V C × ,α τ ∩ V C × ,α τ τ = (cid:104) x x , x , x x , x (cid:105) = (cid:104) x x + x x , x , x x , x (cid:105) . To conclude the proof, note that the intersection of this space with it’s imageunder τ is generated just by x x + x x , and recall that the space generatedby x x + x x is by definition in the intersection of all the V C × ,α ’s. (cid:3) Proof of theorem 2
Lemma 14.
Let X be the intersection of the image of | K C | × | K C | in P V C with P V C,α , then the two dimensional fibers of the projection on the firstcoordinate of the pullback of X to | K C | × | K C | lie exactly over the modulipoints which represent the four hyperplanes through the four possible tripletsof the blow-downs of the strict transforms in S of the four lines l , . . . , l (see Wirtinger’s Theorem in the introduction regarding S and the l i s).Proof. Note that the isomorphism |O S (3 H − E ) | ∼ = | K C | from Wirtinger’stheorem induces an isomorphism between quadric forms over | K C | ∗ , andsextic forms over | K C + α | ∗ containing the six intersection points l i ∩ l j .Moreover, if we let L i be a linear form on | K C + α | ∗ whose null set is l i , thenthis map sends any quadric form on | K C | ∗ which is zero on the four nodesof W C,α to a product of the four L i ’s and a quadric in | K C + α | ∗ .We write the cubic W C,α as the null set: W C,α = Z ( h h h + h h h + h h h + h h h ) , where h , h , h , h are linear forms such that the four points ∩ ≤ i ≤ ,i (cid:54) = j Z ( h i )– for j ∈ { , , , } – are the four nodes of W C,α . We endow the space V C with the coordinates { h i h j } ≤ i ≤ j ≤ . Then the quadrics passing through thefour nodes of W α are exactly the null sets of quadric forms in the span ofthe six forms: { h i h j } ≤ i Pulling the image of the intersection back to | K C | × | K C | (under themap s ), we see that the projection on the first coordinate is a birationalisomorphism between the intersection and | K C | , where the exceptional fibersare of two types: • The fibers over each of the moduli points [ h i ] is a P . • The fiber over each point in the six lines span ([ h i ] , [ h j ]), is a P . (cid:3) Proof of theorem 2. By Lemma 14 the four hyperplanes through triplets ofthe four nodes of W C,α are exactly the projective points represented by a ∈ H ( K C ), such that for all non trivial v ∈ ( V C /V C,α ) ∗ ⊂ V ∗ C = S ( H ( K C )),the null spaces of the operators H ( K C ) → C x (cid:55)→ v · ( a ⊗ x ) , where · is tensor contractionare identical. Equivalently, this means that any two vectors in ( V C /V C,α ) ∗ · a are dependant I.e. the solutions are the solutions of the equation:(( V C /V C,α ) ∗ · a ) ∧ (( V C /V C,α ) ∗ · a ) = 0 , However, since contraction commutes with tensor products (and wedge prod-ucts, which may be viewed as tensor products followed by projections to asubspace),(( V C /V C,α ) ∗ · a ) ∧ (( V C /V C,α ) ∗ · a ) = (( V C /V C,α ) ∗ ∧ ( V C /V C,α ) ∗ ) · ( a ⊗ a ) . Hence the solution set is the intersection of P (( V ∗ C / (( V C /V C,α ) ∗ ∧ ( V C /V C,α ) ∗ )) ∗ ) , and the image of | K C | ∗ under the 2nd Veronese map. Finally, Since thedegree of the 2nd Veronese of a 3 dimensional space is 2 = 8, and since byTheorem C in [Ca], the Galois group acting on the four nodes of W C,α is S ;hence, each of the intersection points has multiplicity 2. 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