The surface of Gauss double points

We study the surface of Gauss double points associated to a very general quartic surface and the natural morphisms associated to it.

0. Introduction 0.1. The result. We work over C, the complex number field. In this paper X ⊂ P 3 is a very general quartic surface. In particular there are no lines inside it. We explicitly describe a singular surface Σ dou naturally associated to X. To the best of our knowledge Σ dou has not been studied yet. In order to introduce and to study Σ dou we revise some aspects of the geometry of quartic surfaces. We ground our exposition on [6]. Indeed [6] contains also both a well written modern account about the projective Gauss map and a very nice description of the geometry of a general quartic surface.
Given the surface X, it is well known that its bitangents describe a smooth algebraic surface S inside the Gassmannian G(2, 4) of lines of P 3 . If Q S is the restriction to S of the universal bundle Q over G (2,4), it is also wellknow that inside its Grothendieck's projectivisation P(Q S ) we can define the surface Y ⊂ P(Q) of contact points associated to S. Indeed a point y ∈ Y is a couple y = ([l], p) where l is a bitangent line of X and p ∈ l ∩ X. The geometry of Y is quite well-known; c.f. see: Proposition 3.1.4 and the references there. Together with S and Y there is a third surface which naturally comes with the geometry of X. To describe it, we consider a point p ∈ X and we denote by T p X the projective tangent space of X at p. It is known that the closure of the loci of those points p ∈ X such that the hyperplane section X p := T p X ∩ X has geometrical genus strictly less than 2 is a 1-dimensional subscheme C dou inside X; c.f. see: Theorem 2.5.2. We will see that for a general point p ∈ C dou there exists another unique singular point p ∈ X p such that T p X = T p X. Clearly both p, p ∈ C dou and since p = p there exists a unique line l p,p := p, p ⊂ P 3 passing throught p, p . Clearly [l p,p ] ∈ S ⊂ G. The new singular surface we have mentioned above is the locus Σ dou ⊂ P 3 obtained by the closure of the surface swept by the lines l p,p where p ∈ C dou is a general point. We give a description of the geometry of Σ dou . The projective bundle P(Q S ) is easily seen to coincide with the following incidence variety: {([l], p) ∈ S × P 3 | p ∈ l}. One of the most basic morphisms associated to the geometry of X is the forgetful one: f : P(Q S ) → P 3 , ([l], p) → p Nevertheless it has not been yet deeply studied. We denote by B(f ) → P 3 its branch locus and by R(f ) → P(Q S ) its ramification one. We prove that there exists a curve C ⊂ S which parameterises the bitangent lines associated to the couples p, p ∈ C dou where T p X = T p X. Geometrically C ⊂ S is birational to the quotient of C dou ⊂ X by the natural involution p → p where [l p,p ] ∈ S.
In particular we can construct the ruled surface π C : Σ → C obtained by the pull-back of the natural morphism π S : P(Q S ) → S via the natural inclusion C → S. We can prove: Main Theorem. If X is a very general quartic surface then S and Y are smooth surfaces and C dou , C ∨ dou are singular curves whose singularities are fully classified. The morphism f : P(Q S ) → P 3 is finite of degree 12. The almost ruled surface of Gauss double points Σ dou has degree 160 and osculates the quartic X along C dou . It holds: Moreover the natural morphism f : P(Q S ) → P 3 induces by restriction two morphisms ρ : Y → X and ρ dou : Σ → Σ dou of degree respectively 6 and 1.
For the proof of the Main Theorem; see: Subsection 3.6. Actually we need a detailed description of the geometry associated to the curves C dou , C ∨ dou . This geometry is shown in Proposition 1.3.2. The proof of Proposition 1.3.2 is postponed in Subsection 3.5.
Finally we like to mention that this work was originally motivated by Diophantine problems: smooth quartic surfaces are particularly interesting in Diophantine geometry, since they lie at the frontier between rational surfaces, for which the distribution of rational points is well understod, and surfaces of general type, for which it is conjecture that their rational points are never Zariski-dense; for quartic surfaces defined over a number field, it is widely believed that their rational points become Zariski-dense after a suitable finite extension of their field of definition, but this is proved only in very particular cases, and no example with Picard number one is known where this density can be proved.
Quartic surfaces are limiting cases also for the problem of integral points on open subsets of P 3 : the complement of a surface of degree ≤ 3 in P 3 is known to have potential density of integral points; for complements of smooth surfaces of degree five or more, Vojta's conjecture predicts degeneracy (but no case is known for the complement of a smooth surface). The complement of quartic surface should have a potentially dense set of integral points, but again this is still a widely open problem. Again by Vojta's conjecture, we expect that removing the union of a quartic surface and any other surface from P 3 produces an affine variety with degenerate sets of integral points. As a by product of this work and our previous work [1], we could, for instance, deduce the finiteness of the set of integral points, over every ring of S-integers, on the complement of the union of a quartic surface X and its associated surface Σ dou to be described in the present work.
We intend to devote a future paper to the arithmetic applications of these geometrical investigations.

The morphism of bitangents
We agree that a general point on a variety M satisfies a property P, if there exists an open dense subset of M satisfying the property P and that a very general point on M satisfies a property P , if there exists a countable union Z of closed proper subsets of X such that all the points outside Z satisfy the property P.
In this section we introduce some of the geometrical objects which appear in the statement of the Main Theorem.
Let V be a complex vector space of dimension 4 and let V ∨ its C-dual. We set P 3 := P(V ∨ ). Let F ∈ Sym 4 V and let X := (F = 0) ⊂ P 3 be the associated quartic surface. In this paper, unless otherwise stated, we assume that X is a very general quartic. Indeed we need to use Yau-Zaslov formula; see [2,Formula: 13.4.2].
The surface X comes naturally with three other surfaces we are going to describe.
1.1. The surface of bitangents. Let G := G(2, V ∨ ) be the Grassmann variety which parameterises the lines of P 3 . Definition 1.1.1. A line l ⊂ P 3 is a bitangent line to X if the subscheme X |l → l is non reduced over each supporting point.
We denote by S ⊂ G the scheme parameterising bitangent lines of X; that is: the variety of bitangents to X.
1.2. The surface of contact points. We have the standard exact sequence of vector bundles on G: We denote by P(Q) the variety Proj(Sym(Q)). By definition P(Q) coincides with the universal family of lines over G: We denote by π G : P(Q) → G the natural projection and following the mainstream we call the universal exact sequence the following one: By the inclusion j S : S → G we can define Q S := j S Q. It remains defined the variety of contact points.
the variety of contact points.
Obviously there is an embedding j Y : Y → P(Q S ) and the natural morphism π S : P(Q S ) → S restricts to a morphism π : Y → S which we call the forgetful morphism.
1.3. The double cover subscheme. We will need to introduce the subscheme of X given by those points p such that the restriction of X to the tangent plane of X at p is a curve X p of geometrical genus less than 2. It is a 1-dimensional subscheme C dou inside X. Following the literature: the double cover subscheme of X.
We will see in Proposition 2.5.1 and in Proposition 2.5.3 that if X is general then C dou is an irreducible singular curve.
1.3.1. The double cover subscheme and the Gauss map. On X it is defined the Gauss map φ Gauss : X → (P 3 ) ∨ , which is a morphism. We set Let ν dou : C dou → C dou and respectively ν ∨ dou : C ∨ dou → C ∨ dou be the normalisation morphisms. The following Proposition is of interest in itself: such that the following diagram is commutative: dou is a 2-to-1 branched covering induced by the restriction of π : Y → S to C dou .
For the proof of the above Proposition 1.3.2; see: Subsection 3.5.
1.4. The almost ruled surface of Gauss double points. We will need to consider also a third surface naturally associated to X. We have found no reference on it. We will see that for a general point p ∈ C dou there exists another unique singular point p ∈ X p such that T p X = T p X. Clearly both p, p ∈ C dou and since p = p there exists a unique line l p,p := p, p ⊂ P 3 passing throught p, p . We will show that [l p,p ] ∈ j ∨ dou C ∨ dou ⊂ S ⊂ G. On the other hand the natural morphism ρ P 3 : P(Q) → P 3 if restricted to P(Q S ) gives a morphism f : P(Q S ) → P 3 . We will show that the image is a surface inside P 3 which is the closure of the surface swept by the lines l p,p as p ∈ C dou is a general point. Definition 1.4.1. We call the subscheme Σ dou → P 3 given by the closure of the surface swept by the lines l p,p where p ∈ C dou is a general point the almost ruled surface of Gauss double points.
1.5. The basic morphism. Note that geometrically P(Q S ) is easily seen as We have introduced above a basic object of this geometry: the morphism f : P(Q S ) → P 3 . It is obtained by projecting to the second factor and it is called the morphism of bitangents. We want to understand its branch locus subscheme B(f ) → P 3 and its ramification one R(f ) → P(Q S ).
2. Special curves on a quartic surface 2.1. Singularities of a plane quartic. We recall a basic fact on irreducible plane quartic.
Lemma 2.1.1. Let C ⊂ P 2 be an irreducible plane quartic. Then C has at most three singularities. If g(C) = 1 then the following cases occurs (1) a point of multiplicity 3, or (2) a tacnode, or (3) two nodes, or (4) a node and a cusp, or (5) two cusps.
If g(C) = 2 then C has exactly one node or one cusp.

2.2.
Classification of points of a general quartic surface. Let F ∈ C[x 0 , x 1 , x 2 , x 3 ] be as above a general homogeneous polynomial of degree 4 and let X := V (F ) ⊂ P 3 be the corresponding quartic surface. Since X is smooth we can define the Gauss map, which maps a point X p to its tangent space seen as a point of the dual projective space: In the rest of this section we strongly rely on [6]. Actually, by the stability property of the Gauss map, c.f. [6, Section 2.1], we know the analytic behaviour of the Gauss map in an analytic neighborhood of any point p ∈ X. It follows that for any point p ∈ X it holds that dφ Gauss, p = 0; see [6,Proposition 2.15].
Proposition 2.2.1. Let X be a general quartic surface. Then: (1) All tangent curves are irreducible (2) Tangent curves have only singularities of multiplicity 2 (and the second fundamental form is non zero for any p ∈ X). It is interesting to stress here that Proposition 2.2.1 (4) means that if X is a general quartic then there are no elliptic hyperplane sections with two cusps. This should be read together with Lemma 2.1.1 (5). This leads to a full classification of the possible hyperplane sections Proposition 2.2.2. Let X be a very general quartic X ⊂ P 3 . Let p ∈ X and X p = T p X ∩ X. The couple (X p , p) is one of the following types: (1) general case: g(X p ) = 2 and X p has only one node.
(2) simple parabolic point: g(X p ) = 2 and X p has only one cusp (3) simple Gauss double point: g(X p ) = 1 and X p has only two nodes (4) parabolic Gauss double point: g(X p ) = 1 and X p has a cusp on p and a node on another point p = p.
(5) dual to parabolic Gauss double point: g(X p ) = 1 and X p has a cusp on p = p and a node on p.
(6) Gauss swallowtail: g(X p ) = 1 and X p has one tacnode.  2.3. The parabolic curve of a general quartic surface. There are three curves on X which contain important information on X. The first one is the following: Definition 2.3.1. We define the parabolic curve C par ⊂ X to be the ramification locus of the Gauss map φ : X → X * . A point p ∈ X is called parabolic if p ∈ C par . 2.3.1. The asymptotic directions. We recall that locally we can consider a neighbourhood of the point p = (0, 0, 0) with coordinates (x, y, z) ∈ C 3 such that locally T p S is given by (z = 0). Hence the germ of S at p is given In particular the local analytic expression of the Hessian of F is Then each principal direction, that is those giving the tangents to the branches of X p at p, is obtainable by the vector v ∈ T p S such that for the induced quadratic form it holds Hess (S,p) (v, v) = 0. Following a notation coming from differential geometry these two directions are called asymptotic directions at p. In particular if X p has a cusp on p then there exists a unique direction v such that Classification of the parabolic points. By a local analysis it follows that Proposition 2.3.2. Let X be a general quartic. A point p ∈ X is parabolic iff p is a cusp or a tacnode of X p . The parabolic curve C par is the zero locus of the determinant of the Hessian of X. Moreover C par is a smooth element of the linear system |8h|. In particular C par has genus 129.
Proof. We have noted above that C par is the locus where the Gauss morphism is not a smooth one. Hence by the local analytic description of the Gauss map the claim follows. See c.f. [6, Proposition 2.2.4].
2.4. The flecnodal curve. The second curve inside X which is useful to understand the geometry of X is the one containing all the hyperflexes. Definition 2.4.1. A line l ⊂ P 3 is called a hyperflex line if the subscheme X |l → l is supported over a unique point p ∈ X ∩ l. In this case the line l is called an hyperflex line of X at p.
there exists a hyperflex line through it. We define the hyperflex curve C hf ⊂ X to be the reduced scheme of hyperflexes.
We sum up the result on C hf we need: Proposition 2.4.3. Let X be a general quartic. Then it holds: (1) C hf is irreducible; (2) if p ∈ C hf is a general point then X has exactly one hyperflex in p; (3) C hf has geometric genus 201; (4) if p ∈ C hf has two distinct hyperflexes then p is a singular point of C hf ; Proof. See c.f. [ It is well-known that the sheme where ν Cpar|L vanishes is the one given by Gauss swallowtails; see c.f. [6, Remark 2.2.7].
Proposition 2.4.4. Every swallowtail is a simple zero of ν Cpar|L . A point p ∈ C par is a Gauss swallowtail iff p ∈ C hf , that is set theoretically Proof. See c.f. [ 2.5. The double cover curve. By the classification of points of a general quartic surface it is natural to consider the closure of the locus of simple Gauss double points. This gives the third curve on X whose importance to understand the geometry of X has been recognised by many authors; see c.f. [5] and the bibliography of [6]. We have introduced in Definition 1.3.1 the double cover curve C dou ⊂ X as the subset of points x ∈ X such that g(X p ) ≤ 1. By Theorem 2.2.2, as a set C dou consists exactly of simple Gauss points, parabolic Gauss double points, dual to parabolic Gauss double points, Gauss swallowtails and Gauss triple points. Note that there exists a rational involution which sends the node p ∈ X p ∩ C dou to the other node p ∈ X p ∩ C dou if p is a general point of C dou . The computation of the degree of C dou requires a certain amount of work on its image C ∨ dou through the Gauss morphism. Using the apolarity theory we can easily see that letting Pol p (X) the polar cubic surface to X with respect to the point p, we can define the curve . In other words we can interpret geometrically the Gauss morphism as induced by the morphism P 3 → (P 3 ) ∨ given by the sublinear system of the polar cubics. This gives The curve C dou is singular. We sketch the local analysis necessary to understand the local geometry of C dou , but we stress that it requires the deep Yau-Zaslov formula which says that there are exactly 3200 nodal rational curves inside the linear system |O X (1)| if X is general. By Proposition 2.2.2 (7) we know that each one of these rational nodal curves is a tangent section with three nodes. Each node determines the tangent section and since there are only nodes then there are exactly 3 nodes. By our previous notation this means that these 3200 nodal rational curves are exactly the hyperplane sections with Gauss triple points. This implies that there are exactly 9600 Gauss triple points. By the local analysis which uses the local stability of the Gauss morphism we have that C dou and C par intersect only at the 320 Gauss swallowtails, see: Corollary 2.4.5, with multiplicity two and with multiplicity one at the parabolic Gauss double point, see: [6, Proposition 2.5.15]. Hence we obtain that there are 1920 parabolic Gauss double points and so there are also 1920 dual to a parabolic Gauss double point. The above analysis leads to the following: Proposition 2.5.3. The double cover curve C dou ⊂ X is irreducible and it has only ordinary singularities. More precisely (1) C dou has a node at each point of any Gauss triple; (2) C dou has a cusp at each dual to a parabolic Gauss double point; (3) Gauss swallowtails are smooth points of C dou ⊂ X; (4) C dou is smooth at any parabolic Gauss double point. Finally there are precisely 9.600 Gauss triple points and 1920 parabolic Gauss double points. In particular the genus of C dou is 1281. 3. The almost ruled surface 3.1. Numerical invariants associated to the surfaces of bitangents. We need to recall briefly some results mainly taken from [4] and [5]. We stress that we have set P 3 = P(V ∨ ) and that X ⊂ P 3 = P(V ∨ ) is a very general quartic surface. In particular there are no lines contained inside X. The next Proposition is well-known, possibly since very long time ago, but we include a proof of it because in the sequel we need analogue techniques and notation to write the ramification divisor R(f ).
Proposition 3.1.1. The scheme S ⊂ G which parameterises bitangents to a smooth quartic surface X ⊂ P 3 with no lines is a smooth surface.
We consider an open neighbourhood U ⊂ G of [l] and let (u 0 , u 1 , u 2 , u 3 ) be a regular parameterisation of U of [l] inside G; this means that for points [r] close to [l] inside U we can write

We look for conditions on the tangent vector
then it must exist a polynomial q ∈ C[x 0 , x 1 , u 0 , u 1 , u 2 , u 3 , ] of degree at most 2 in the variables x 0 , x 1 with f = q 2 . Since f (x 0 : x 1 ; u 0 , u 1 , u 2 , u 3 , ) = 3.1.1. Basic diagrams. We need a description of S ⊂ G and of its invariants. Denote by H G the hyperplane section of the Plücker embedding G → P( 2 V ∨ ). We have the universal exact sequence, its restriction over S and we stress that O G (H G ) = detQ = detS. Now we consider the standard conormal sequence of X inside P 3 : We partially maintain the notation of [4] to help the reader to check some of our assertions. Following [4] we can build the following diagram: (3.2) J X : P(Ω 1 where the inclusion J X : P(Ω 1 X (1)) → P(Ω 1 P 3 |X (1)) is given by the sequence (3.1) and the morphism P(Q S ) → P(Ω 1 P 3 |X (1)) is the restriction over S of the standard diagram: and ρ : P(Ω 1 P 3 |X (1)) → P(V ∨ ) is the obvious restriction.
3.1.2. Geometrical interpretation. By construction the P 1 -bundle π G : P(Q) → G is the universal family of G, and the P 2 -bundle ρ : P(Ω 1 P 3 (1)) → P 3 is the projective bundle of the tangent directions on P 3 ; that is: ρ −1 (p) = P(T P 3 ,p ) where T P 3 ,p is the vector space given by the tangent space to P 3 at the point p. The isomorphism P(Ω 1 P 3 (1)) ∼ = P(Q) is well-known. We denote by N the divisor on P(Ω 1 P 3 (1)) and by R the divisor on P(Q) such that Since no confusion can arise we denote by R also the restriction to P(Q S ) of R, hence π S O P(Q S ) (R) = Q S We also denote by T the divisor on P(Ω 1 X (1)) = P(Ω 1 X ) such that: Lemma 3.1.2. For the 3-fold P(Ω 1 X (1)) it holds: Proof. Trivial since X is smooth.
By the diagram (3.2) and by a slight abuse of notation we obtain the basic diagram: The fact that Y is a divisor both in P(Ω 1 X (1)) than in P(Q S ) makes possible to link the geometry of X to the geometry of S via the one of Y . In particular it is noteworthy that we can try to obtain special curve on S via special curves on X and viceversa. First we recall that since X is general Pic(X) = [h]Z, where we recall that h := H P 3 |X . Hence 3.1.5. The class of S in the Chow ring of G. It is quite natural to introduce the following classes inside the Chow ring ⊕ 4 i=1 CH i (G) associated to the fundamental ladder, point, line, plane, p ∈ l ⊂ h ⊂ P 3 : It is well known that σ 2 l = σ h + σ p and that the divisorial class of σ l is the class H G .
Lemma 3.1.6. The following identity holds in the Chow ring of G: 3.1.6. Numerical invariants.
3.2. The dual geometry. We consider the following open set of C dou : . Lemma 3.2.1. The restriction of the Gauss morphism to C 0 dou induces a 2-degreeétale covering τ : C 0 dou → (C 0 dou ) ∨ Proof. If p is a simple Gauss double point, it comes together with a unique other pointp ∈ T p X where X p has the other node exactly onp and TpX = T p X; that is φ Gauss (p) = φ Gauss (p). By definition of the double cover curve there are no other points of C 0 dou over φ Gauss (p). We define Σ 0 dou ⊂ P 3 the open surface swept by the lines l p,p := p,p ⊂ P 3 where p ∈ C 0 dou . We notice that: 1) [l p,p ] ∈ S and 2) if p 1 = p then l p 1 ,p 1 ∩ l p,p = ∅. This implies that we can define an injective morphism . By the normalisation morphism ν ∨ dou : C ∨ dou → C ∨ dou we obtain a global morphism generically of degree 1: . In Definition 1.4.1 we have called Σ dou → P 3 the almost ruled surface of Gauss double points.
Lemma 3.2.2. If Σ 0 dou ⊂ P 3 is the projective closure of Σ 0 dou then it holds that Σ dou = Σ 0 dou ⊂ P 3 . Moreover C dou → Σ dou . Proof. By Proposition 2.5.3 C dou is irreducible, then the first claim follows by the universal property of G. By construction it holds that C 0 dou → Σ dou then C dou → Σ dou since C dou is the closure inside P 3 of C 0 dou . Let π C : Σ → C be the ruled surface obtained by the pull-back of π S : P(Q S ) → S via the natural inclusion C → S.
Lemma 3.2.3. The curve C is irreducible and the surface Σ is irreducible.
Proof. By Proposition 2.5.3 and by its construction the curve C is irreducible. Since π S : P(Q S ) → S is a fiber bundle then the claim follows.
Consider again the surface Σ dou → P 3 of Gauss double points. Note that the pull-back Σ 0 → (C 0 dou ) ∨ of π C : Σ → C via the inclusion (j 0 dou ) ∨ : (C 0 dou ) ∨ → S is a smooth open ruled surface. Now we obviously have: Corollary 3.2.4. There exists a rational map µ : Σ dou C ∨ dou which induces the natural structure of smooth ruled surface on Σ 0 → (C 0 dou ) ∨ Proof. The pull-back π : Σ → C ∨ dou of π S : P(Q S ) → S via the morphism j ∨ dou : C ∨ dou → S is a smooth surface which is mapped birationally onto Σ dou in a way compatible with the morphism φ Gauss|C dou : C dou → C ∨ dou . Lemma 3.2.5. The almost ruled surface of Gauss double points is irreducible and it osculates X along C dou ; that is the restriction Σ dou|X is a subscheme of X which contains 2C dou .
Proof. By construction we have seen that Σ dou = f (Σ). Then the first claim follows by Lemma 3.2.3. We show that the open ruled surface Σ 0 osculates X along C 0 dou ; but this follows by definition since any fiber l of Σ 0 → (C 0 dou ) ∨ osculates X along the corresponding two simple Gauss points p,p such that X |l = 2p + 2p ∈ Div(l).
3.3. The geometry of the morphism of bitangents. Now we start the study of the morphism f : P(Q S ) → P 3 . Proof. Let p ∈ P 3 such that the f -fiber is of positive dimension. This means that there are infinite bitangent lines through p. Then the polar cubic Pol p (X) has at least a component swept by lines through p. The restriction of Pol p (X) to X is non reduced. This implies that this restriction is a divisor of type 2D + A. Since we are assuming that Pic(X) = [H P 3 |X ]Z, X does not contain curves of degree ≤ 3. Then the only possibility is that D is an hyperplane section. This implies that S contains the rational curve which parameterises the pencil of lines contained in a plane and passing through p. This is a contradiction. Indeed there exists a unamified covering S X → S where S X is an irregular surface; the details of the proof are in [4,Proposition 2.4]and in [4,Proposition 3.1]. Another self-contained proof is in [5,Lemma 1.1]. See also §3 of [1]. Finally by [5,Corollary A. 3,p. 53] it is known that the Albanese morphism alb : S X → Alb(S X ) is a closed immersion. In particular there are no rational curves on S.
3.3.1. The ramification divisor. Proof. By definition f −1 (X) = Y . Since any bitangent line through a point p ∈ X is contained in T p X, any bitangent line through p gives a ramification point for the morphism X p → P 1 given by the pencil of lines inside T p X with focal point p. This and Lemma 3.3.1 imply that the induced morphism ρ : Y → X is finite of degree 6. Since f −1 (X) = Y then Chow groups projection formula implies that 12 Proof. This follows easily by a local count. We use notation of Proposition 3.1.1. We consider a line l ⊂ P 3 which is bitangent to X. W.l.o.g. we can assume that l := (x 2 = x 3 = 0) and that the two points where l is tangent to X are P = (1 : 0 : 0 : 0), P λ (1 : λ : 0 : 0). For a while assume λ = 0. The general point p ∈ l is then given by a parameter t ∈ C and p = (1 : t : 0 : 0). We easily can write the tangent space to P(Q S ) at the point ([l], p). Indeed let x 0 such that (w 1 , w 2 , w 3 ) is a system of local coordinates around the point p ∈ P 3 and let (u 0 , u 1 , u 2 , u 3 ) be a local system of coordinates at the point [l] ∈ G, that is a line r near to l is parameterised by Then (locally) inside P 3 ×G, where we take coordinates (w 1 , w 2 , w 3 , u 0 , u 1 , u 2 , u 3 ) the tangent space T ([l],p) P(Q S ) is (locally) given by since the proof of Proposition 3.1.1. The morphism f : P(Q S ) → P 3 is given by the restriction to P(Q S ) of the natural projection ρ : P(Q) = P(Ω 1 P 3 ) → P 3 , which is, locally, (w 1 , w 2 , w 3 , u 0 , u 1 , u 2 , u 3 ) → (w 1 , w 2 , w 3 ). Now suppose for a while we are in the general case where h(1 : 0) = H(1 : 0 : 0 : 0) = 0 and h(1 : λ) = H(1 : λ : 0 : 0) = 0. We take (w 1 , u 0 , u 1 ) as local coordinates for the threefold P(Q S ) around the point ([l], p) . The matrix of the differential d ([l],p) f is then given by is independent of t. This last condition means that when it is satisfied this occurs for all the points which belong to the bitangent line l. Finally it is a trivial verification on the equation of X to see that generically the condition 0 = det g(1:0) h(1:0) g(1:λ) h(1:λ) occurs iff T P X = T P λ X and X P = T P X ∩ X is a quartic with two nodes respectively on P and on P λ . Analogue computations hold if we are in more special cases  Proof. By Theorem 3.1.7 K S = 3H G|S + σ. Since the first Chern class of Q S is H G|S then by the standard formula of the canonical class divisor we conclude that K P(Q S ) ∼ −2R + π S (4H G|S + σ). By Proposition 3.3.1 we By Proposition 3.3.2 we know that sheaf theoretically f O P 3 (X) = O P(Q S ) (2Y ) and by Proposition 3.1.4 we know that Y ∈ |2R + π S (σ)|. This implies that f (X) − (2R + π S (σ)) ∼ Y . Hence by substitution inside Equation (3.7) we have 3.3.2. The Branch divisor. We now study the branch divisor of f : P(Q S ) → P 3 . The claim follows by a delicate computation on Chow groups. Since there is no possibility of misunderstanding we will indicate the Chow class [A] of the cycle A simply by A. We recall here that a line l inside P 3 is given as l = H P 3 · H P 3 = H 2 P 3 . Proposition 3.3. 5. B(f ) = X + Σ dou Proof. By projection formula . It remains to compute the natural number: f (H 2 P 3 )·π S (4H G|S ). We can find it by divisors restriction on Y . Inside Pic(P(Q S ) the class Y is numerically equivalent to 2f (H P 3 ). Then in the Chow algebra of P(Q S ) it holds that: f (H 2 P 3 )·π S (4H G|S ) = f (H P 3 )·f (H P 3 )·π S (4H G|S ) = 2·f (H P 3 )·Y ·π S (H G|S ). We point out that the number f (H P 3 ) · Y · π S (H G|S ) coincides with the intersection number of the following two divisors on Y : (f (H P 3 )) |Y and (π S (H G|S )) |Y . Now we carry on this computation on Y seen as a divisor inside P(Ω 1 X (1))), where we can use the conversion rules recalled in Lemma 3.1.3. By Proposition 3.1.5 we know that as a class inside Pic(P(Ω 1 X (1))), Y ∈ |6T + 8ρ X (h)|. Then (f (H P 3 )) |Y · (π S (H G|S )) |Y = (6T + 8ρ X (h)) · ρ X (h) · (T + 2ρ X (h)) and since on Pic(P(Ω 1 X (1))) it holds that (ρ X (h)) 3 = 0, T · (ρ X (h)) 2 = 4 and T 2 · (ρ X (h)) = 0 it follows that (6T + 8ρ X (h)) · ρ X (h) · (T + 2ρ X (h)) = 6T 2 · ρ X (h) + 20T · ρ X (h) 2 = 80.
This implies that b = 160 and that X + Σ dou is exactly the divisor B(f ).

3.4.
Geometrical consequences on the geometry of Gauss curves. By Proposition 3.3.5 and by its proof we can complete the geometrical picture behind Proposition 1.3.2.
Lemma 3.4.1. R(f ) = Y + Σ. In particular Σ ∈ |π S (4H G|S )| Proof. Consider the two morphisms f : P(Q S ) → P 3 and π S : P(Q S ) → S. The divisor Y + Σ is a subdivisor of R(f ) and by Lemma 3.3.4 R(f ) ∼ Y + π S (4H G|S ). By construction Σ ∈ |π S (C)|. Hence C is a subdivisor of a divisor D ∈ |4H G|S | but looking to the f -direction, by the same technique used in the proof of Lemma 3.3.5 we have that f (H 2 P 3 ) · π S (4H G|S ) = f (H 2 P 3 ) · Σ hence C ∈ |4H G|S |, and the claim follows.
3.5. The proof of Proposition 1.3.2. By the proof of Lemma 3.4.1 C ∈ |4H G|S | and by Lemma 3.2.3 C is irreducible. Hence by Lemma 3.1.6 and by Theorem 3.1.7 (1) it holds that C is an irreducible curve of arithmetical genus ρ a (C) = 561. By Proposition 2.5.4 we know that C ∨ dou has genus 561. Then the morphism j ∨ dou : C ∨ dou → C ⊂ S is an embedding. By Proposition 3.1.4 we know that π : Y → S is branched on the curve of hyperflexes which is in |2H G|S |. Then an analogue argument shows that j dou : C dou → Y too is an embedding.
By construction we have that the diagram (1.6) is commutative. We have shown Proposition 1.3.2.
3.6. The proof of the Main Theorem. We have shown in Proposition 3.1.1 that S is smooth. By Proposition 3.1.4 we know that Y is smooth. By Proposition 2.5.3 we have a full classification of the singularities of the double cover curve C dou ⊂ X. By Proposition 2.5.4 we have a full classification of the singularities of the dual curve C ∨ dou . By Lemma 3.3.1 the morphism f : P(Q S ) → P 3 is finite of degree 12. By the proof of Proposition 3.3.5 we know that Σ dou is a surface of degree 160. By Lemma 3.2.5 and by the proof of Proposition 3.3.5 we know that Σ dou |X is exactly 2C dou . Finally by proposition 3.3.5 B(f ) = X + Σ dou , by Lemma 3.4.1 R(f ) = Y + Σ, by Lemma 3.3.2 the morphism f |Y = ρ : Y → X is of degree 6 and by the proof of Lemma 3.4.1 and by construction the morphism f |Σ : Σ → Σ dou is birational. We have shown the Main Theorem.