On Morin configurations of higher length

This paper studies finite Morin configurations $F$ of planes in $\mathbb P^5$ having higher length. The uniqueness of the configuration of maximal cardinality $20$ is proven. This is related to the stable canonical genus $6$ curve $C_{\ell}$ union of the $10$ lines of a smooth quintic Del Pezzo surface $Y$ in $\mathbb P^5$ and to the Petersen graph. Families of length $\geq 16$, previously unknown, are constructed by smoothing partially $C_{\ell}$. A more general irreducible family of special configurations of length $\geq 11$, we name as Morin-Del Pezzo configurations, is considered and studied. This depends on $9$ moduli and is defined via the family of nodal and rational canonical curves of $Y$. The special relations between Morin-Del Pezzo configurations and the geometry of special threefolds, like the Igusa quartic or its dual Segre primal, are focused.


Introduction
Let G be the Grassmannian of n-spaces of P 2n+c , c ≥ 1. For any u ∈ G we denote by P u its corresponding n-space and by σ u the codimension c Schubert variety (1.1) σ u := {e ∈ G | P u ∩ P e = ∅}.
A scheme of incident n-spaces is a closed scheme F ⊂ G satisfying the condition This implies P u ∩ P v = ∅, ∀u, v ∈ F . We say that F is complete if the equality holds.
Definition 1.1. A Morin configuration is a complete scheme F of incident n-spaces.
Integral Morin configurations F of planes in P 5 were classified in 1930 by Morin himself if dim F > 0, see [14]. In the same paper the following problem is posed: Problem 1.2. Classify finite Morin configurations of planes in P 5 .
Notice that, as Zak points out, the analogous classification in P 2n+c is elementary in the case 2n + c + 1 = n+2 2 , see [21]. Morin problem in P 5 , which is specially related to hyperkähler geometry, was readdressed in [5] by Dolgachev and Markushevich. They construct and study configurations of minimal cardinality 10 and their families. In [17] O' Grady proved the existence of configurations of cardinality k for any 10 ≤ k ≤ 16. Next he showed that a finite Morin configuration of planes in P 5 has length k ≤ 20 and asked about the missing cases. The main result of [6] is the construction of a finite Morin configuration of planes in P 5 of cardinality 20. In this paper we contribute to Morin problem and to describe the geometry of the configurations in several ways. We work over the complex field, let us summarize our results as follows.
Along the paper we construct in P 5 an irreducible family of Morin configurations F of length k between 11 and 20. This family depends on 9 moduli and defines a divisor in the moduli space of finite Morin configurations. A general configuration has instead length 10. For reasons soon to be evident, the members of our family will be called Morin-Del Pezzo configurations. Relying on the geometry of singular genus 6 canonical curves, we describe these configurations of length k ∈ [11,20]. We prove that any smooth configuration F of length k ≥ 16 is Morin-Del Pezzo and moreover that: Theorem 1.3. Up to Aut P 5 a unique Morin configuration of planes in P 5 exists having maximal cardinality 20.
See sections from 6 to 9. The central core of the paper is dedicated to show several relations connecting Morin configurations of planes to the beautiful geometry of some classical projective varieties. Our methods relies indeed on these relations, which seem to be of independent interest. This includes: (1) The geometry related to a quintic Del Pezzo surface and the Segre primal.
(3) Highly singular canonical curves of genus 6 and possibly higher.
(1) To reasonably summarize these relations and our further work, let us consider a smooth quintic Del Pezzo surface Y ⊂ P 5 . The linear system P 4 Y , of the quadrics through Y , is a 4-space. By the way we prove the following result, see 5.6.
Theorem 1.4. The discriminant hypersurface in P 4 Y is twice the Segre cubic ∆ Y . Then we consider the union of the ten lines of Y . This is a stable canonical curve of genus 6. The linear system P 5 of the quadrics through C is a 5-space and P 4 Y is a hyperplane in it. It is known that the locus in P 4 Y of all quadrics of rank ≤ 4 is union of five planes P 1 . . . P 5 of ∆ Y . Moreover, Sing C is a set of 15 nodes and, for each z ∈ Sing C , the linear system P z := {Q ∈ P 5 | z ∈ Sing Q} is a plane. P z is not in P 4 Y . Let z 1 , z 2 ∈ Sing C , one can show that P z 1 ∩ P z 2 = ∅. Then it is possible to deduce as in section 8 that the mentioned planes define a Morin configuration (1.4) F := {P 1 . . . P 5 , P z z ∈ Sing C }.
In particular, theorem 1.3 can be also stated as follows.
Theorem 1.5. F is the unique Morin configuration of cardinality 20 up to Aut P 5 .
(2) The Lagrangian Grassmannian LG (10,20) and Morin configurations are strictly related. To be more precise let us fix some conventions, to be used throughout all the paper. We assume P 5 = P(W ) and denote the natural symplectic pairing of ∧ 3 W as (1.5) w : Let G ⊂ P(∧ 3 W ) be the Grassmannian of planes of P 5 . As is well known a closed scheme F ⊂ G is a Morin configuration iff its linear span is P(A), where A belongs to the Lagrangian Grassmannian LG(10, ∧ 3 W ) and (1.6) F = P(A) · G.
We fix a point u ∈ G: for any finite F considered u will be a smooth point of F . Let P ⊥ u be the net of hyperplanes through the plane P u , we also fix the notation: (1.7) P 2 × P 2 := P u × P ⊥ u . This paper also relies on the construction, given in section 2, where we associate to a Morin configuration F a hypersurface of bidegree (2,2) in P 2 × P 2 . Indeed, F spans P(A) as above and F is pointed by u. We show that the pair (A, u) uniquely defines a hypersurface V A ⊂ P 2 × P 2 and prove the following theorem. Theorem 1.6. There exists a natural biregular map between F − {u} and Sing V A .
This relates the study of Morin configurations F of higher length to the study of hypersurfaces V of bidegree (2,2) in P 2 × P 2 such that Sing V is finite. In particular let V ⊂ P 2 × P 2 be the threefold associated to the maximal configuration F . In the paper we describe its very interesting geometry as follows.
• V contains a configuration of eight planes a i × P 2 and P 2 × b j , 1 ≤ i, j ≤ 4, such that the sets α = {a 1 . . . a 4 } and β = {b 1 . . . b 4 } are in general position in P 2 .
• Let p : V → P 2 be the first, (second), projection and Γ p ⊂ P 2 its branch sextic. Then Γ p is the union of the singular conics of the pencil whose base locus is α, (β).
Let I be the ideal sheaf of the set {(a 1 , b 1 ) . . . (a 4 , b 4 )} ⊂ P 2 × P 2 . Then |I(1, 1)| defines a degree 2 rational map π : P 2 × P 2 → P 4 , recently considered in [9,3]. Its branch divisor is the Igusa quartic threefold, that is the dual of the Segre cubic. As a consequence of the mentioned results and of our description, it follows: Theorem 1.7. V is the ramification divisor of π and π(V ) is the Igusa quartic.
(3) Going back to the quintic Del Pezzo Y , let C ∈ |C | be a reduced singular curve and P 5 C the 5-space of the quadrics through C. As in the case of C we can reconstruct from Sing C, in the Grassmannian of planes of P 5 C , the family of planes (1.8) where P 1 . . . P 5 are the nets of rank 4 quadrics through Y and P z ⊂ P 5 C is the net of quadrics which are singular at z. The next theorem is proven in section 8. Theorem 1.8. Let Sing C be not in a hyperplane then F C is a Morin configuration.
The special feature of F C is that {P 1 , . . . , P 5 } is a smooth linear section of the Grassmannian G Y of planes of P 4 Y , see [4, 8.5.3]. Then the corresponding points p 1 . . . p 5 only span a 3-space. Since a finite Morin configuration spans a 9-space, it follows that F C has length k ≥ 11 and, moreover, Sing C necessarily spans P 5 C . Definition 1.9. A Morin-Del Pezzo configuration is a finite Morin configuration which contains with multiplicity one a 5-tuple projectively equivalent to {P 1 . . . P 5 }.
In sections 6, 7, 8 we construct an integral family whose members are the Morin-Del Pezzo configurations and describe their properties. Let F = P(A) · G be one of these and V A ⊂ P 2 × P 2 the bidegree (2, 2) hypersurface defined by (A, u). We prove that: F = F C for some C ∈ |C | and that V A contains a plane. Then V A is rational and is reconstructed from C as follows. In the ambient space of C the base locus of the net P 5 is a Segre product P 1 × P 2 . Let J be the ideal sheaf of C in it, then |J (2, 2)| defines a rational map q : P 1 × P 2 → P 2 . Let p : P 1 × P 2 → P 2 be the projection map, then: These results are used to deduce theorem 1.3 and enumerate configurations. This is quickly done in section 9. Then some concluding remarks follow: we note that a stable canonical C of genus g ≥ 7 defines an analogous scheme of incident (g − 4)-spaces in the dual of the space I C of quadrics through C. That is F C := {P z , z ∈ Sing C}, where P z is the orthogonal of I z := {Q ∈ I C | z ∈ Sing Q}. The involved dimensions satisfy the mentioned Zak's equality. This makes interesting the question: (1.9) when F C is a Morin configuration and has maximal cardinality?
Here canonical graph curves, like C , could come into play. These are union of 2g − 2 lines and have 3g − 3 nodes. Each is uniquely defined by its dual associated graph. For C this is the well known Petersen graph.
We discuss some example generalizing C and chances that 3g − 3 be the maximal cardinality. In this paper we also revisit O'Grady's bound for g = 6 and discuss realizations of singular plane sextics as 3×3 determinant of quadratic forms, see section 3, remark 7.12 and [17].
Further notations X linear span of X.
[X] vector space generated by X.

Morin configurations of planes in P 5 and V -threefolds
In this section we start studying finite Morin configurations of planes in P 5 = P(W ). We keep our conventions and begin from the point u, such that [u] = U , we have previously fixed in the Grassmannian G. Notice that u defines a natural filtration of ∧ 3 W , say defined by the wedge product. Notice that W 2 u /W 3 u is naturally isomorphic to the Zariski tangent space to G at u. Hence its projective completion is embedded as . By definition the tangential projection of G from u is the linear map P(∧ 3 W ) → P 9 of center the 9-space T u . We will be more interested to its restriction (2.4) τ : H u → P 8 to the hyperplane H u := P(W 1 u ) ⊂ P(∧ 3 W ). We point out that the target space of τ is . Moreover H u cuts on G the codimension 1 Schubert cycle G u defined by P u , that is It will be useful to describe τ |G u . Let e ∈ G u and e / ∈ T u , then e = [u 1 ∧ e 2 ∧ e 3 ] so that U ∩ E = [u 1 ] and {x} = P u ∩ P e with x = [u 1 ]. This defines a rational map (2.6) γ : G u → P u such that γ(e) := x. Now let P ⊥ u ⊂ |O P 5 (1)| be the net of hyperplanes through P e . It is also clear that e uniquely defines an element y ∈ P ⊥ u . This is the hyperplane in P 5 generated by P u and the points [e 2 ] and [e 3 ]. Notice also that we have This defines a rational map Leaving some details to the reader, we conclude that s is the Segre embedding of P u × P ⊥ u . Hence the next lemma follows. Proposition 2.1. τ : G u → P 8 factors as in the next diagram: Let (x, y) be as above then x = [u 1 ]. Moreover y defines in W the codimension 1 vector space W y = [U, e 2 , e 3 ]. Then the fibre of τ |G u at (x, y) is the family of planes (2.9) {P | x ∈ P ⊂ P(W y )}.
With some more effort, one can show that such a fibre is naturally embedded as the Plücker quadric of the Grassmannian of lines of P(W y /[u 1 ]).
Now we start dealing with maximal isotropic spaces A of w.
Since A is maximal isotropic then ∧ 3 U ⊂ A and u ∈ P(A). The converse is obvious.
Next we fix the following assumptions on the maximal isotropic space A. • Equivalently u ∈ P(A) and the intersection scheme P(A)·G is smooth and 0-dimensional at u. That is a cheap restriction with respect to our goals. We will be mainly interested in the following loci in LG (10,20), to be repeatedly considered.
Under our assumptions P(A) contains u, now we consider the restriction (2.10) τ A : P(A) → P 8 of τ to P(A). Since we have P(A) ∩ T o = {u}, it is clear that τ A is just the projection of P(A) from its point u and that the image of τ A is P 8 . Since A is isotropic, a quadratic section of P u × P ⊥ u is intrinsically associated to A as follows. Let y ∈ P ⊥ u then y = [(e 2 mod U ) ∧ (e 3 mod U )] for some vectors e 2 , e 3 ∈ W − U . It is easy to describe the 3-space τ −1 A (P u × {y}). Indeed, let a ∈ A then, as for any vector of W 1 u , we can write a = a 1 + a 2 + a 3 , where a i is in the image of the previously considered pairing ∧ i U × ∧ 3−i W → ∧ 3 W , i = 1, 2, 3. It is therefore clear that so that a = a 1 + a 2 + a 3 and a = a 1 + a 2 + a 3 , then (2. 12) a ∧ a = a 1 ∧ a 1 + a 1 ∧ a 2 + a 2 ∧ a 1 = 0. Let Since this vector space has dimension 5, its vectors v 1 , v 2 , v 3 , v 1 , v 2 , v 3 are linearly dependent. This implies a 1 ∧ a 1 = 0 so that Let A y ⊂ A be the subspace such that P(A y ) = τ −1 A (P u × {y}) and let a, a ∈ A y . Then the above equality 2.13 defines a symmetric bilinear form (2.14) <, > y : In the same way, putting < a, a > x := a 1 ∧ a 2 , we obtain a symmetric bilinear map We omit some details. Since < a, a > x =< a, a > y , the construction defines a vector v A ∈ H 0 (O Pu×P ⊥ u (2, 2)) whose restrictions to P u × {y} and {x} × P ⊥ u respectively are the quadratic forms <, > y and <, > x .
Following some use we say that a bidegree (2, 2) hypersurface in P 2 × P 2 is a Verra threefold, for short a V -threefold. As we will see, v A is not zero so that div(v A ) is a V -threefold of P u × P ⊥ u . Let us introduce the following definitions. Definition 2.4. V A := div(v A ) is the V -threefold associated to A. From now on we will assume, up to different advice, that F A is finite. Now we describe F A in terms of the singular locus of V A . Let F A := F A − {u} and let e ∈ P(A) − {u}. We consider representations of e as e = [a 1 + a 2 + a 3 ], with a 1 , a 2 , a 3 as above. It is clear that the following condition are equivalent: (1) the line joining u to e intersects F A , (2) a representation of e satisfies a 2 = 0.
Notice also that: To prove that τ |F A is biregular to τ (F A ) consider any scheme ζ ⊂ F A of length 2. We have ζ ⊂ P(A). Moreover the restriction of τ to P(A) is the projection from u. Hence τ |ζ is not biregular to its image iff the line ζ contains u. Since u / ∈ ζ, this is equivalent to say that the scheme ζ · G has length ≥ 3. Then it follows ζ ⊂ G, because G is intersection of quadrics, and hence F A is not finite: against our assumption. This implies the statement. Now let us consider the cone C(F A ) := τ * |P(A) τ (F A ). Then C(F A ) is a cone of vertex o over F A and it is defined by the equation a 2 = 0, that is Of course the condition defines as well the embedding τ (F A ) ⊂ P u × P ⊥ u . Now we study τ (F A ). To this purpose let e = [a 1 + a 2 + a 3 ] ∈ P(A) as usual. If e ∈ C(F A ) then we have τ (e) = (x, y) ∈ P u × P ⊥ u . At first we remark that the condition a 2 = 0 is precisely equivalent to the property that the polar forms (2.17) < ·, a 2 > y : A x → C and < ·, a 2 > x : A y → C of the vector a 1 + a 2 + a 3 are identically zero. This immediately translates in the following simple condition on a point o := (x, y) ∈ P u × P ⊥ u : (2.18) both the planes P u × {y} and {x} × P ⊥ u are tangent to V A at o. Since these planes generate the embedded tangent space in the Segre embedding of In order to have more precision let us write explicitly the equations of τ (F A ). Under our notation we have P u × P ⊥ u = P 2 × P 2 ⊂ P 8 . On P 2 × P 2 we fix projective coordinates (x 1 : x 2 : x 3 ) × (y 1 : y 2 : y 3 ) defining the point (x, y) and then we consider the equation f of V . Therefore we have where the a ij 's are quadratic forms in y. By the condition 2.18 the partials , so the next theorem follows.
Theorem 2.7. F A is biregular to the scheme defined by the above derivatives i.e. to the singular locus of V A .
We are grateful to M. Kapustka for discussions around this result. We remark that τ (F A ) fits in the standard exact sequence of tangent and normal sheaves realizing the singular locus of V A . This complete the proof of theorem 1.6.

V -threefolds with isolated singularities
Continuing in the same vein we consider now a V -threefold V ⊂ P 2 × P 2 such that Sing V is finite. We want to discuss more on Sing V . Let us consider the projections It is clear that R x and R y are the ramification schemes respectively of π y and π x . Their supports are the loci where the tangent maps dπ y and dπ x have rank ≤ 1. Now, in the Chow ring CH * (P 2 × P 2 ), let h x and h y be respectively the classes of the pull-back of a line by π x and π y . Then it is very easy to see that f x,i and f y,i define divisors The next properties we show for R x are true for R y with the same arguments.
Clearly the tangent space T Rx,o is defined by the latter two equations so that dim T Rx,o ≥ 2. By theorem 3.4 the only irreducible surfaces in R x are planes P 2 × y. Since the only one through o is not, it follows that o ∈ Sing R x . 3 of class h x + 2h y . Hence one computes that R x has arithmetic genus 10 and class 6h 2 x h y + 12h x h 2 y in CH * (P 2 × P 2 ). Since Sing V is finite, no component of R x is a fixed component of the net of divisors generated by D y,1 , D y,2 , D y,3 . Hence an element D of this net intersects R x properly and Sing V is embedded in the finite scheme R x · D. By lemma 3.1 each point o ∈ Sing V is singular for R x . Then, since R x · D has length 30 and its multiplicity is Proof. Let f = π y |V . From the finiteness of Sing V and generic smoothness it follows that the discriminant of f is a curve. Assume B is a non reduced, irreducible component of it. Let y ∈ B be a general point then Sing V ∩ f * (y) = ∅. Moreover it follows from [2] that f * (y) is a conic of rank 1. Let S ⊂ V be the closure of the union of the lines f −1 (y), where y ∈ B is general. Then f : S → B red is a P 1 -bundle and f * B red has multiplicity 2 along S. In particular there exists an affine open set U = Spec R ⊂ P 2 so that U ∩B = ∅ and the equation of are forms respectively of degree 1 and 2 in (x 1 , x 2 , x 3 ). Since d / ∈ (b) and V is irreducible, we can assume d = 1 up to shrinking U . Now consider in U × P 2 the set Z = {a 2 = b = c = 0}. It is clear that Z is non empty and hence of dimension 1. Moreover, Z is contained in Sing V : this contradicts the finiteness of Sing V .
It easily follows that the only surfaces possibly contained in R x ∪ R y are planes.
Proof. π y (S) is an irreducible component of the discriminant curve Γ of π y |V . By the lemma Γ is reduced Assume π y (V ) is a curve, then the previous lemma and its proof imply that the general fibre of V over π y (S) is a conic of rank 2. This is impossible because implies dim S = 1. Hence π y (S) is a point and S is a plane, fibre of π y |V . Now we assume that R x contains a plane P := P 2 × o. Proof. We can assume that P = {y 1 = y 2 = 0}. Then the equation of V is: f = q 11 y 2 1 + q 22 y 2 2 + q 12 y 1 y 2 + q 13 y 1 y 3 + q 23 y 2 y 3 . Restricting the derivatives f y,1 , f y,2 , f y,3 to P u we conclude that Hence b is the base locus of the pencil of conics λq 13 + µq 3 = 0.
Remark 3.6. The locus b is the complete intersection {q 13 = q 23 = 0}. Assume for simplicity that b is smooth and let σ : Y → P 2 be the blowing up of o = π y (P ). Then a standard resolution φ :Ṽ → V of Sing V at b is provided by the Cartesian square LetP be the strict transform of P and E = σ −1 (o). Thenπ y (P ) = E and the morphism π y :P → E is defined by the pull-back of the pencil of conics of λq 13 + µq 23 = 0.
It is now useful to define the sets Definition 3.7. t x (V ) and t y (V ) are the cardinalities of P x , and P y .
Lemma 3.8. Both P x and P y have at most four points and no three are collinear.
Proof. Assume P x contains five points o 1 . . . o 5 and let C a conic through these. It is easy to see that then V properly contains P 2 × C: against the irreducibility of V . The same proof applies to V and P 2 × L, where L is a line through three points of P x .
It follows from the previous results that R x and R y admit the decompositions where R 1 x , R 2 y are curves and R 2 x , R 2 y disjoint union of planes respectively of class h 2 x , h 2 y . In what follows we assume, for each plane P in R 2 x , that b = P · Sing V is smooth. Assuming this we now give an alternative proof of O'Grady's bound | Sing V | ≤ 20, [17]. This will be useful for further purposes. Dropping the smoothness assumption for b, one has to extend the argument of the proof to any b which is complete intersection of two conics. Since the bound is known, we avoid to address the singular cases. In particular the next theorem suggests that a V -threefold V such that | Sing V | = 19 has to contain four disjoint planes.
Consider a general member D of the net generated by D y,1 , D y,2 , D y,3 . As in the proof of theorem 3.2, we can assume that D intersects the curve R 1 x properly and that B : x . We can also assume that B is disjoint from R 2 x · R 1 x . Indeed, as follows from remark 3.6, this consists of the singular points of the singular conics through b. Now we consider the intersection scheme D · R x . This is defined by 3 divisors of class h x + 2h y and one of class 2h x + h y . In 3.4 this intersection was proper and hence of length 30. Here it is not proper and B is the excess intersection scheme. Since b is smooth one can check that B is a smooth irreducible component of D · R x . Applying excess intersection formula to B, [11, 6.3], one computes that D · R x = B ∪ Z, where Z has length 24 and B ∩ Z = ∅. Now, arguing as in the proof of 3.4, each o ∈ Z has multiplicity ≥ 2. This implies that the cardinality of Sing V is ≤ 12 + deg b = 16 and proves the statement for t = 1. The argument easily extends to t ≤ 4. Remark 3.10. Consider a general D of the net generated by D x,1 , D x,2 , D x 3 . Then S = D·D is a complete intersection of class 2h 2 x +5h x h y +2h 2 y . Assume S is integral with at most isolated nodes, which is the general case. Then S is a K3 surface through R 1 x ∪B. Let σ : S → S be its minimal desingularization and B and H the pull-back of B and H ∈ |O S (1, 2)|. It is easily seen that Z has length (H − B ) 2 = 30 + B 2 − 2H B = 24. This recovers the above excess intersection formula for B, cfr. [7, 13.3.6].

Highly singular V-threefolds and the Igusa quartic
Before introducing the main family of finite Morin configurations to be considered, and explicitly reconstruct in it the unique one of maximal cardinality, we already use the previous results to describe its associated V -threefold and its relation to a well known threefold in P 4 , namely the Igusa quartic.
Moreover we denote by t(V ) the number of these singularities on V .
Then, by the latter theorem, This gives a constructive way to produce families of finite Morin configurations of higher length in the range 16 ≤ ≤ 20. Indeed, let t := t(V ) ≥ 1 then Sing V is necessarily endowed with a set of tangential singularities such that the projection maps π x : O t → P 2 and π y : O t → P 2 are injective. In particular V contains 2t distinct planes, say Then a set O t as above is {o i := (u i , v i ), i = 1 . . . t}. By lemma 3.8 the sets are sets of distinct points so that no three are collinear. To construct configurations of length ≥ 16 we consider the union of planes and its ideal sheaf I t in P 2 × P 2 . This defines the linear system of V -threefolds The case t = 4 leads to Morin configurations of length ≥ 16, in particular to the maximal one with 20 planes. In what follows we assume t = 4. Since the points u 1 . . . u 4 and v 1 . . . v 4 are in general position, we can fix coordinates (x, y) on P 2 × P 2 so that (u i , v i ) is in the diagonal {x − y = 0}. We can also assume that Let q 1 (x) and q 2 (x) be quadratic forms in x generating the ideal of {u 1 . . . u 4 }. Then q 1 (y) and q 2 (y) generate the ideal of {v 1 . . . v 4 } and the next theorem easily follows.
Let V ∈ |I 4 (2, 2)| be general then Sing V is the set of 16 tangential singulartities Later in this paper we will see that the branch sextic Γ of π y : V → P 2 is the union of three conics of the pencil λq 1 (y) + µq 2 (y) = 0. The most interesting case of V arises when Γ is the union of the three singular conics of the pencil, that is Then it turns out that V has 19 ordinary double points: the 16 tangential singularities and 3 other points, one over each double point of Γ. We will also show that a unique V satisfies | Sing V | = 19 and that it is defined by a complete Morin configuration of 20 planes in P 5 , which is unique as well. For reasons to be made clear in the end of this section, we fix for such a V the notation V ram . Its equation is We continue this section by some constructions useful to put V ram in its due geometric perspective. Let O 4 ⊂ P 2 ×P 2 be the set of four points as above and let B 4 be the linear system of V -threefolds singular at O 4 . We consider the linear projection of center O 4 of the Segre embedding P 2 × P 2 ⊂ P 8 . The map φ is defined by the linear system Then, since the Segre product P 2 × P 2 has degree six, it follows deg φ = 2.
The ramification divisor of φ is strictly related to the subject of this paper and to a very well known threefold and its dual. We recall that the Segre primal is the unique, up to projective equivalence, cubic threefold ∆ whose singular locus consists of ten double points, which is the maximum for a cubic hypersurface with isolated singularities in a 4-space. Of equivalent interest is its dual hypersurface This is in turn a quartic threefold which is very well known. It is the Igusa quartic, see e.g. [3] and [4]. In particular, in the recent paper [3], it is shown that: Now let us consider as in 4.2 the threefold V ram , which defines the unique complete Morin configuration of 20 planes in P 5 . Relying on its equation, and on the equations of φ, it is not difficult to compute the image of V ram by φ and conclude as follows.
Theorem 4.4. V ram is the ramification of φ and φ(V ram ) is the Igusa quartic.

Del Pezzo 5-tuples of planes and the Segre primal
In what follows G P 4 is the Grassmannian of planes of P 4 embedded by its Plücker map, then deg G P 4 = 5. Let us consider any transversal 0-dimensional linear section then h spans a 3-space. It is known that its points are in general position in h .
Definition 5.1. We say that h is a Del Pezzo 5-tuple of planes of P 4 .
All Del Pezzo 5-tuples are projectively equivalent. So it is not restrictive fixing a Del Pezzo 5-tuple of special geometric interest as follows. Let Y ⊂ P 5 be a smooth quintic Del Pezzo surface and I Y its ideal sheaf. Then |I Y (2)| is a 4-space endowed with a natural Del Pezzo 5-tuple: see lemma 5.4. We restart from Y assuming that P 4 is (5.2) H := |I Y (2)| and h is the Del Pezzo 5-tuple considered in lemma 5.4. More precisely H is a 4-space of quadrics of P 5 and the locus of its quadrics of rank ≤ 4 is the union of five nets of quadrics. These planes of H are the elements of h. From now on we fix the notation (5.3) G H and G H * respectively for the Grassmannians of planes and of lines of H in their Plücker spaces. At first we want to describe the discriminant sextic hypersurface of H, that is the scheme of the singular quadrics Q ∈ H. Omitting the most standard steps, let us summarize this description as follows. Consider the correspondencẽ together with its natural projection maps and notice that q 1 :∆ → Y is a P 1 -bundle. Indeed, any projection π z : Y → P 4 from a point z ∈ Y defines an integral complete intersection of two quadric hypersurfaces The pencil of quadrics through Y z pulls back to a pencil of quadrics Proof. The projection from Sing Q defines a rational map f : Y → P r so that f (Y ) ⊂ Q, where Q is a smooth quadric and r ≤ 4. Let Sing Q ∩ Y = ∅ then f is a morphism. But then f (Y ) is a quintic surface in Q, which is impossible. Hence Sing Q ∩ Y = ∅.
Since a general Q ∈ H is smooth then ∆ is a hypersurface and the support of the sextic discriminant of H. The name of ∆ is well known, see [4, 8.5]. Before of coming to it we recall more on its geometry, which is determined by Y . The surface Y has exactly five pencils of conics. Each of these defines a distinct Segre embedding of P 1 × P 2 in P 5 , let us say Σ i is union of the supporting planes of the conics of a pencil. As is well known Let I i be the ideal sheaf of Σ i . Notice also that |I i (2)| is the net of quadrics (5.9) We observe that the previous P 1 -bundle q 1 :∆ → Y defines a morphism (5.11) ι : Y → G H * sending z to the parameter point of the pencil L z . We point out the following: Proof. Let π z : Σ i → P 4 be the projection from z. Since z ∈ Σ i and Σ i is smooth of degree 3 then Q i := π z (Σ i ) is a quadric in P 4 . Let Q i be its pull-back by π z , then Q i is singular at z and contains Σ i . Hence Q i ∈ P i ∩ L z and ι(z) ∈ H 1 ∩ · · · ∩ H 5 .
Let h ⊥ be the orthogonal of the linear span h in the Plücker space of G H * , then Then we can consider ι as a morphism ι : Y → h ⊥ with image in h ⊥ · G H * . The next statement, essentially well known, will be also useful in the next sections. Proof. Consider the Euler sequence of Then its dual defines a monomorphism υ : where we have put S := Sym 2 T P 5 (−1). Let z ∈ Y , then S * |Y,z is the vector space of quadratic forms singular at z and υ z : S * |Y,z → H 0 (O P 5 (2)) is the inclusion map. Now let U be the pull-back by ι of the universal bundle of G H * , observe that P(U) =∆ and that Finally we go back to the hypersurface ∆.
Theorem 5.6. ∆ is the Segre primal and 2∆ is the sextic discriminant of H.
Proof. In the Chow ring of the Grassmannian of lines of a 4-space a 2-dimensional linear section has class (2, 3). Since q 2 :∆ → ∆ is a birational morphism it follows deg ∆ = 3. Since Y h is smooth, it is well known that ∆ is the Segre cubic primal.
Remark 5.7. As remarked the planes P 1 . . . P 5 are in ∆. It is easy to see that a unique quadric Q ij satisfies P i ∩ P j = {Q ij }, i < j. This implies that Q ij ∈ Sing ∆ and describes the ten singular points of ∆. Notice also that Sing Q ij is one of the ten lines in Y and that the obvious map {Q ij , i < j} → {lines of Y } is bijective.
Remark 5.8. The previous statement has somehow a classical flavor, however we are not aware of any reference for it. We thank Igor Dolgachev for his useful comments.

Morin-Del Pezzo configurations
Now we use Y and the natural Del Pezzo 5-tuple of planes P 1 . . . P 5 ⊂ H to describe an interesting family of special Morin configurations. We fix a linear embedding (6.1) H ⊂ P(W ), the choice of it is irrelevant up to Aut P(W ). We fix the notation W Y := H 0 (I Y (2)) so that it follows P(∧ 3 W Y ) ⊂ P(∧ 3 W ) and G H ⊂ G. We will also assume that The space ∧ 3 W Y is obviously isotropic. We recall that a Morin configuration F ⊂ G is, by definition, a configuration of incident planes which is finite and complete. As we know, this is equivalent to say that F is finite and, moreover, that there exists a maximal isotropic space A ∈ LG(10, ∧ 3 W ) such that F = P(A) · G and F = P(A). Definition 6.2. Let F be a Morin configuration: we say that F is a Morin-Del Pezzo configuration if F contains h and h · G = h.

Let us point out that
This follows because, counting dimensions, the intersection L ∩ G H is not finite for any space L ⊂ P(∧ 3 W Y ) which contains h properly. Let then the condition h · G = h is equivalent to say that (6.5) F = F ∪ h.
We fix the notation F for the subscheme of F occurring in this decomposition.
Remark 6.3. In this part of the paper we describe Morin-Del Pezzo configurations and give a method for their explicit construction in any possible length. As we will see, these configurations are strictly related to the family of V -threefolds containing a plane and to the Severi variety of quadratic sections C of Y such that Sing C has length ≥ 6. We stress however that our construction only gives Morin configurations of special type. The reason is that h spans a 3-space. Since any Morin configuration spans a 9-space, otherwise it is not complete, it follows that the length of F ∪ h is at least 11, while a general configuration has length 10. Nevertheless this construction recovers most families of Morin configurations for any length k ∈ [11,20]. As we will see, the family of Morin-Del Pezzo configurations is irreducible and depend on 9 moduli.
To begin let us fix since now a vector f / ∈ W Y and the decomposition where F is generated by f . Moreover we fix the identification So far we then have In particular any two vectors v, v ∈ ∧ 3 W are uniquely decomposed as v = a + f ∧ b and v = a + f ∧ b , where a, a ∈ ∧ 3 W Y and b, b ∈ ∧ 2 W Y . Therefore we have Notice that w is induced by the natural pairing ∧ 3 W Y × ∧ 2 W Y → ∧ 5 W Y , up to a non zero factor the choice of f is irrelevant. The proof of the next lemma is immediate.
Lemma 6.4. The subspaces ∧ 3 W Y and ∧ 2 W Y are isotropic spaces of w.
Let r : ∧ 3 W → ∧ 2 W Y be the map sending a + f ∧ b to b, then r has a geometric meaning. Indeed r defines the projection of center P(∧ 2 W Y ) (6.9) r : P(∧ 3 W ) → P(∧ 2 W Y ).
Now let G H * ⊂ P(∧ 2 W Y ) be the Grassmannian of lines of H, then we have: Hence P o ∩ H is the line defined by the vector −kb = r(v) and the statement follows.
Remark 6.6. In particular the fibre of r|G at r(u) is the P 3 of planes of I containing the line P o ∩ H and the next commutative diagram solves the indeterminacy of r|G: In itG is the correspondence defined below and γ,r are its projections.r is a P 3 -bundle. since Ker r = ∧ 3 W Y we have the exact sequence of vector spaces Furthermore, under the previous pairing, we have the equality: Then, since H Y is 6-dimensional, the next lemma follows.

Lemma 6.7. Let A be maximal isotropic then r(
Let H 1 . . . H 5 ⊂ P(∧ 2 W Y ) be the hyperplanes respectively defined by h 1 . . . h 5 . As in 5.5, P(H Y ) is the 5-space spanned by the smooth Del Pezzo quintic surface Now assume that F := (P(A) − h ) · G is finite, then we have: Lemma 6.8. r restricted to F is an embedding.
Proof. Let ζ ⊂ F be a scheme of length 2 such that r|ζ is not an embedding. Then the line ζ intersects h and ζ is contained in a fibre of r. This, by remark 6.6, is a 3-space linearly embedded in G. It is the family of planes containing a fixed line of H. But then ζ is a pencil of planes contained in F and F is not finite: a contradiction.
Now we concentrate on Morin-Del Pezzo configurations. We start more in general from a maximal isotropic A. Keeping our notation we assume where h · G = h and F is finite. Let V A be the V -threefold defined by A, we want to reconstruct it explicitly and see that it is rational. Notice that u ∈ h. We put u = h 5 and consider the projection map from which V A is constructed. We know that this is the restriction to P(A) of the tangential projection of P(∧ 3 W ) from the embedded tangent space to G at u. This is just the projection from u, we denote since now as (6.17) p : P(A) → P 8 , see 2.3. P 8 is the space of the Segre embedding P 2 × P 2 of P u × P ⊥ u and we know that Since h spans a 3-space containing u we can add to our play the plane (6.18) P h := p( h ).
Moreover we will also consider the set of four points These are in general position in P h , since the same is true in h for h. Theorem 6.9. V A contains the plane P h , in particular V A is rational.
Proof. We know that V A has bidegree (2, 2) and isolated singularities. Now assume that P h ⊂ P 2 × P 2 . Then, since V A is singular at the four points of h u , it is clear that P h · V A cannot be a conic. This implies that P h ⊂ V A . Hence it suffices to show that Assume P h is not in P 2 × P 2 and consider the scheme D := P h · (P 2 × P 2 ). Then D contains the set h u of four points in general position but D = P h . We claim that then D is a conic. Let us prove this fact: the variety Σ of bisecant lines to P 2 × P 2 is a well known cubic hypersurface and a Severi variety. In particular Σ contains the six lines joining two by two the points of h u . Hence P h is in Σ, though not in P 2 × P 2 . It is known that every such a plane cuts exactly a conic of bidegree (1, 1) on P 2 × P 2 , cfr. [18, chapter 5]. Then D is a conic and its projections in the factors are lines L 1 and L 2 . We have D ⊂ L 1 × L 2 and L 1 × L 2 is embedded in P 2 × P 2 as a quadric. Assume is a quadratic section of L 1 × L 2 , singular at the set of coplanar points h u . This implies V A · (L 1 × L 2 ) = 2D. Notice also that D spans P h . Now we can fix coordinates (x 1 : x 2 : x 3 ) × (y 1 : y 2 : y 3 ) on P 2 × P 2 so that (6.20) d being a form of bidegree (1,1) in (x 1 : x 2 ) × (y 1 : y 2 ). Then 2D is the complete intersection {x 3 = y 3 = d 2 = 0} and the equation of V A is ax 3 + by 3 + kd 2 = 0, where a and b are forms of bidegrees (1, 2) and (2, 1) and k = 0. If L 1 ×L 2 is in V A we have k = 0. One computes that Sing V A ∩ D is defined by the equations a = b = d = x 3 = y 3 = 0. Moreover a, b, d define in L 1 × L 2 curves C a , C b , D of bidegrees (1, 2), (2, 1), (1, 1) and Sing V A is finite. Since C a D = C b D = 3, it follows that Sing V A ∩ D contains at most three singular points of V A . Since h u has cardinality 4 this is a contradiction. Hence we can conclude that P h ⊂ V A . Finally, the rationality of V A follows from the explicit birational map V A → P 1 × P 2 we construct in the next section.
Remark 6.10. Let F = P(A) · G be any Morin configuration, smooth at u as we assume in this paper, and V A its associated V -threefold. If F has length ≥ 16 then theorem 3.2 implies that V A contains a plane. Up to Aut P 2 × P 2 we can assume that this is P h . Thus Morin configurations of length ≥ 16 are basically Morin-Del Pezzo configurations.

The V -threefold of a Morin-Del Pezzo configuration
In this section we construct V -threefolds associated to Morin-Del Pezzo configurations. Let F = P(A) · G be a Morin-Del Pezzo configuration and let be the plane contained in the threefold V A . Now we consider the projection of P 8 from P h and study p h |V A . Let us point out that p h factors as in the diagram where r is as in section 6. Indeed, r |P(A) is the projection from h , while p and p h are the projections from u and h u . Then, since h u = p(h), it follows r |P(A) = p h • p.
Remark 7.1. Notice that P 5 = P(H Y ) ⊂ P(∧ 2 W Y ) and that P(H Y )·G H * is the quintic Del Pezzo surface defined by < h > ⊥ . This, by the definition of r, is the locus where o is the pencil of quadrics of H singular at o. See 6.14 and also lemma 6.8.
Let σ : P → P 2 × P 2 be the blowing of P h then we have the commutative diagram wherep h is a P 1 -bundle and the bottom arrow is the Segre embedding. Let us consider the projection p o : P 2 → P 1 from the point o, then we have Moreover, let E ⊂ P be the exceptional divisor of σ. Since P h has trivial normal bundle the morphismp h : E → P 1 × P 2 is biregular and its inverse defines a regular section (7.7) s : We want to study the diagram more in detail with respect to V A . Denoting byṼ A the strict transform of V A via σ, and by p A the restriction of p h to V A , we have: It is clear thatṼ A is rational, because it is an integral member of (7.9) 1). Since 2H − E has degree one on the fibres of p h theñ is a biregular map. Proof. We have E = P 1 × P h so that σ |E : E → P h is the natural projection. Let us compute the bidegree (m, n) of E h in P 1 × E h . SinceṼ A has degree one on the fibres ofp h , it follows m = 1. Now notice that Sing V A · P h = h u , because h · G = h. Then, writing a local equation for a V -threefold containing a plane like P h , it is easy to deduce that the pencil of conics through h u lifts, by σ |E h , to a pencil of conics. This is cut by the ruling of planes of E. Hence n = 2 and E h is the blowing up of P h . Since h u is a set of four points in general position, then E h is a smooth quintic Del Pezzo surface.
Notice also that O E h (1, 1) ∼ = ω −1 E h , therefore the Segre embedding of E restricts to the anticanonical map of E h . Moreover the next theorem follows. Theorem 7.3. σ :Ṽ A → V A is the small contraction of four disjoint copies of P 1 . Let us fix the notation Y h :=p A (E h ). This is a smooth quintic Del Pezzo surface (7.11) Y h ⊂ P 1 × P 2 ⊂ P 5 .
Now we describe the birational morphismp A :Ṽ A → P 1 × P 2 in order to invert it. To this purpose it is useful to consider the conic bundle π : V A → P 2 defined by the projection of P 2 × P 2 onto the second factor. We have the commutative diagram whereπ is the projection map. Indeed, let t ∈ P 2 then π * (t) is V A · (P 2 × {t}) and P h · π * (t) = (o, t). Moreover,p A • σ −1 |π * (t) is precisely the projection from (o, t) p o,t : π * (t) → P 1 × {t}.
Notice that (o, t) ∈ π * (t) ⊂ P h = {o} × P 2 . It is clear that the tangent space to V A at (o, t) has dimension 4 if (o, t) ∈ Sing π * (t). This implies the next lemma.
Let Γ ⊂ P 2 be the discriminant sextic of π and t ∈ P 2 − Γ, then π * (t) is a smooth conic. Let π * (t) be its strict transform by σ, then π * (t) = (p A •π) * (t) and p A |π * (t) : π * (t) → P 1 × {t} is biregular and induced by p o,t . Moreover,p A is regular on π * (t) . In E we define: C h is a curve embedded in the Del Pezzo surface E h = E ·Ṽ A . Let (7.14) be the linear isomorphism such that s o (t) = (o, t) and let Γ h := s o (Γ). Then the next lemma is standard, we omit further details.
Lemma 7.5. C h is the strict transform of Γ h by the blowing up σ |E h : E h → P h . In particular C h is a quadratic section of the anticanonical embedding of E h .
Finally let us define and consider the following varieties Definition 7.6. C := p h * C h and F := p * h C. F is a P 1 -bundle over C and C is the biregular to C h via p h . We have We recall that C is complete intersection in P 1 × P 2 of Y h and a quadratic section.
Proof. Let ζ ⊂Ṽ A be a scheme of length 2. Assume that the morphismp A is not an embedding on ζ. Then ζ ⊂ f for a fibre f of p h : P → P 1 ×P 2 . SinceṼ A has intersection index 1 with f , it follows f ⊂Ṽ A . Notice also that, as every fibre of p h , σ * f is a line in a plane P 2 × {t}. Hence the fibre π * (t) cannot be a smooth conic, since it contains the line f . Then we have σ * f ⊂ π * Γ and f ⊂ F . This implies the statement.
Remark 7.8. In a more descriptive way let t be a point of Γ such that t / ∈ h u . Then π * (t) is a rank 2 conic and it is not singular at (o, t), as remarked. Let f + f ⊂Ṽ A be its strict transform by σ. Then a summand, say f , is a fibre of p h and intersects F h . For the other summand the mapp A : f → P 1 × {t} is a linear isomorphism. Remark 7.9. Let g :Γ → Γ be the degree 2 cover defined by π : V A → P 2 . ThenΓ parametrizes the lines in π * (t), t ∈ Γ. At a general t we have g * (t) = {f, f }. Since f and f are distinguished by the intersection with P h , thenΓ is split over Γ. If Γ is nodal one can see that g is a Wirtinger cover of Γ, in the sense of [1, section 5].
We can now reconstruct V A , describing explicitly the inverse map (7.15) σ •p −1 A : P 1 × P 2 −→ P 8 and its image V A ⊂ P 2 × P 2 ⊂ P 8 . Let J P h be the ideal sheaf of P h in P 2 × P 2 , then the rational map p h is defined by |J P h (1, 1)|. Since we have it follows where Y h + |O P 1 ×P 2 (1, 1)| denotes the linear system of divisors of bidegree (2, 3) having Y h as a fixed component. This is contained in the linear system J, defining the rational map σ •p −1 A : P 1 × P 2 → P 8 . Let J C be the ideal sheaf of C. Then we have just because C is in the indeterminacy of σ •p −1 A . Now the target space of this rational map is P 8 , since V A is not contained in a hyperplane. This implies dim J = 8 and makes our reconstruction much simpler.
Theorem 7.10. σ •p −1 A : P 1 × P 2 → P 8 is defined by the linear system |I C (2, 3)|. Proof. It suffices to show that dim |J C (2, 3)| = 8. This follows, with the usual notation, from the standard exact sequence of ideal sheaves It is easy to see that this is actually the sequence Passing to the associated long exact sequence it follows h 0 (J C (2, 3)) = 9.
Proof. Consider the standard exact sequence of ideal sheaves of P 1 × P 2 Since Y h has bidegree (1, 2) this is just Tensor it by L ⊗ O P 1 ×P 2 with L := H 0 (O P 1 ×P 2 (0, 1)). Passing to the corresponding long exact sequences, one obtains the exact commutative diagram where µ 1 , µ 2 are multiplication maps and isomorphisms. Then µ is an isomorphism.
Finally, we conclude this section by the following remark. Remark 7.12. As above let Γ ⊂ P 2 be the discriminant sextic of π : V A → P 2 . The set Sing Γ contains the set of four points π(h u ). Let I be its ideal sheaf in P 2 , then the product map H 0 (I(2)) ⊗ H 0 (O P 2 (1)) → H 0 (I P 2 (3)) is an isomorphism. Moreover, the pencil of conics |I(2)| defines a rational map q : P 2 → P 1 and hence the birational embedding q × id P 2 : P 2 −→ P 1 × P 2 ⊂ P 5 , whose image in P 1 × P 2 is Y h . We know that C is the strict transform of Γ by q × id P 2 . Composing q×id P 2 with the product map π 1 ×π 2 we obtain the plane P h . Moreover, the image of π 1 × π 2 in P 2 × P 2 is the V -threefold V A and we retrieve Γ as the discriminant curve of its projection π : V A → P 2 . Clearly this construction always works: under the only assumption that the sextic Γ contains four singular points in general position. This shows the next property.
As in remark 7.9, π defines a double cover g :Γ → Γ which is split over Γ. If Γ is nodal π is a Wirtinger cover. HenceΓ is the gluing, according to the prescriptions, of two copies of the partial normalization of Γ at the above mentioned four nodes.

Geometry of Morin-Del Pezzo configurations
Now we describe the truly geometric construction of a Morin-Del Pezzo configuration like F . We infer that such configurations form an irreducible family It turns out that F is determined by the curve C and Sing C as follows. Let ν : C n → C be the normalization map, then Sing C is defined by the exact sequence as usual. Restricting to F the commutative diagram of linear maps 7.3, we obtain Here r |F is an embedding by lemma 6.8 and p |F embeds F in U := V A − P h . Hence F is biregular to p(F ) and p h embeds p(F ) in P 5 . On the other hand let R ⊂ V A be the ramification scheme of π : V A → P 2 . Then σ * R is contained in the fundamental divisor Z ofp A :Ṽ A → P 1 × P 2 . More precisely we havep A (Z) = C so thatp A : Z → C is a P 1 -bundle. Then σ * R is a birational section of it cutting on F · U the locus of the singular points of the singular fibres of π. Then theorem 2.7 implies that Since σ −1 : U →Ṽ A is an open embedding and p h |U =p A • σ −1 |U , it follows: Lemma 8.1. The rational map r embeds F in Sing C.
Let F h := r(F ) ⊂ P 5 , the next lemma will be useful.
Proof. We have F = P(A) since F = F ∪ h is complete. Moreover r |F : F → P 5 is an embedding. Assume h 0 (I F h (1)) ≥ 1, then F h is contained in a hyperplane L. But then the pull-back of L by r | P(A) contains F : a contradiction. Now we assume W = H 0 (I C (2)) for our usual vector space W and that the inclusion of H = |I Y (2)| in P(W ) is induced by the standard exact sequence of global sections As already remarked this is not restrictive up to projective equivalence. As in the proof of lemma 5.4 let S = Sym 2 T P 5 (−1). Then S * o ⊂ H 0 (O P 5 (2)) is the space of quadratic forms singular at o ∈ P 5 and this inclusion defines a monomorphism Restricting υ to Sing C we then construct the Cartesian square N is a rank 3 vector bundle over the finite scheme Sing C. Indeed, we have (8.4) N o = H 0 (I C (2)) ∩ H 0 (I 2 o (2)) and we know that L o := H 0 (I Y h (2))∩H 0 (I 2 o (2)) has dimension 2. Since C is a quadratic section of Y h singular at o, the above exact sequence implies dim N o = 3. Let N o := P(N o ), then N o is the net of quadrics through C singular at o. In particular it is clear that the map associated to N is the embedding sending o to N o , say (8.5) f N : Sing C → G. After these remarks we can describe explicitly Morin-Del Pezzo configurations and construct an irreducible family which includes all these configurations. To this purpose we invert now the previous construction and start from a reduced C ∈ |O Y h (2)| such that Sing C spans P 5 . We define the embedding f N : Sing C → G as above and set: Then A S is generated by s 1 , . . . , s 5 , n, n 1 , . . . , n k . By induction s 1 , . . . , s 5 , n 1 , . . . , n k generate an isotropic space. Moreover n is isotropic. Hence A S is isotropic if w(n, s i ) = w(n, n j ) = 0 for i = 1 . . . k and j = 1 . . . 5. Since the tangent space to G at any point is isotropic, we have w(n, n j ) = 0 for every n j such that ζ j is tangent to G at o. Otherwise we have o = o j and we are left to show that w(n, n j ) = 0. To prove this we argue as follows, leaving some details to te reader. Let N o and N j respectively be the net of singular quadrics defined by o and o j as above. To prove w(n, n j ) = 0 it suffices to show that N o ∩ N j is non empty. Let β : Y h → P 3 be the projection from the line b := oo j . If b is not in Y h then β(Y h ) is an integral cubic surface. Moreover β(C) is a 4-nodal canonical curve. In particular it follows that β(C) = Q · β(Y ), where Q is a quadric surface. Let Q = β * Q, then Q is a quadric of rank 4, singular along the line oo j and contains C. In what follows we will denote by C the family of curves like C, that is (1)) = 0 and C is reduced}.
Notice that then Sing C has length ≥ 6. Now let V be the family of all reduced curves D ∈ |O Y h (2)| such that Sing D has length ≥ 6, it is known that V is integral, [19]. Moreover, it is easy to see that a general D in the family is an integral nodal curve such that Sing D consists of six nodes in general position in Y h ⊂ P 5 . In particular the conditions defining C are open and not empty on V, so that C is integral. In a similar way we can define and use the universal singular point over C, that is the family Now, to globalize slightly, we fix our notation as follows. Let C ∈ C, then we set W C := H 0 (I C (2)) and consider the rank 6 vector bundle π : W → C, whose fibre at C is W C . Passing to wedge product, we have the Grassmann bundle whose fibre G C is the Grassmannian of planes of P(H 0 (I C (2)), and the P 9 -bundle (8.11) P ⊂ P(∧ 3 W) whose fibre P C is P(A C ) and A C ⊂ ∧ 3 W C is the isotropic space A as above. Let Some comments now are due. Let f : S → P be the morphism defined by the assignment (C, o) → N o , where N o is a net of quadrics as above. It is clear that (8.13) Z := f (S) is an irreducible component of Z. Z contains as well the five irreducible components (8.14) where s i : C → P is the section such that s i (C) : Let H := ∪H i , so far we have Z ∪ H ⊆ G ∩ P. In the next theorem we show that the latter is an equality. Of course this implies that each fibre of the family (8.15) ∧ 3 π : G ∩ P −→ C is a finite and complete configuration of incident planes, in particular a Morin-Del Pezzo configuration. This completes our description of these configurations.
To prove the theorem we proceed at follows. Let z ∈ G ∩ P be a point in the fibre over C ∈ C. Then z is the parameter point of a net of quadrics N ⊂ |I C (2)| and we have to show that z ∈ Z ∪ H. If z is in h then there is nothing to show. Hence we can assume that z is not in h, in other words that N is not in the hyperplane H of quadrics through Y h . Then L := N ∩ H is a pencil of quadrics singular at some point v ∈ Y h . Its base scheme is a cone B v of vertex v over an integral complete intersection of two quadrics in P 4 , see 5.4. Hence the base scheme of N is Lemma 8.7. Sing C ⊂ Sing X.
Proof. Let o ∈ Sing C, we can assume o = v. Since N defines a point of P(A) and A is isotropic we have N ∩ N o = ∅. Hence there exists a quadric Q o ∈ N which is singular at In this case Sing Q o contains z, v and the line E := ov . Let π E : Y → P 3 be the projection from E then π E (Y ) is a quadric. Moreover, it is easy to deduce that then E ⊂ Y and that the cone B v is singular along E. Hence we have o ∈ X ∩ Sing B v ⊂ Sing X.
Lemma 8.8. Let v be as above then v ∈ Sing C.
Proof. Let X = Q · B v and let q v ∈ H 0 (O P 5 (1)) be the polar form of v with respect to Q. If q v = 0 then Q is singular at v and the statement follows. If q v (v) = 0 then v is not in X nor in Q. In this case consider the projection π v : X → P 4 from the vertex v of B v . Then π v is a finite double covering of π v (X), which is an integral complete intersection of two quadrics. Since X = Q · B v the ramification divisor of π v is the hyperplane section of X by {q v = 0}. In particular q v vanishes on Sing X. But then, by the previous lemma, q v vanishes on Sing C. Since we are assuming h 0 (I Sing C (1)) = 0, we have a contradiction. If q v (v) = 0 and q v = 0 assume v / ∈ Sing C and observe that the line vp is in Q for each p ∈ Sing C. Indeed, vp is tangent to Q at v and contains p. Then q v vanishes in Sing C and the same contradiction follows. Hence v ∈ Sing C.
The lemma implies that the net N , corresponding to z ∈ G ∩ P, is the net N v of all quadrics through C singular at v. Hence z ∈ Z and the proof of the theorem follows. Remark 8.9. We point out that, as a consequence of our description, a general Morin-Del Pezzo configuration is obtained from a nodal, integral canonical curve C ⊂ Y h with exactly 6 nodes. Notice also that C ⊂ P 1 × P 2 so that its projection in P 2 is a nodal sextic with 10 nodes.

Morin configurations of higher length via canonical curves
Finally we apply the previous results and constructions to deduce the uniqueness, up to projective equivalence in P 5 , of the finite Morin configuration having maximal cardinality 20. We also outline the simple description of those families of configurations of length k ≥ 16 having the one of maximal cardinality as a limit. We rely as previously on stable, highly singular canonical curves of genus 6.
Let F = P(A) · G be a finite Morin configuration of planes in P 5 of length k ≥ 16. By theorem 3.2 the V -threefold V A of A contains a plane, say P h = {o} × P 2 as in 7.1. By proposition 3.5 b := P h · Sing V A is the base scheme of a pencil of conics and it is finite. Assume that F has maximal cardinality 20, then F is smooth since its length is bound by 20. Since F − {u} is biregular to Sing V A and Sing V A contains b, it follows that b is a smooth complete intersection of two conics. Now we know from section 7 that V A is the birational image of the product map considered in 7.20, namely π 1 × π 2 : P 1 × P 2 → P 2 × P 2 .
We recall its definition. We have C ⊂ Y ⊂ P 1 × P 2 ⊂ P 5 , where Y is a smooth quintic Del Pezzo surface and C ∈ |O Y (2)| is a canonical curve. Then π 1 : P 1 × P 2 → P 2 is defined by the net of divisors |J (2, 2)|, where J is the ideal sheaf of C, and the map π 2 : P 1 × P 2 → P 2 is the obvious projection. Moreover, F is a Morin-Del Pezzo configuration and we have shown so far that Sing C is biregular to Sing(V A − P h ).
Therefore a smooth F of cardinality 20 is defined, up to Aut P 5 , by a nodal curve C ⊂ Y such that | Sing C| = 15. Finally it is well known, and easy to see, that the unique curve C such that Sing C is smooth of cardinality 15 is the curve C union of the 10 lines of Y . This proves the next uniqueness theorem. Notice that C is invariant under the action of Aut Y , which is the symmetric group S 5 . Actually C is a stable graph curve which is uniquely defined by its associated graph Γ. This has 10 vertices corresponding to the 10 lines of C . Each edge of Γ corresponds to a node o ∈ Sing C and joins the vertices corresponding to the two lines through o. In our situation Γ is the famous Petersen graph Γ.
We do not address a systematic study of the stratification by their length of Morin-Del Pezzo configurations. We simply outline here some simple ways of smoothing partially C so to obtain some of the missed families of length k ∈ [16,19]. To this purpose just consider suitable connected subgraphs λ of arithmetic genus zero and consider the family of graph curves defined by the graph Γ λ , obtained from Γ after contracting λ to a point. Let L ⊂ C be the curve defined by λ, that is L = L 1 +· · ·+L n where the summands correspond to the vertices of λ. Then the linear system |L| is very ample. We have L 2 = n − 2 and C D = 2n. Let D ∈ |L| be general and (9.1) C = C − L + D.
It is easy to see that C is nodal and that | Sing C| = 15 − n. Moreover for 1 ≤ n ≤ 4 the construction provides a curve C such that Sing C spans P 5 . Let P 5 C be 5-space of quadrics through C, then Sing C defines in it, as usual, a Morin-Del Pezzo configuration 20 − n planes. Iterating the contraction to a point of a genus 0 subgraph, one can describe all the irreducible families of Morin configurations of length k ≥ 16 and their quotients by Aut Y . Hopefully this matter will be reconsidered elsewhere.
Concluding remarks Some constructions in this paper, involving singular canonical curves of genus 6, admit natural extensions to higher genus. Indeed let W g be a vector space whose dimension is the triangular number g−2 2 . We can assume that W g is the dual of the space of quadratic forms vanishing on a nodal canonical curve (9.2) C ⊂ P g−1 .
Then the equality considered by Zak in [21] has, as a special case, the following one and this makes interesting to consider Morin configurations of (g − 4)-spaces in the projective space P(W g ). Let F be a finite Morin configuration in the Grassmannian G of (g − 4)-spaces of P(W g ). Among many other questions it is natural to ask: What one can say about the maximal length of F ?
Stable canonical curves C of genus g with many nodes provide interesting examples of finite families of incident (g−4)-spaces. Indeed let I C be the linear system of quadrics through a stable C and let I z := {Q ∈ I | z ∈ Sing Q, z ∈ Sing C}. It turns out that the orthogonal P z ⊂ P(W g ) is a subspace of dimension g − 4. Then the family is an example of family of incident g − 4-spaces. Indeed let z 1 , z 2 ∈ Sing C be distinct points and let P z 1 , P z 2 ⊂ P(W g ) be the orthogonal (g − 4)-spaces respectively of I z 1 , I z 2 . Then, with the same argument used in genus 6, the codimension of the space spanned by I z 1 ∪ I z 2 turns out to be g−4 2 = dim W g−2 . Equivalently P z 1 ∩ P z 2 is a point. Hence F C is a finite family of incident (g − 4)-spaces of P ( g−2 2 )−1 . Now stable canonical curves C which are union of lines are 3g − 3-nodal and provide smooth families F C of cardinality 3g − 3. Each curve C of this type is uniquely defined by a suitable graph as in the case of the Petersen graph. For instance a generalized Petersen graph G(2k − 1, 1) , see [10], uniquely defines a stable canonical curve C g, ⊂ P 1 × P k−1 ⊂ P g−1 of even genus g = 2k. Omitting the discussion of the odd genus case and several details, this is readily constructed in P 1 × P k−1 as follows. In the Segre embedding P 1 × P k+1 consider the lines L i := P 1 × t i , i = 1 . . . k + 1, where t := {t 1 . . . t k+1 } is a set of points in general position in P k−1 . On the other hand one can construct in P k−1 three nodal rational normal curves R 1 , R 2 , R 3 which are union of lines, have no common component and contain t as a subset of smooth points. Then we can define the curve where R j := u j × R j , j = 1, 2, 3 and u j ∈ P 1 . Let F g, := {P z , z ∈ Sing C g, } be the set of incident (g − 4)-spaces defined by Sing C g, . For g ≥ 8 it is natural to ask wether F g, is a Morin configuration and has maximal cardinality. In any case the study of graph curves like C g, seems to be interesting in order to study Morin configurations and their relations to the geometry of canonical curves.