Families of minimal involutive surfaces in projective space

Journal of Algebra 461 (2016) 201–225

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Journal of Algebra www.elsevier.com/locate/jalgebra

Families of minimal involutive surfaces in projective space S.C. Coutinho a,∗,1 , C.C. Saccomori Jr. b a

Departamento de Ciência da Computação, Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, 21945-970 Rio de Janeiro, RJ, Brazil b Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal Rural do Rio de Janeiro, 23890-000 Rio de Janeiro, RJ, Brazil

a r t i c l e

i n f o

Article history: Received 30 November 2015 Communicated by J.T. Stafford MSC: primary 14Q10 secondary 37F75, 32S65

a b s t r a c t We present an algorithm that can be used to determine if a given surface in complex projective 3-space is minimal involutive and we apply it to the construction of explicit examples of families of minimal involutive surfaces of degrees 3 and 4. © 2016 Elsevier Inc. All rights reserved.

Keywords: Algebraic geometry Symplectic geometry Involutive varieties

1. Introduction Homogeneous involutive varieties were introduced by J. Bernstein and V. Lunts in [4] as part of their solution of a problem in the representation theory of the Weyl algebra. This is the subalgebra An of C-endomorphisms of C[x1 , . . . , xn ] generated by the partial

* Corresponding author. E-mail addresses: [email protected] (S.C. Coutinho), [email protected] (C.C. Saccomori). URL: http://www.dcc.ufrj.br/~collier (S.C. Coutinho). 1 The first author was partially supported by CNPq (grant number 302585/2013-3) and PRONEX (Holomorphic foliations). http://dx.doi.org/10.1016/j.jalgebra.2016.04.025 0021-8693/© 2016 Elsevier Inc. All rights reserved.

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derivatives ∂1 , . . . , ∂n and the operators defined by multiplication by x1 , . . . , xn . In order to describe their results in more detail we need a few basic notions concerning the Weyl algebra. Our starting point is the Bernstein filtration, which is defined by giving degree one to each of the 2n generators of An . The corresponding graded ring gr(An ) is a polynomial ring in 2n variables. Moreover, a finitely generated left An -module M admits a filtration whose associated graded module gr(M ) is finitely generated as a gr(An )-module. The most important geometric invariant of M is its characteristic variety Ch(M ) = V(anngr(An ) (gr(M )) ⊂ A2n . This variety is involutive (co-isotropic) with respect to the standard symplectic structure of A2n ; see subsection 2.2 for details. The involutivity of the characteristic variety implies that n ≤ dim(Ch(M )) ≤ 2n. The An -modules whose characteristic varieties have dimension n are called holonomic. Much is known about these modules, as they play a very important role in many applications; see [5] or [7], for example. The same cannot be said of nonholonomic modules. Indeed, although irreducible holonomic An -modules are easily constructed, the first example of a nonholonomic irreducible An -module was obtained by Stafford [15] in 1985. Stafford’s example was quite explicit but its proof was entirely ad-hoc. In [4] Bernstein and Lunts approached the same question from a geometric and far more general point of view. They called an involutive homogeneous variety of A2n minimal if it does not contain any proper involutive homogeneous subvarieties and proved that if M is a cyclic An -module and anngr(An ) (gr(M )) is a prime ideal such that Ch(M ) ⊂ A2n is minimal involutive homogeneous, then M is irreducible. Moreover, they showed that a generic homogeneous hypersurface of A4 of degree greater than 3 is minimal involutive homogeneous; a result that was later generalized by Lunts to A2n [11] and by T. C. McCune [12] to polynomials of degree 3. As a corollary, we have that if n ≥ 2 is an integer and d is a generic operator of degree at least 3 in An , then An /An d is a nonholonomic irreducible module of dimension 2n − 1 > n ≥ 2. Although the result of Bernstein and Lunts implies that the module A2 /A2 d is nonholonomic irreducible for most operators d of A2 , its proof is not at all constructive. The first concrete example of a homogeneous polynomial that defines a minimal involutive homogeneous hypersurface of A4 seems to have been the rather unwieldy one presented in [1]. In this paper we describe an algorithm capable of determining that a given homogeneous polynomial of degree 3 or 4 gives rise to a minimal involutive homogeneous hypersurface of A4 . We use this algorithm to give examples of polynomials that give rise to infinite families of homogeneous minimal involutive hypersurfaces of A4 . Following Bernstein and Lunts, we translate the problem in terms of direction fields of P3 to which we apply the theory of holomorphic foliations. Unfortunately, the minimal involutivity criterion obtained with this approach works only for polynomials of degree 3 or 4; cf. Theorem 3.1.

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The plan of the paper is as follows. In section 2, we collect a number of results concerning holomorphic foliations and symplectic geometry. It also contains two elementary, but important, propositions that will be often used throughout the paper. The main criterion for proving that a given hypersurface is minimal involutive homogeneous is given in section 3, where it is also applied to the construction of a very simple example. This criterion is extended, in section 4, to cover families of polynomials with integer coefficients. In this final section, we also give examples of minimal involutive families of degrees 3 and 4. All the computations in this paper that were too long to be done by hand were performed with the help of the computer algebra system Singular [9]. Finally, a few comments on notation and terminology. Let K be a subfield of C. Given an ideal I in the polynomial ring K[x0 , . . . , xn ], we denote by V(I) the affine algebraic subset of An = AnC defined by I. If I is homogeneous, then V+ (I) denotes the algebraic subset of Pn = PnC defined by I. We will say that a variety or foliation is defined over K if it is defined by polynomials with coefficients in K. 2. Preliminaries In this section, we introduce a number of concepts and results from symplectic geometry and foliation theory that will be required later on in the paper. 2.1. Foliations Throughout this section, X denotes a smooth irreducible complex projective variety, Ω1X its sheaf of Kähler differentials and ΘX its tangent sheaf. A foliation of X is a sheaf homomorphism ξ : Ω1X → L, where L is an invertible sheaf on X, called the cotangent sheaf of ξ. The singular locus Sing(ξ) is the set of closed points of X at which ξ is not surjective. It is a closed subset of X. A closed subvariety Y of X is invariant under ξ if there exists a map Ω1Y → L|Y such that the diagram Ω1X |Y

ξ|Y

L|Y

Ω1Y is commutative. Dualizing ξ and tensoring the result with L we find that a foliation can also be defined by a global section of L ⊗OX ΘX . We will switch between these definitions without further comment whenever convenient. We also use the same symbol to denote both the map Ω1X → L and the corresponding global section of L ⊗OX ΘX . All the foliations considered in this paper are defined on n-dimensional complex projective space Pn , so we analyse this case in more detail. Applying the functor Hom(·, OPn ) to the Euler sequence [10, Theorem 8.13, p. 176] we deduce that a foliation

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ξ : Ω1Pn → OPn (k − 1) is induced by a homogeneous vector field δ = G0 ∂0 + · · · + Gn ∂n , where the Gi are homogeneous polynomials of degree k in C[x0 , . . . , xn ], for all 0 ≤ i ≤ n. The integer k is the degree of the foliation ξ. If I is a homogeneous ideal of the polynomial ring C[x0 , . . . , xn ], then V+ (I) is invariant under ξ if and only if δ(I) ⊂ I; see [11, Lemma, p. 534]. Although a foliation defined on a smooth projective variety does not always have such a nice global description, it can always be locally defined by a vector field. A more precise statement requires the following definition. Let p ∈ X and let U be an open affine neighbourhood of p in X. We say that x1 , . . . , xn ∈ OX (U ) are local coordinates at p ∈ X if dx1 , . . . , dxn is a basis of Ω1X (U ). Let ∂i ∈ ΘX (U ) be the dual of dxi . Given a foliation ξ : Ω1X → L and a point p ∈ X, an open set U of X is adapted to ξ if it admits local coordinates and L(U ) is a free OX (U )-module of rank one. In this case, ξ|U is defined by a derivation a1 ∂1 + · · · + an ∂n ∈ ΘX (U ), which we also denote by ξ|U , and p is a singular point of ξ if and only if ai (p) = 0, for all 1 ≤ i ≤ n. The jacobian of ξ at p is the matrix  Jξ (p) =

∂ai ∂xj

 1≤i,j≤n

and p is said to be a non-degenerate singular point of ξ if det(Jξ (p)) = 0. When all the singular points of a foliation are non-degenerate, the foliation itself is said to be non-degenerate. If U = {Uj }j∈J is a finite open cover of X by open sets adapted to ξ, then a subvariety Y of X is invariant under ξ if and only if ξ|U (I(U )) ⊂ I(U ), where I is the ideal sheaf of Y . In the special case X = Pn , the open sets D+ (xi ) are adapted to any foliation of Pn , for 0 ≤ i ≤ n. Furthermore, if the foliation ξ is defined by the homogeneous vector field G0 ∂0 + · · · + Gn ∂n , then its restriction to D+ (xk ) is given by the vector field ξ|D+ (xk ) =



(Gi − xi Gk )|x

k =1

∂i .

(2.1)

i=k

A famous theorem of P. Baum and R. Bott [2, Theorem 1, p. 280] gives a characterization of non-degenerate foliations of Pn in terms of the number of its singular points; see [13, p. 69]. Proposition 2.1. Let ξ be a foliation of Pn of degree d with isolated singularities. Then, #Sing(ξ) ≤ dn + dn−1 + · · · + 1 and equality holds if and only if ξ is non-degenerate. From now on we will restrict ourselves to the case X is a surface in order to introduce the notation that will be used throughout the remaining of the paper. We begin with another consequence of the Baum–Bott Theorem. Let ξ be a foliation of X. The Baum–Bott index of a non-degenerate singularity p of ξ is BB(ξ, p) =

λ1 trace(Jξ (p))2 λ2 = + + 2, det(Jξ (p)) λ2 λ1

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where λ1 and λ2 are the eigenvalues of Jξ (p). It follows from the Baum–Bott Theorem that if ξ is a non-degenerate foliation then 

 BB(ξ, p) =

p∈Sing(ξ)

c1 (ΘX /L∗ ),

(2.2)

X

where ΘX /L∗ denotes the corresponding virtual bundle; see [2, Theorem 1, p. 280]. If the eigenvalues of Jξ (p) are nonzero at some singularity p of ξ, then their ratio ρξ (p) (taken in whatever order) plays a very important role in our algorithms because, as the next lemma shows, it controls the local behaviour of any curves that may be invariant under ξ. Lemma 2.2. Let ξ be a foliation of a smooth surface S and let C be an invariant curve of ξ that passes through a non-degenerate singular point p of ξ. If the ratio ρξ (p) of the eigenvalues of Jξ (p) is not a rational number, then C is either smooth or has a normal crossing at p. A proof of this lemma can be found in [4, Proposition 3, p. 232] or [14, Lemma 5.1, p. 156]. Finally, we need the following result about foliations defined by polynomials with coefficients in a subfield of C. A proof of this theorem can be found in [6, Theorem 2.3]. Theorem 2.3. Let K be a subfield of C and let ξ be a foliation of Pn defined by homogeneous polynomials with coefficients in K. If there exists a projective subvariety of Pn of dimension d invariant under ξ, then there exists a subvariety of Pn of dimension d, defined over K, that is also invariant under ξ. 2.2. The hamiltonian vector field Although most of the results in this subsection hold in greater generality, we restrict our exposition to A4 and P3 because that allows us to introduce the terminology that will be in force throughout the whole paper. The Poisson bracket of the polynomial ring C[u, x, y, z] is given by  {f, g} =

∂f ∂g ∂g ∂f − ∂u ∂y ∂u ∂y



 +

∂g ∂f ∂f ∂g − ∂x ∂z ∂x ∂z

 ,

where f, g ∈ C[u, x, y, z]. It defines a Lie algebra structure on C[u, x, y, z] and a symplectic structure on A4 . The hamiltonian vector field defined by a polynomial f ∈ C[u, x, y, z] is given by hf (g) = {f, g}, for all g ∈ C[u, x, y, z]. An ideal I of C[u, x, y, z] is involutive if it is closed with respect to the Poisson bracket; that is, {I, I} ⊆ I. In particular, such an I is invariant under the hamiltonian vector field hf , for all f ∈ I. Let F ∈ C[u, x, y, z] be a homogeneous polynomial and assume that S = V+ (F ) is a smooth surface of P3 . In this case hF defines a foliation of P3 that leaves S invariant,

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because hF (F ) = 0. Thus, hF defines a foliation of S that we denote by ξF . Since Sing(ξF ) ⊂ S, by [4, Lemma 2, p. 228], it follows that I(Sing(ξF )) is the radical of the ideal generated by the minors of the matrix 

Fy MF = u

Fz x

−Fu y

 −Fx . z

(2.3)

The ideal of any surface of P3 is principal, hence involutive. We say that a surface in P3 is minimal involutive if its ideal is not properly contained in any other involutive homogeneous ideal. Of course, V(I) ⊂ A4 is minimal involutive homogeneous if and only if V+ (I) is minimal involutive. In particular, if S does not contain any curve invariant under ξF then it must be minimal involutive. Moreover, it is easier to prove that S does not have such invariant curves, because this is a problem that can be approached using techniques from the theory of holomorphic foliations. Our next result is obtained by applying the Baum–Bott formula (2.2) to the foliation ξF of the surface V+ (F ). See [1, Theorem 3.4, p. 305] for a proof. Theorem 2.4. Let F be a homogeneous polynomial of degree k ≥ 2. If the surface S = V+ (F ) ⊂ P3 is smooth and ξF is non-degenerate as a foliation of V+ (F ), then 

BB(ξF , p) = 4k.

p∈Sing(ξF )

We now analyse the properties of the eigenvalues of ξF at a point p ∈ Sing(ξF ) through which passes a smooth invariant curve. The following notation will be used throughout the paper. Given p = [u0 : x0 : y0 : z0 ] ∈ D+ (u) ⊂ P3 (C), let Kp be the smallest extension of Q that contains x0 /u0 , y0 /u0 and z0 /u0 . To simplify the notation, we will denote the jacobian matrix of ξF at a singular point p by JF (p) rather than JξF (p). Proposition 2.5. Let F ∈ Q[u, x, y, z] be a homogeneous polynomial and let p ∈ Sing(ξF ) ∩ D+ (u). Assume that C ⊂ V+ (F ) is a curve defined over Q that is invariant under ξF . If p is a smooth point of C and V+ (F ), then the eigenvalues of JF (p) belong to Kp . Proof. Set S = V+ (F ) and let f = F |u=1 be the dehomogenization of F with respect to u. Since C is defined over Q, there exists a generator g ∈ Kp [x, y, z] of the ideal of C in the local ring OS,p . Thus, all the coordinates of ∇f (p) and ∇g(p) belong to Kp and there exists a nonzero vector v ∈ K3p , such that ∇f (p)v = ∇g(p)v = 0. Moreover, v generates Tp C because the curve C is smooth at p. Let w ∈ K3p be a nonzero vector such that ∇f (p)w = 0 but ∇g(p)w = 0. Since the surface S is smooth at p, it follows that v and w form a basis of Tp S. On the other hand, both Tp S and Tp C are invariant under JF (p), because S and C are invariant under ξF . Hence, there exists an invertible 2 × 2 matrix M with coefficients in Kp , such that M −1 JF (p)M is an upper triangular matrix. In particular, the eigenvalues of JF (p) belong to Kp , as claimed. 2

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The next result is a version of the well-known Shape Lemma, adapted to the requirements of this paper. Before we state it, we need a few definitions. Given a ring R and a non-constant homogeneous polynomial F ∈ R[u, x, y, z], we denote by SF (R) the ideal of R[x, y, z] generated by the minors of the matrix MF |u=1 . If R ⊆ C, then V(SF (R)) = Sing(ξF ) ∩ D+ (u) will be called the affine singular set of the foliation ξF . We will drop R from the notation whenever it is clear from the context. It follows from Proposition 2.1 that if the singular set of the foliation of P3 induced by hF is finite, then it cannot have more than m(k) = (k − 1)3 + (k − 1)2 + (k − 1) + 1 points, where k = deg(F ). Since all the singularities of this foliation belong to V+ (F ) by [4, Lemma 2, p. 228], this is also the maximal number of singularities of ξF . Proposition 2.6. Let F ∈ Q[u, x, y, z] be a homogeneous polynomial of degree k. If Sing(ξF ) ⊂ D+ (u) and SF ∩ Q[y] is generated by an irreducible polynomial f of degree m(k), then (1) there exist polynomials f1 , f3 ∈ Q[y], such that SF = (f, x − f1 , z − f3 ); (2) #Sing(ξF ) = m(k); (3) ξF is non-degenerate and its singularities have the form (f1 (β), β, f3 (β)), for some root β of f . Proof. We begin by showing that Sing(ξF ) must be a finite set. Suppose, by contradiction, that this is false. Then Sing(ξF ) contains a curve C all of whose points are singularities of ξF . But, by [10, Theorem 7.2, p. 48], ∅ = C ∩ V+ (u) ⊆ Sing(ξF ), which contradicts Sing(ξF ) ⊂ D+ (u). Therefore, #Sing(ξF ) ≤ m(k) by Proposition 2.1. Let ϕ : Q[y] → Q[x, y, z]/SF be the canonical projection. Since ker(ϕ) = SF ∩ Q[y] = (f ), it follows that ϕ induces an injective homomorphism ϕ : Q[y]/(f ) → Q[x, y, z]/SF . However, m(k) = deg f = dimQ

Q[y] Q[x, y, z] C[x, y, z] ≤ dimQ = dimC ≤ m(k), (f ) SF C[x, y, z]SF

where the last inequality follows from Proposition 2.1. In particular, #Sing(ξF ) = m(k), which proves (2). Moreover, dimQ (Q[y]/(f )) = dimQ (Q[x, y, z]/SF ), so that ϕ is an isomorphism. Using bars to denote images in the corresponding factor rings, let f1 , f3 ∈ Q[y] be such that ϕ(f1 ) = x and ϕ(f3 ) = z. Then (f, x − f1 , z − f3 ) ⊂ SF . Conversely, given h ∈ SF , let  h be the polynomial obtained substituting f1 for x and f3 for z in h. Hence,  h ∈ SF ∩ Q[y] = (f ). Therefore,

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h ∈ (f, x − f1 , z − f3 , h) = (f, x − f1 , z − f3 ,  h) = (f, x − f1 , z − f3 ). Thus, SF = (f, x − f1 , z − f3 ), which proves (1). Finally, (3) is an immediate consequence of (1) and (2). 2 Our next result is a consequence of Theorem 2.3 and Proposition 2.6 and is one of the crucial ingredients in the proof of the effective criterion, proposed in the next section, for showing that a given surface of P3 is minimal involutive. Corollary 2.7. Let F ∈ Q[u, x, y, z] be a homogeneous polynomial of degree k > 2, such that Sing(ξF ) ⊂ D+ (u) and V+ (F ) is smooth. Let C ⊂ V+ (F ) be an algebraic curve defined over Q and invariant under ξF . If the generator of SF ∩ Q[y] is irreducible of degree m(k), then Sing(ξF ) ⊂ C. Proof. It follows from Proposition 2.6 that ξF has exactly m(k) distinct singularities, which can be written in the form (f1 (β), β, f3 (β)), where f1 , f3 ∈ Q[y] and β is a root of the generator f of SF ∩ Q[y]. Let L be the splitting field of f over Q and let G be the Galois group of the extension L/Q. We define an action of G on Sing(ξF ) by σ((f1 (β), β, f3 (β))) = (σ(f1 (β)), σ(β), σ(f3 (β))) = (f1 (σ(β)), σ(β), f3 (σ(β))), where the last equality follows from the hypothesis that the coefficients of f1 and f3 belong to Q. Moreover, this action is transitive, because f is irreducible over Q. On the other hand, by [8, Lemma 5.2, p. 129], there exists at least one point p0 ∈ C ∩ Sing(ξF ). But, if g is an element of I(C ∩ D+ (u)) with rational coefficients, then 0 = σ(g(p0 )) = g(σ(p0 )). Hence, all the singularities of ξF are zeroes of g. The desired result follows from the fact that I(C ∩ D+ (u)) is generated by polynomials with rational coefficients. 2 3. A minimality criterion In this section we prove a criterion, suitable for use with a computer algebra system, that will enable us to give, in subsection 3.4, a simple example of a minimal involutive surface of degree 3. In order to simplify the notation, we will write ρF (p) instead of ρξF (p) for the ratios of the eigenvalues of JF (p). See the previous section for this and any other unexplained notation. 3.1. Sufficient conditions for minimality The following conditions on a polynomial F ∈ Q[u, x, y, z] will often recur throughout the paper, so we list then here for ease of reference:

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C.1: C.2: C.3: C.4:

209

Sing(ξF ) ⊂ D+ (u); V+ (F ) is smooth; SF ∩ Q[y] is generated by an irreducible polynomial of degree m(k); / Kp , for all p ∈ Sing(ξF ). ρF (p) ∈

Note that C.1 implies that F is irreducible, because if F = GH, with G, H ∈ Q[u, x, y, z] \ Q, then V+ (G) ∩ V+ (H) ⊂ Sing(ξF ). Our first result shows that the above conditions imply minimality. Theorem 3.1. If F ∈ Q[u, x, y, z] is a homogeneous polynomial of degree k = 3 or k = 4 that satisfies conditions C.1 to C.4, then V+ (F ) is a minimal involutive surface. Proof. Suppose, by contradiction, that S = V+ (F ) is not minimal involutive. Then, by definition, I(S) = (F )  J, for some involutive homogeneous ideal J of C[u, x, y, z]. By C.2, the ideal I(S) is prime. Thus, if C = V+ (J), 2 = dim(S) > dim(C) ≥ 1, where the last inequality follows from the involutivity of J. Hence, dim(C) = 1. Moreover, by Theorem 2.3, we can assume that C is defined over Q. Since hF (G) = {F, G} ∈ I(C), for all G ∈ I(C), it follows that C is a curve in S, defined over Q, and invariant under ξF . Thus, by Corollary 2.7, Sing(ξF ) ⊂ C. However, Proposition 2.5 and C.4 imply that C cannot be smooth at any singularity of ξF . There/ Q, for all p ∈ Sing(ξF ), so fore, Sing(C) = Sing(ξF ). But C.4 also implies that ρF (p) ∈ all the singularities of C must be nodes by Lemma 2.2. Since C has m(k) singularities it follows from [3, Remarks I.16, p. 8] that its arithmetic genus is equal to g(C) + m(k), where g(C) is its geometric genus. Hence, by the genus formula [3, I.15, p. 8] C 2 = 2(g(C) + m(k)) − 2 − deg(C)(k − 4),

(3.1)

because the canonical sheaf of S is OC (k − 4). / Q and Sing(C) = Sing(ξF ), it follows from the index formula However, since ρF (p) ∈ in [16, Theorem 2.5, p. 2995] and Theorem 2.4, that C 2 = 4k. Combining this with (3.1) and taking into account that m(k) = k3 − 2k 2 + 2k, we get 2g(C) + deg(C)(4 − k) = −2k3 + 4k 2 + 2. But the right hand side of this equation is negative for all k ≥ 3, while the left hand side is positive when k = 3 and k = 4. This contradicts our assumption on the existence of J and proves that S is minimal involutive. 2 Even though, in principle, Theorem 3.1 can be used to prove that a given hypersurface of P3 is minimal, it depends on condition C.4, which must be checked for each one of

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the singularities of the hamiltonian foliation. Thus, in order to apply it, one must first find a way to check this exactly. In practice this means that we must replace C.4 by a condition in which the coordinates of the singularities do not figure explicitly. However, before we do this, we need a description of the jacobian of ξF at each of its singular points. 3.2. Local analysis Let F ∈ C[u, x, y, z] be a homogeneous polynomial of degree k ≥ 2, such that V+ (F ) is smooth and let p = [1 : α : β : γ] be a singularity of ξF . As Lunts showed in [11, §2.3.4, p. 536] the jacobian matrix of ξF has a very simple form when the coordinates of the singularities are q = [1 : 0 : 0 : 0]. In order to benefit from this, we change coordinates using the symplectic automorphism τp whose inverse is given by τp−1 (u, x, y, z) = (u, γu − z, −βu − γx + y + αz, −αu + x). As one readily checks, τp−1 (p) = q. Set P (u , x , y  , z  ) = F (τp (u , x , y  , z  )) = F (u , αu + z  , βu + αx + y  + γz  , γu − x ). Note that the coefficients of P depend on the choice of singularity p. Since τp is a symplectic isomorphism, the foliation HP of P3 induced by the hamiltonian vector field hP is singular at q. This should not be confused with ξF = HF |V+ (F ) . Moreover, • HP is non-degenerate at q if and only if HF is non-degenerate at p, and • JHF (p) and JHP (q) have the same eigenvalues. Since τp−1 maps V+ (P ) isomorphically onto V+ (F ), it follows that JF (p) and JP (q) also have the same eigenvalues. In particular, ρF (p) = ρP (q). In order to simplify the notation in the computations that we are about to perform, we drop the dashes from the variables and denote by the same letter both a point of D+ (u) and its image in A3 under the usual isomorphism. Set P = P (1, x, y, z) and let HP |D+ (u) be the dehomogenization of Hp at D+ (u) as defined in (2.1). The jacobian of the dehomogenization of HP at D+ (u) can be written in the form JHP (q) = Jq − Py (q) · I,

(3.2)

where I is the identity 3 × 3 matrix and Jq is the jacobian at q of the vector field z (q), −P u (q), −P x (q)). Let ζ, a, b, c ∈ Kp be such that (P P ≡ ζuk + (ax + by + cz)uk−1 Since HP |D+ (u) (q) = 0, it follows that

(mod (x2 , y 2 , z 2 )).

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a = Px (q) = 0 and c = Pz (q) = 0.

211

(3.3)

Thus,



P ux (q) = (k − 1)a = 0 and Puz (q) = (k − 1)c = 0. Taking into account that V+ (P ) is smooth, we also have from (3.3) that (∇P)(q) = (0, Py (q), 0) = 0;

(3.4)

 P uy (q) = (k − 1)b = (k − 1)Py (q) = 0.

(3.5)

so that

Denoting the coefficients of x2 uk−2 , xzuk−2 and z 2 uk−2 in P by a2,2 , a2,4 and a4,4 , respectively, we have that Pxx (q) = 2a2,2 ,

Pzx (q) = a2,4

and Pzz (q) = 2a4,4 .

Thus, 

a2,4 − b ∗ 0 −kb JHP (q) = ∗ −2a2,2

2a4,4 0 . −a2,4 − b

But by (3.4), Tq (V+ (P )) is generated by ∂/∂x and ∂/∂z, so that  JP (q) = (JHP )(q)|Tq V+ (P ) =

a2,4 − b −2a2,2

2a4,4 −a2,4 − b

 (3.6)

whose determinant is b2 − (a22,4 − 4a2,2 a4,4 ). 3.3. An effective minimality criterion In this subsection we determine the conditions that will replace C.4 in the effective criterion that we use to prove minimal involutivity. Note that, from now on, we will be assuming that all surfaces are defined over the rational numbers. Algorithm 3.2. Given a homogeneous polynomial F ∈ Q[u, x, y, z] of degree k ≥ 2, such that V+ (F ) is smooth, Sing(ξF ) ⊂ D+ (u) and SF ∩ Q[y] is generated by an irreducible polynomial f of degree m(k), the algorithm returns a polynomial μF ∈ Q[y] with the property that ρF (p) ∈ / Kp , for all p ∈ Sing(ξF ), if and only if t2 − μF is irreducible over Q[y]/(f ).

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Step 1: compute the reduced Gröbner basis G = {f, x − f1 , z − f3 } of the ideal SF in Q[x, y, z], where f1 , f3 ∈ Q[y]; Step 2: adjoin the variables α, β and γ to Q[u, x, y, z]; Step 3: compute the coefficients A2,2 , A2,4 , A4,4 ∈ Q[α, β, γ] of the monomials x2 uk−2 , xzuk−2 and z 2 uk−2 in PF (α, β, γ, u, x, y, z) = F (u, αu + z, βu + αx + y + γz, γu − x); Step • • Step

4: set Δ ← A22,4 − 4A2,2 A4,4 ; μF ← Δ(f1 (y), y, f3 (y)); 5: return μF .

Proof. The notation used in the description of the algorithm will be in force throughout the proof. We will also denote by B the coefficient of yuk−1 in the polynomial PF defined in Step 3. It follows from Proposition 2.6 that the reduced Gröbner basis of SF has the form of Step 1. Thus, if p ∈ Sing(ξF ), its coordinates in A3 ∼ = D+ (u) are (f1 (βp ), βp , f3 (βp )) for some root βp of f (y) = 0. Hence, the coefficients of yuk−1 , x2 uk−2 , xzuk−2 and z 2 uk−2 in PF (p, u, x, y, z) = PF (f1 (βp ), βp , f3 (βp ), u, x, y, z), which we denote by the corresponding lower case letters, are obtained by taking α = f1 (βp ),

β = βp

and γ = f3 (βp )

into B, A2,2 , A2,4 and A4,4 . By (3.6), the eigenvalues of JF (p) are the roots of λ2 + 2bλ + (b2 − a22,4 + 4a2,2 a4,4 ) = 0.

(3.7)

Let Δp = a22,4 − 4a2,2 a4,4 and let Δp be one of the roots of x2 = Δp . Computing the roots of (3.7) and taking their ratio we find that ρF (p) =

−b + −b −



b2 − 2b Δp + Δp = . b2 − Δp Δp Δp

Note that the denominator of this fraction must be nonzero because ξF is non-degenerate by Proposition 2.6. Taking into account that V+ (F ) is smooth, we have from equation (3.5) that b = 0. Since Δp ∈ Kp and b ∈ Kp \ {0}, it follows that ρF (p) ∈ / Kp if and

only if Δp ∈ / Kp . But this last condition is equivalent to saying that the polynomial t2 − μF (y) is irreducible in Kp [t] ∼ = Q[y, t]/(f ), proving that the algorithm is correct. 2 Combining Theorem 3.1 with the above algorithm, we get the following effective test to determine whether a given surface of P3 is minimal involutive.

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Corollary 3.3. Let F ∈ Q[u, x, y, z] be a homogeneous polynomial of degree k = 3 or k = 4. Assume that V+ (F ) is smooth, that Sing(ξF ) ⊂ D+ (u) and that SF ∩ Q[y] is generated by an irreducible polynomial f of degree m(k). Let μF be the polynomial in Q[y] computed by the procedure of Algorithm 3.2. If t2 − μF is irreducible in Q[y, t]/(f ), then V+ (F ) is a minimal involutive subvariety of P3 . 3.4. An example of degree three In this subsection, we illustrate how Algorithm 3.2 and Corollary 3.3 can be used to prove that a surface of degree 3 is minimal involutive. We begin with a lemma that will also be required in subsection 4.4. Lemma 3.4. If d ∈ Z \ {0} and F = y 3 + u2 x + 2x2 u + dz 3 then (1) V+ (F ) is a smooth surface of P3 ; (2) Sing(ξF ) ⊂ D+ (u). Proof. Suppose, by contradiction, that [u : x : y : z] is a singularity of V+ (F ). The ideal generated by F and its partial derivatives contains Fy = 3y 2 and Fz = 3dz 2 . Since d = 0, it follows that y = z = 0. Therefore, F (u, x, 0, 0) = ux(2x + u), which implies that u = 0, x = 0 or 2x + u = 0. But Fu (0, x, 0, 0) = 2x2 and Fx (u, 0, 0, 0) = u2 . Hence, u = 0 implies x = 0 and vice-versa. Finally, if 2x + u = 0 but ux = 0, then Fu (u, x, 0, 0) = 2x(x + u) = 0, so x + u = 0, a contradiction. Hence, u = x = y = z = 0, which proves (1). In order to prove (2), note that the ideal generated by the 2 × 2 minors of the matrix MF |u=0 , contains the polynomials 3y 3 , 3dz 3 and 3dyz 2 +2x3 , from which we immediately deduce that y = z = x = 0, whenever d = 0, proving (2). 2 Suppose now that d = 3. It follows from Lemma 3.4 that F satisfies conditions C.1 and C.2 of subsection 3.1. Using the computer algebra system Singular, we find that the reduced Gröbner basis G computed at Step 1 of Algorithm 3.2 has the form {f, x − f1 , z − f3 }, where f = 9y 15 + 386y 6 − 96y 3 + 6; f1 = (−216y 12 − 9y 9 + 576y 6 − 9240y 3 + 382)/3074; f3 = (6903y 13 + 864y 10 + 36y 7 + 293758y 4 − 36672y)/9222. Moreover, by Eisenstein’s criterion, f is irreducible of degree 15 = m(3). Thus, condition C.3 of Theorem 3.1 holds for ξF . The polynomial μF ≡ (−81540y 13 − 10314y 10 − 3888y 7 − 3524988y 4 + 437772y)/1537 (mod f ),

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is computed as in Algorithm 3.2. With the help of Singular’s function factorize we check that the image of t2 − μF in Q[y, t]/(f ) is irreducible. Hence, V+ (F ) is a minimal involutive surface by Corollary 3.3. 4. Families of minimal involutive surfaces In this section, we combine the effective criterion of the previous section with a reduction modulo a prime argument to construct families of minimal involutive surfaces with integer coefficients. 4.1. q-equivalence In this subsection, we introduce an equivalence relation that allows us to construct infinite families of minimal involutive surfaces using reduction modulo a prime. To make the reduction step possible we assume, from now on, that the given polynomial has integer coefficients. Let F ∈ Z[u, x, y, z] be a homogeneous polynomial of degree k and let G = {hi ∈ Q[x, y, z] : i ∈ I} be the reduced Gröbner basis of the singular ideal SF (Q) with respect to the lexicographic order for which x > z > y. For each i ∈ I, let ai be the rational number such that ai hi is a primitive polynomial in Z[x, y, z]. We will call {ai hi : i ∈ I} the Z-reduction of G. A Gröbner basis that is equal to its Z-reduction is said to be Z-reduced. We implicitly assume, from now on, that this refers to the monomial order mentioned above. Suppose that F satisfies conditions C.1 and C.2 of subsection 3.1. Given a prime number q > 1, we say that F is q-Gröbner if the Z-reduced basis of SF (Q) has the form GF = {f, nF x − f1 , mF z − f3 },

(4.1)

and f, f1 , f3 ∈ Z[y] satisfy (1) f is irreducible of degree m(k) over Z; (2) q does not divide nF · mF · lc(f ); where lc(f ) denotes the leading coefficient of the polynomial f and mF , nF ∈ Z. Let Oq be the localization of Z at the maximal ideal qZ and let π : Oq [y] → Fq [y]

(4.2)

be the homomorphism of polynomial rings defined by the canonical projection of Oq onto the field Fq = Oq /qOq . Suppose now that F, G ∈ Z[u, x, y, z] are q-Gröbner homogeneous polynomials, such that

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GF = {f, nF x − f1 , mF z − f3 } and GG = {g, nG x − g1 , mG z − g3 }, are the Z-reduced bases of SF and SG . We will say that F and G are q-equivalent, which we denote by F ∼q G, if: (1) F ≡ G (mod q) and deg F = deg G; (2) π(f ) and π(g) are associated polynomials in Fq [y]; (3) π(f1 /nF ) = π(g1 /nG ) and π(f3 /mF ) = π(g3 /mG ). Note that this equivalence relation is only defined for q-Gröbner polynomials. Lemma 4.1. Let q be a prime number and F, G ∈ Z[u, x, y, z] be q-Gröbner homogeneous polynomials. If F ∼q G, then π(μF ) = π(μG ). Proof. Let {f, nF x −f1 , mF z −f3 } and {g, nG x −g1 , mG z −g3 } be the Z-reduced bases of the affine singular sets of ξF and ξG , respectively, and define PF , PG ∈ Z[α, β, γ][u, x, y, z] as in Step 3 of Algorithm 3.2. Clearly, F ≡ G (mod q) implies that PF ≡ PG (mod q). In particular, the coefficients in PF and PG of any given monomial in u, x, y, z, are equal as polynomials in Fq [α, β, γ]. Thus, if ΔF and ΔG are constructed as in Step 4 of Algorithm 3.2, then ΔF ≡ ΔG (mod q). Since π(f1 /nF ) = π(g1 /nG ) and π(f3 /mF ) = π(g3 /mG ), because F ∼q G, it follows that           f1 f3 g1 g3 , y, π = ΔG π , y, π = π(μG ), π(μF ) = ΔF π nF mF nG mG which completes the proof of the lemma. 2 Let q > 1 be a prime number and let F ∈ Z[u, x, y, z] be a q-Gröbner homogeneous polynomial whose singular ideal SF has Z-reduced basis {f, nx − f1 , mz − f3 }. We say that F is q-minimal if (1) π(f ) is irreducible of degree m(k); (2) the image of t2 − π(μF ) is an irreducible polynomial of Fq [y, t]/(π(f )). As the next result shows, Algorithm 3.2 and Lemma 4.1 can be combined to prove that some of the conditions required in Theorem 3.1 are common to all polynomials that are q-equivalent to a given q-minimal polynomial. Proposition 4.2. Let q be a prime number and F, G ∈ Z[u, x, y, z] be q-Gröbner homogeneous polynomials. Assume that F is q-minimal and that F ∼q G. If SG (Q) ∩ Z[y] = (g) then the image of t2 − μG is irreducible in Q[y, t]/(g). Proof. Let {f, nF x −f1 , mF z−f3 } and {g, nG x −g1 , mG z−g3 } be the Z-reduced Gröbner bases of SF and SG , respectively. Since π(f ) is associated to π(g), the identity map

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induces an isomorphism K = Fq [y]/(π(g)) ∼ = Fq [y]/(π(f )). However, by Lemma 4.1, the 2 images of the polynomials t − π(μF ) and t2 − π(μG ) coincide in K[t] ∼ = Fq [y, t]/(π(f )). Thus, by the q-minimality of F , the image of t2 − π(μG ) is irreducible in K[t]. Taking into account that Oq [t, y]/(g) maps onto Fq [t, y]/(π(g)), it follows that t2 − μG must be irreducible over Oq [y]/(g), hence also in Q[y, t]/(g), by Gauss’s Lemma. 2 Corollary 4.3. Let q be a prime number and let F, G ∈ Z[u, x, y, z] be q-Gröbner homogeneous polynomials of degree k = 3 or k = 4. If F is q-minimal and F ∼q G, then V+ (G) is a minimal involutive surface. Proof. We must check that hypotheses C.1 to C.4 of subsection 3.1 hold for G. Let G = {g, nG x − g1 , mG z − g3 } be the Z-reduced basis of SG . Since g is irreducible and g ∈ SG ∩Q[y], it follows that SG ∩Q[y] = (g). All the other conditions in subsection 3.1 follow from the definition of a q-Gröbner polynomial, Proposition 4.2 and Algorithm 3.2. 2 4.2. Specialization of parameters In this subsection, we prepare the ground for the introduction of families of minimal involutive surfaces by studying the behaviour of a Gröbner basis of the singular ideal of a family of hamiltonian foliations under specialization of parameters. Let t = {t1 , . . . , t } be a set of parameters. Given d = (d1 , . . . , d ) ∈ Z , let Ψd : Q[t][u, x, y, z] → Q[u, x, y, z] be the Q-algebra homomorphism that takes the variables u, x, y and z into themselves but maps ti onto di for each 1 ≤ i ≤ . We will use the same symbol to denote the restriction of this homomorphism to subrings of Q[t][u, x, y, z]. Let F ∈ Z[t][u, x, y, z] be homogeneous with respect to the variables u, x, y and z. Recall that SF (Q[t]) is the ideal generated by the coefficients of the dehomogenization of ξF in the polynomial ring in x, y and z with coefficients in the ring Q[t]. In order to simplify the notation, we will write [t]

SF = SF (Q[t])

and

(t)

[t]

SF = SF (Q(t)) = Q(t)SF . (t)

We will be concerned with the case that SF satisfies the following condition. (t)

[t]

Condition 4.4. SF admits a Gröbner basis of the form G = {f, rx − f1 , sz − f3 } ⊂ SF , where 0 = r, s ∈ Z[t], f , f1 and f3 are polynomials in Z[t][y] such that degy f1 < degy f , degy f3 < degy f and f is primitive as a polynomial in y. If c(t) ∈ Z[t] is the leading coefficient of f , set B = V(rsc) ⊂ A . The following lemma shows that a Gröbner basis for which the above condition holds specializes under Ψd to a Gröbner basis of the singular ideal of ξΨd (F ) .

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Lemma 4.5. Let F ∈ Z[t][u, x, y, z] be homogeneous with respect to the variables u, x, y, z. (t) If SF satisfies Condition 4.4, then {Ψd (f ), Ψd (rx − f1 ), Ψd (sz − f3 )}, is a Gröbner basis of SΨd (F ) , for all d ∈ Z \ B. Proof. For any d ∈ Z \ B, the polynomials Ψd (f ), Ψd (rx − f1 ) and Ψd (sz − f3 ) clearly form a Gröbner basis in Q[x, y, z]. Thus, it is enough to prove that they generate SΨd (F ) when d ∈ Z \ B. Let d ∈ Z and let P1 , . . . , P6 be the 2 × 2 minors of the matrix [t] MF |u=1 , defined at (2.3). For each i = 1, . . . , 6, set pi = Ψd (Pi ). Since f ∈ SF , there  exist g1 , . . . , g6 ∈ Q[t][x, y, z], such that f = gi Pi . In particular, Ψd (f ) =



Ψd (gi )pi ∈ SΨd (F ) ,

because the polynomials p1 , . . . , p6 generate SΨd (F ) as an ideal of Q[x, y, z]. Applying the same reasoning to rx − f1 and sz − f3 , we conclude that (Ψd (f ), Ψd (rx − f1 ), Ψd (sz − f3 )) ⊂ SΨd (F ) .

(4.3)

In order to prove the other inclusion, let P (t, x, y, z) =



ai,j xi z j ,

0≤i,j≤m

be a 2 × 2 minor of MF |u=1 , where ai,j ∈ Z[t][y] for 0 ≤ i, j ≤ m. Multiplying both sides of this equation by rm sm , we get rm sm P =



ai,j · rm−i · (rx)i · sm−j · (sz)j ∈ Z[t][x, y, z].

But from rm sm P =



ai,j · rm−i · ((rx − f1 ) + f1 )i · sm−j · ((sz − f3 ) + f3 )j ,

it follows that rm sm P = φ + Φ1 · (rx − f1 ) + Φ2 · (sz − f3 ),

(4.4)

where φ=



ai,j · rm−i · f1i · sm−j · f3j ∈ Z[t][y] [t]

and Φ1 , Φ2 ∈ Z[t][x, y, z]. Since P, (rx − f1 ), (sz − f3 ) ∈ SF by hypothesis, it follows (t) (t) that φ ∈ SF ∩ Q[t][y]. But, G ⊂ Z[t][x, y, z] is a Gröbner basis of SF with respect to

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(t)

the lexicographic order for which x > z > y. Hence, SF ∩ Q(t)[y] is generated by f and there exist a=

n 

αi (t)y i ∈ Q[t][y]

b ∈ Q[t],

and

i=1

such that φ = (a/b)f and gcd(a, b) = 1. Thus, by (4.4), brm sm P = af + bΦ1 · (rx − f1 ) + bΦ2 · (sz − f3 )

(4.5)

belongs to the ideal of Q[t][x, y, z] generated by the polynomials of G. Before we apply Ψd to (4.5), we must make sure that Ψd (brs) = 0. Since d ∈ / B = V(crs), it is enough to show that b(d) = 0. Suppose, by contradiction, that this is not the case. Then, V(b) ⊂ V(c). Hence, by the Nullstellensatz, there is an irreducible factor h of b that does not divide c. But it follows from (4.5) that V(b) ⊂ V(αi c), for all 1 ≤ i ≤ n. Hence, using the Nullstellensatz again and taking into account that the polynomials b,

αi and c have rational coefficients, we conclude that αi c ∈ Q[t]b, for all i. Therefore, h must divide αi c, for all i. Since h does not divide c, it follows that it divides gcd(a, b) = 1, a contradiction. Therefore, Ψd (brs) = 0, for all d ∈ Z \ B. Let d ∈ Z \ B. Since, as we have just shown, Ψd (brm sm ) is not zero, it follows that Ψd (P ) =

1 Ψd (af + bΦ1 · (rx − f1 ) + bΦ2 · (sz − f3 )). Ψd (brm sm )

Thus, Ψd (P ) belongs to the ideal of Q[x, y, z] generated by Ψd (f ), Ψd (rx − f1 ) and Ψd (sz − f3 ). Since P is an arbitrary minor of MF |u=1 , we conclude that SΨd (F ) = (Ψd (P1 ), . . . , Ψd (P6 )) ⊂ (Ψd (f ), Ψd (rx − f1 ), Ψd (sz − f3 )), for all d ∈ Z \ B, completing the proof of the lemma. 2 (t)

Unfortunately, the cost of computing the basis G of SF is very high, so much so that the computation will, most often, not successfully terminate when F has degree four. The next proposition gives us an indirect way to determine the polynomials f , r and s which will be required in Algorithm 4.8. Proposition 4.6. Let F ∈ Z[t][u, x, y, z] be homogeneous as a polynomial in u, x, y, z. If there exist f, f3 ∈ Z[t, y], f1 ∈ Z[t, y, z] and 0 = r, s ∈ Z[t], such that [t]

{f, rx − f1 , sz − f3 } ⊂ SF , where f is irreducible in Q(t)[y], then SF admits a Gröbner basis {f, rx − f1 , sz − f3 } ⊂ [t] SF which satisfies Condition 4.4. Moreover, r and s have the form r = lc(f )i sm r and s = lc(f )j s, for some choice of i, j, m ∈ N. (t)

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Proof. Let m be the degree of f1 = f1 (t, z, y) as a polynomial in z. Substituting f3 for [t] sz in sm (rx − f1 ), we get a polynomial rx − f1 ∈ SF , where r = sm r and f1 depends only on t and y. In order to have a homogeneous notation we will write f3 = f3 and s = s. (t) Let P be an element of SF and denote by P ∈ Q(t)[y] the polynomial obtained substituting f1 / r for x and f3 / s for z in P . Then, P belongs to the ideal (P, rx − f1 , sz − f3 ) of Q(t)[x, y, z]. However, since f is irreducible, (t) P ∈ Q(t)[y] ∩ SF = (f ).

Thus, P ∈ (P, rx − f1 , sz − f3 ) ⊂ (f, rx − f1 , sz − f3 ), (t) so that SF ⊆ (f, rx − f1 , sz − f3 ). Since the opposite inclusion is clear, we conclude that [t] (t) G = {f, rx − f1 , sz − f3 } ⊂ SF is a Gröbner basis of SF . However, G may not satisfy Condition 4.4, because nj = degy (fj ) need not be smaller than degy (f ), for j = 1, 3. To get around this problem, let μj = max{nj − degy (f ) + 1, 0} and denote by fj the remainder of the division of cμj fj by f , where c = lc(f ). Then, [t] {f, cμ1 rx − f1 , cμ3 sz − f3 } ⊂ SF (t)

is a Gröbner basis of SF that satisfies Condition 4.4. Thus, to get the statement of the proposition, it is enough to take s = cμ3 s = cμ3 s and r = cμ1 r = cμ1 sm r. 2 4.3. Families of minimal involutive surfaces In the next theorem, we combine Corollary 4.3 with Lemma 4.5 to produce an effective criterion that can be used to prove that a given family of surfaces is minimal involutive. Recall that t denotes the set of parameters t1 , . . . , t . If d0 ∈ Z and q > 1 is a prime number, we write d0 + qZ = {d0 + qv : v ∈ Z }. Theorem 4.7. Let F ∈ Z[t][u, x, y, z] be homogeneous of degree k = 3 or k = 4 as a polynomial in u, x, y, z and assume that it satisfies Condition 4.4. If d0 ∈ Z is chosen such that (1) q does not divide r(d0 ) · s(d0 ) · Ψd0 (lc(f )); (2) Ψd0 (F ) is q-minimal, then V+ (Ψd (F )) is a minimal involutive surface for d ∈ (d0 + qZ ), whenever Ψd (F ) satisfies conditions C.1 and C.2 of subsection 3.1.

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Proof. Let [t]

G = {f, rx − f1 , sz − f3 } ⊂ SF (t)

be a Gröbner basis of SF as in Condition 4.4. Let H = Ψd0 (F ) and choose n ∈ Z such that G = Ψd0 +qn (F ) satisfies conditions C.1 and C.2. By Corollary 4.3, it is enough to check that G is q-Gröbner and that H ∼q G. Since, for any n ∈ Z, the specializations of r · s · lc(f ) at d0 and d0 + qn are congruent module q, it follows from hypothesis (1) that d0 , d0 + qn ∈ / V(r · s · lc(f )). Thus, the specializations of G to d0 and d0 + nq are themselves Gröbner bases in Q[x, y, z]. Let GH = {h, rH x − h1 , sH z − h3 } and GG = {g, rG x − g1 , sG z − g3 }

(4.6)

be the Z-reductions of the Gröbner bases that result from specializing G at d0 and d0 +nq, respectively. Since GH and GG generate the same ideals of Q[x, y, z] as the Gröbner bases of which they are the reductions, it follows from Lemma 4.5 that GH and GG generate SH and SG , respectively. In order to prove that G is q-Gröbner, note first that Ψd0 +qn (f ) ≡ Ψd0 (f ) (mod q). Thus, π(Ψd0 +qn (f )) = π(Ψd0 (f )). But q does not divide Ψd0 (lc(f )) by (1), so π(Ψd0 (f )) = 0. Hence, the polynomials π(g) and π(h) are associated in Fq [y]. Moreover, since q does not divide Ψd0 (lc(f )), it follows that deg(π(h)) = deg(h). Therefore, deg(g) ≥ deg(π(g)) = deg(π(h)) = deg(h) = m(k). On the other hand, since GG generates SG , we also have that m(k) ≥ dimC (C[x, y, z]/SG ) ≥ dimQ Q[y]/(g) = deg g. Thus, deg(g) = deg(π(g)) = m(k) and q does not divide lc(g). But we have already seen that π(g) is associated to π(h), which is irreducible because Ψd0 (F ) is q-minimal; so π(g) is also irreducible. Hence, from deg(g) = deg(π(g)), we conclude that g is irreducible in Oq [y] and, a fortiori, in Q[y]. Finally, since q divides r(d0 + qn) − r(d0 ) but not r(d0 ), it cannot divide rG , which is a factor of r(d0 + qn), and a similar argument shows that q does not divide sG . Turning now to the q-equivalence of H and G, note that we have already shown that π(h) and π(g) are associated of degree m(k) in Fq [y]. In particular, this implies that deg(H) = deg(F ) = deg(G). Moreover, G ≡ Ψd0 +qn (F ) ≡ Ψd0 (F ) ≡ H

(mod q).

In order to prove that π(h1 /rH ) = π(g1 /rG ), recall that, by construction, r(d0 )h1 = Ψd0 (f1 )rH and r(d0 + qn)g1 = Ψd0 +qn (f1 )rG . Thus, the desired equality follows from Ψd0 (f1 ) ≡ Ψd0 +qn (f1 ) (mod q) and r(d0 ) ≡ r(d0 + qn) ≡ 0 (mod q). A similar argument

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shows that π(h3 /sH ) = π(g3 /sG ), concluding the proof that H ∼q G. Therefore, as noted above, V+ (G) is a minimal involutive surface by Corollary 4.3. 2 Combining our previous results, we obtain the following algorithm for proving that a given polynomial of Z[t][u, x, y, z] defines a family of minimal involutive surfaces in P3 . Algorithm 4.8. Given a polynomial F ∈ Z[t][u, x, y, z] of degree 3 or 4 homogeneous in the variables u, x, y and z, a prime q > 1 and d0 ∈ Z , the algorithm returns one of two messages: minimal involutive, which means that V+ (Ψd (F )) is minimal involutive for all d ∈ d0 + qZ such that • V+ (Ψd (F )) is smooth; • Sing(ξΨd (F ) ) ⊂ D+ (u); or the algorithm failed. [t]

Step 1: compute a Gröbner basis G of SF with respect to the lexicographic order for which x > z > y are greater than all the ts; Step 2: check that there exist r, s ∈ Z[t] and f, f3 ∈ Z[t][y], f1 ∈ Z[t][y, z], such that f, rx −f1 , sz −f3 ∈ G, f is irreducible and q does not divide r(d0 ) ·s(d0 ) ·Ψd0 (lc(f )); Step 3: set H = Ψd0 (F ) and check that V+ (H) is smooth and that the singular set of ξH is contained in D+ (u); Step 4: compute the Z-reduced Gröbner basis GH = {h, rH x − h1 , sH z − h3 } of SH , where rH , sH ∈ Z and h, h1 , h3 ∈ Q[y]; Step 5: check that h is irreducible of degree m(k); Step 6: compute μH using Algorithm 3.2; Step 7: check that π(h) is irreducible in Fq [y] and that the image of t2 − π(μH ) is irreducible in Fq [y, t]/(π(h)); Step 8: return minimal involutive if all the tests return true; otherwise, return the algorithm failed. Proof. We need only show that if all the tests performed by the algorithm return true, then the conditions of Theorem 4.7 are satisfied. To begin with, the existence of a Gröbner (t) basis for SF satisfying Condition 4.4 follows from Proposition 4.6 and Steps 1 and 2 of the algorithm. Condition (1) of Theorem 4.7 is also checked at Step 2. Thus, by Lemma 4.5, {Ψd0 (f ), r(d0 )x −Ψd0 (f1 ), s(d0 )z −Ψd0 (f3 )} is a Gröbner basis of SH , whose Z-reduction is GH . In particular, since lc(h) · rH · sH divides Ψd0 (lc(f )) · r(d0 ) · s(d0 ), it is not divisible by q. Finally, in Steps 5, 6 and 7 we check whether H is q-minimal. Note that, when constructing μH , we should use  h1 = h1 /rH and  h3 = h3 /sH , because Algorithm 3.2 requires that the corresponding generators of SH are monic in x and z, respectively. 2

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4.4. Families of degree 3 with one parameter As in subsection 3.4, all the computations below were performed with the help of the computer algebra system Singular. For our first example we apply Algorithm 4.8 to F = y 3 + u2 x + 2x2 u + tz 3 ,

(4.7)

with q = 7 and d0 = 3. A Gröbner basis computation in the ring Q[t][x, y, z] shows that [t]

G = {f, 4x + 3zy 2 + 1, (1024t3 + 2t2 )z − f3 } ⊂ SF , where f = 27y 15 + (128t2 + 2t)y 6 − 32t2 y 3 + 2t2 is irreducible and f3 is equal to (6912t − 27)y 13 + 864ty 10 + 36ty 7 + (32768t3 − 384t2 − 2t)y 4 + (−4096t3 + 64t2 )y. Setting r(t) = 4 and s(t) = 1024t3 + 2t2 , we find that r(3) · s(3) · Ψ3 (lc(h)) ≡ 4 · (1024 · 27 + 2 · 9) · 27 ≡ 6

(mod 7),

which confirms that the conditions of Step 2 of Algorithm 4.8 hold for this choice of F . The conditions of Step 3 are satisfied by Lemma 3.4. The Gröbner basis of H = SΨ3 (F ) required in Step 4 is {h, 3074x − h1 , 9222z − h3 }, where h = 9y 15 + 386y 6 − 96y 3 + 6; h1 = −216y 12 − 9y 9 + 576y 6 − 9240y 3 + 382; h3 = 6903y 13 + 864y 10 + 36y 7 + 293758y 4 − 36672y. Next, using Singular’s factorization function, we check that π(h) = 2y 15 +y 6 +2y 3 −1 is irreducible of degree m(3) = 15 over F7 , so that Step 5 holds for H. Computing μH with the help of Algorithm 3.2, we are left with the task of proving that t2 + y 13 − y 10 − y 7 + 3y 4 + 2y is irreducible as an element of the ring F7 [y, t]/(π(f )), which can be done, once again, with the help of Singular’s factorization function. Therefore, by Algorithm 4.8 and Lemma 3.4, V+ (Ψd (F )) is a minimal involutive surface for all d ∈ 3 + 7Z. Using Singular it is also easy to show that if there exists a symplectic automorphism σ of A4 such that σ(Ψ3+7d (F )) = Ψ3+7e (F ), for integers e = d and  ∈ C, then 3136ed + 1344e + 1344d + 575 = 0; which does not have any integer solutions, because 575 is odd and all the other coefficients are even. Thus, V+ (Ψ3+7Z (F )) is a family of minimal involutive surfaces, no two of which are linearly symplectic isomorphic. Using a similar argument one can prove that V+ (Ψd0 +qZ (F )) is a family of minimal involutive surfaces for each pair (q, d0 ) in Table 1.

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Table 1 Pairs (q, d0 ) for the polynomial defined in (4.7). q 7 13 19 31 37 43 61 67 d0 3 4 15 11 24 24 29 5

73 3

79 7

223

97 4

4.5. Families of degree 4 with one parameter In this subsection, we study families of minimal involutive surfaces defined by the polynomial F = y 4 + u3 x + 2x3 u + tz 4 + x2 y 2 + u3 y.

(4.8)

We begin by finding the values of e, for which the polynomial Ψe (F ) defines a smooth surface and a foliation whose singularities belong to D+ (u). Lemma 4.9. If F = y 4 + u3 x + 2x3 u + tz 4 + x2 y 2 + u3 y, then V+ (Ψe (F )) is smooth and Sing(ξΨe (F ) ) ⊂ D+ (u), for all e = 0, 16. Proof. Let I be the ideal of Q[t][u, x, y, z] generated by F and its partial derivatives with respect to u, x, y and z. Using Singular to compute a Gröbner basis of I, we find that √ √ y 7 ∈ I. Thus, u3 = Fy |y=0 ∈ I, from which we deduce that 2x3 = Fu |u=y=0 ∈ I. √ Finally, 4tz 3 = Fz ∈ I. Therefore, if e = 0, then y, u, x and z belong to the radical of

√ Ψe ( I), which is Ψe (I). On the other hand, the singular locus of Ψe (F ) is V+ (Ψe (I)), because the partial derivatives with respect to u, x, y and z commute with Ψe . Hence, V+ (Ψe (F )) is smooth whenever e = 0. Turning now to the singular set of the foliation, let J be the ideal of Q[t][u, x, y, z] generated by u and the 2 ×2 minors of the matrix MF , defined at (2.3). Using Singular,

we find that y 8 (t − 16) ∈ J. Hence, y, u ∈ Ψe (J) when e = 16. Substituting u = y = 0

in MF , we discover that if e = 16, then 4ez 4 and −2x4 also belong to Ψe (J), which implies that there are no singularities of ξΨe (F ) at V+ (u) when e = 16. 2 [t]

Applying now Algorithm 4.8 to F , we must first compute a Gröbner basis of SF . The basis returned by Singular has six elements, three of which satisfy the conditions of Step 2. Using the notation of the algorithm we denote these polynomials by f , rx − f1 and sz − f3 , where r = 1875965850 and s is the product of 185315271772840448 · t with an irreducible polynomial of degree 74 in Z[t]. Moreover, f is an irreducible polynomial in Q(t)[y] of degree 40 = m(4), whose leading coefficient is c = 65536(t − 16)2 . Note that, once again, we must have e = 0, 16 or the algorithm will fail at Step 2. With the help of an implementation of Algorithm 4.8 in Singular and taking into account Lemma 4.9 we have been able to show that V+ (Ψd (F )) is minimal involutive whenever d ∈ d0 + qZ for the pairs (q, d0 ) in Table 2.

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Table 2 Pairs (q, d0 ) for the polynomial defined in (4.8). q 43 61 67 79 83 89 d0 6 3 51 32 4 48

4.6. A family with several parameters Our final example consists of a family with several free parameters defined by the polynomial F = t1 y 3 + t2 u2 x + t3 x2 u + t4 uz 2 + t5 xyz. We check first for what values of d = (d1 , . . . , d5 ) ∈ Z5 the variety V+ (Ψd (F )) is smooth. Computing a Gröbner basis of the ideal I generated by F and its partial derivatives with respect to u, x, y and z in the ring Q[t1 , . . . , t5 ][u, x, y, z], we find that by 5 ,

t24 t25 z 5 + (9t21 t3 t4 z − 9t21 t2 t5 y)y 4 ,

3t23 x3 + 3t3 t4 xz 2 − 5t2 t5 xzy, t2 u2 + 2t3 ux + t5 zy ∈ I,

where b = 110592t31 t53 t34 − 3125t1 t42 t65 . Thus, if d is chosen so that b(d) · d1 d2 d3 d4 d5 = 0,

then Ψd (I) = (u, x, y, z). Hence, V+ (Ψd (F )) is smooth for these values of d. Similarly, if J is the ideal of Q[t1 , . . . , t5 ][u, x, y, z] generated by u and the 2 × 2 minors of MF , then t1 y 3 ,

t4 t25 z 4 − t35 z 2 y 2 + 9t21 t3 y 4 ,

t3 x3 + t4 xz 2 + t5 xy 2 ∈ J.

Thus, if d satisfies d1 d2 d3 d4 d5 = 0 then Ψd (J) = (u, x, y, z), which implies that Sing(ξΨd (F ) ) ∩ V+ (u) = ∅. Summing up, we have shown that if g = b · t1 t2 t3 t4 t5 does not vanish at d ∈ Z5 , then V+ (Ψd (F )) is smooth and Sing(ξΨd (F ) ) ⊂ D+ (u). In particular, taking q = 101 and d0 = (1, 5, 1, 1, 1), we conclude that these two conditions hold for every d ∈ d0 + qZ5 , because g(d) ≡ g(d0 ) ≡ 0 (mod 101). Applying Algorithm 4.8 to F with q = 101 and d0 = (1, 5, 1, 1, 1) we find that [t]

G = {f, t2 x + 2t1 y 3 , (729t41 t72 t34 − 2t52 t24 t75 )z − f3 } ⊂ SF , where f3 ∈ Q[y] has degree 14 and f , as a polynomial in y with coefficients in Q[t1 , . . . , t5 ], is irreducible of degree 15, with leading coefficient c = 16t51 t3 t25 . Setting r = t2 and s = 729t41 t72 t34 − 2t52 t24 t75 we get that Ψd0 (rsc) ≡ 65 (mod 101). Performing Steps 4 to 8 of Algorithm 4.8 with the help of Singular, we find that all the required tests return true. Therefore, we have proved that V+ (Ψd (F )) is minimal involutive for all d ∈ d0 + 101Z5 .

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