Classification of Bertini s series of varieties of dimension less than or equal to four.

Classification of Bertini s series of varieties of dimension less than or equal to four. ENRICO ROGORA January 14, 2005 Author s address: Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale A. Moro 5, I-00185 Roma e-mail: Rogora@mat.uniroma1.it Abstract In this paper I give a classification of irreducible projective varieties of dimension less than or equal to four, according to a new classification scheme. No assumption is made about singularities. Mathematics Subject Classifications (1991): Primary, 14N05; secondary, 51N35. Key words: Bertini s series, dual variety. 1 Introduction In this paper, by a variety I shall mean a projective subvariety of some projective space over the complex numbers. No assumption about singularities is made. Let X P(V ) be an irreducible variety and let X P(V ) be its dual variety. The defect of X, denoted by δ(x), is δ(x) = N dim(x ) 1. For any hyperplane H, the contact locus of H with X, denoted by C H, is the closure of the set of smooth points x of X for which the embedded tangent space T X,x is contained in H. I shall denote by h the point in P(V ) representing a hyperplane H. Let V be an N + 1 dimensional linear space. For every integer k there exists a natural isomorphism ( ) : G(N 1 k, P(V )) G(k, P(V )), (1) The author has been supported by the Italian C.N.R. and M.U.R.S.T. 1

which sends w to the k dimensional subspace (w) P(V ), intersection of all hyperplanes of P(V ) which belong to w. Let X P(V ) be any variety. For the general tangent hyperplane H, it is well known that the contact locus C H is a linear space of dimension equal to the defect of X, and that C H = (T X,h), where T X,h denotes the embedded tangent space to X at the point h corresponding to H (see for example [[12]; par. 4 p.to 16] or [[6]; p. 394]). A linear space L X which is the contact locus for some hyperplane is called a contact space. The Bertini map of X, is the rational map β : X G(δ(X), P(V )) P( δ(x)+1 (V )) which sends the general hyperplane h X to its contact locus C H. Note that the Bertini map is the composition of the Gauss map for the dual variety with the natural isomorphism (1). The first Bertini s image of X, denoted by B 1 (X), is the image of β. A variety is said to be ordinary if its defect is zero, that is, if its dual is a hypersurface. A variety is said to be 1-ordinary if it is not ordinary and its first Bertini s image is ordinary. The second Bertini s image of X, denoted by B 2 (X), is by definition the first Bertini s image of B 1 (X). The n-th Bertini s image of a variety and the property of being n-ordinary are defined inductively. A variety is said to be essentially not ordinary or -ordinary if all its Bertini s images have positive defect. It is natural to associate to an n-ordinary variety X the power series Bert(X) = n δ(b j (X)) t j, j=0 where B 0 (X) = X, and n could be infinite. Bert(X) is called the Bertini s series of X. Bert(X) is a polynomial if and only if X is not -ordinary. If a variety is n-ordinary and n <, then the dual of B n (X) is a hypersurface. Its associated form is called the Bertini s form of X. Note that if B k (X) has defect zero, then B k+1 (X) = B k (X). The above notions have been introduced and discussed in [7] and [9], following an idea of Eugenio Bertini (see [11]). This paper is mainly concerned with the investigation of Bertini s images and Bertini s series of varieties of low dimension. The main results of this paper are: an upper bound for the dimension of B 1 (X) for varieties of defect 1 (theorem 3.2.6), a classification of 3-dimensional varieties with nontrivial Bertini s series (theorem 4.2.2), an analysis of bidimensional family of lines whose focal scheme is degenerate (theorems 4.3.2 and 4.3.3), a structure theorem for 2

3-dimensional varieties with degenerate Gauss map (theorem 4.3.4), a classification of 4-dimensional varieties of defect one whose Bertini s image has maximal dimension (theorem 5.2.2), a classification of 4-dimensional varieties with nontrivial Bertini s series (theorem 6.1.1), a classification of two dimensional family of planes with defect one (proposition 7.3.5). All these classification theorems could be further refined, and some refinements are already discussed in this paper. Many problems remain open. In particular it would be desirable to classify varieties of defect one whose Bertini s image has maximal dimension (see theorem 3.2.6 and subsection 3.3) and to refine the classification given in theorem 6.1.1 (see also the remark in subsection 6.1). The remarks in subsection 6.1 may help to clarify the direction of some possible refinements. The classification of varieties with nontrivial Bertini s series includes in particular the classification of varieties with degenerate dual variety. Since no assumption is made about singularities, theorem 6.1.1 is also an improvement of the result in [4] about classification of (possibly singular) varieties with degenerate dual variety. Of course it is well known that, if smoothness is assumed, much more can be said about this classification, by using adjunction theory. For proving the results contained in this paper, it is necessary to study the local differential geometry of families of linear spaces. This will be done by using the concepts of foci (introduced in 2), of second fundamental form of a projective variety (introduced in 5) and of Segre cone of a family of linear spaces (introduced in 7). These ideas have been developed by Corrado Segre. The interested reader is strongly reccomended to refer directly to his original works, so stimulating and rich of beautiful geometric ideas; in particular to the fundamental memory [12]. I would like to thank Ciro Ciliberto for all his help and constant encouragement, and Paola Contardi for many discussions about section 2. 2 First order foci of a family of linear spaces If the defect of X is positive, then X is covered by a family of linear spaces of positive dimension, namely the contact spaces. Therefore, for classifying varieties with non zero Bertini s series we need to deal with families of linear spaces. One of the main tools for dealing with the infinitesimal aspects of these families is the theory of foci, which will be summarized in this section. 2.1 Families of P k s Let Σ G(k, N) be a reduced subvariety of pure dimension α > 0 of the grassmannian of k-dimensional linear subspaces of P N, let I Σ Σ P N be the incidence variety of pairs (σ, q) such that q σ, and let p 1 : I Σ Σ and p 2 : I Σ P N be the maps obtained by restricting to I Σ the projections 3

of Σ P N to its factors. The morphism p 1 : I Σ Σ is said to be an α- dimensional family of k-dimensional linear subvarieties of P N. The variety Σ is the parameter space of the family, but for brevity I shall often refer to it as to the family itself. Obviously, dim(i Σ ) = k + dim(σ). Let me suppose that Σ is irreducible. I shall denote by V (Σ) the image of I Σ under p 2, and I shall say for brevity that V (Σ) is α P k. If Σ is complete, then V (Σ) is a complete subvariety of P N. Let us assume that the map p 2 : I Σ V (Σ) is birational. This means that there is a unique P k of the family Σ through the general point of V (Σ). Let us suppose moreover that the general point y V (Σ) is not contained in any (k + 1)-plane of V (Σ). In this case I shall say that (Σ, V (Σ)) is a scroll in P k s, or a scroll of type (α, k). For brevity I shall also refer to V (Σ) as to a scroll. Let p 1 : I Σ Σ P N Σ be an α-dimensional family of P k s in P N, let σ Σ be a general point and let G = p 2 p 1 1 (σ) = V (σ). The characteristic map χ(σ) : T Σ,σ O G H 0 (G, N G P N ) describes the first order deformations of G inside the family. Let α + k N. The (first order) focal scheme over G = V (Σ) of the family Σ, denoted by F σ, is the subscheme of G defined by the equations rk(χ(σ)) < α (2) We shall see in section 2.3 how to represent χ(σ) with an easily computable matrix. A point of F σ, for some σ Σ, is called a (first order) focus of the family Σ. Intuitively, a focus is the intersection of an element of the family with another, infinitely close to it. Let Σ 0 Σ be the subset of smooth points σ in Σ such that F σ has minimal dimension, and let F be the closure of the union, for σ Σ 0, of the underlying topological spaces of the focal schemes F σ. We shall call F, (with its reduced scheme structure), the focal variety of the family Σ. Theorem 2.1.1 Let Σ be an α-dimensional family of k-dimensional linear spaces in P N such that F = V (Σ). Then, dim(v (Σ)) < α + k. To prove this theorem it is enough to observe that under the hypothesis the differential of the map p 2 has rank less than α + k at the general point of I Σ, since, as is proved in [2], the map χ(σ) comes from a global carachteristic map χ which drops rank at the points where dp 2 does. 4

2.2 Local parametric representation of families of P k s Let q : M(k + 1, N + 1) G(k, N) be the rational map which sends a rank k + 1 matrix to the (k + 1) dimensional subspace of C N+1 spanned by its rows, i.e. the k-dimensional linear subspace of P N spanned by the k + 1 points P 1,..., P k+1, where the i-th row of the above matrix is a set of homogeneous coordinates for P i. For any f : Σ G(k, N), after possibly shrinking Σ, there exists a lifting g : Σ M(k + 1, N + 1) of f (not unique) such that q g = f. I shall call g a local parametric representation of f. In particular, given Σ G(k, N) with dim(σ) = α, and given any regular point α Σ, there always exists an open disc C α containing zero, an open subset Σ Σ containing σ and a map f : G(k, N) which induces a bianalytic isomorphism between and Σ, such that f(0) = σ. I shall call any local parametric representation of such an f, local parametric representation of the family Σ around σ. The rows of a local parametric representation of a family Σ give local parametric representations of k + 1 variable points in P N such that the linear span of the points corresponding to the value δ is the linear space represented by f(δ). I shall always identify a local parametric representation of a family Σ, i.e. a matrix valued function of α parameters, with k + 1 local parametric representation of points in P N, i.e. k + 1 vector valued functions, each of which is a row of the above matrix valued function. 2.3 Equations for the scheme of foci of a family of α P k s (α + k N) A useful piece of notation which I shall always adopt is the following: let f(u 1,..., u α ) be a given function defined around the origin of C α, then I shall denote f(0,..., 0) simply by f. Moreover I shall denote f u i by f ui Let a N k (u 1,..., u α ),..., a N (u 1,..., u α ) be a local parametric representation of Σ around G. Then, Let G =< a N k,..., a N >. (a N k+i ) uj = βi,jq 0 0 +... + β N k 1 i,j q N k 1 + β N k i,j a N k +... + βi,ja N N. We can define the following two matrices: β 0 i,1 λ i... β 0 i,α λ i F (λ 0,..., λ k ) =....., (3) β N k 1 i,1 λ i... β N k 1 i,α λ i 5

and β 0 0,j h j... β 0 k,j h j M(h 1,..., h α ) =...... (4) β N k 1 0,j h j... β N k 1 k,j h j The matrix F (λ 0,..., λ k ) is an (N k) α matrix, called the focal matrix, while the matrix M(h 1,..., h α ) is an (N k) (k + 1) matrix, called the deformation matrix. It is easy to see that the system of equations (2) is equivalent to rk(f (λ 0,..., λ k )) < α, (5) If we have a solution λ 0,..., λ k of rk(f (λ 0,..., λ k )) < α, then there exists a solution h 1,..., h α of the equation F (λ 0,..., λ k ) (h 1,..., h α ) t = 0 The focus λ 0,..., λ k is said to be a focus for the focal deformation h 1,..., h α. As an easy application of the theory developed so far we can prove the following result, due to Corrado Segre (see [14]). Theorem 2.3.1 Let Σ be N 1 P 1 P N, let r be a general element of Σ and let F r be the first order focal scheme of Σ over r. Then, either F r = r, or F r is a zero dimensional subscheme of F r of degree N 1. Proof. It follows immediatly by the fact that the focal matrix is an (N 1) (N 1) matrix of linear forms. q.e.d. The last piece of notation that I want to introduce in this section is that of tangent space to a deformation of a linear space, defined as follows. Let α G(k, N). An element of P(Θ G(k,N),α ) is called a first order projective deformation of α (briefly, deformation). Let Φ be a deformation of α and let Γ Φ be a smooth curve in G(k, N), containing α and such that Θ Γφ,α = Φ. It is easy to prove that the linear space spanned by the union of all tangent spaces to V (Γ Φ ) at regular points of V (α) depends only on Φ. This space will be called the tangent space to Φ. 2.4 First order foci and Bertini s images of 1 P k P N (1 + k N) I recall here how to build one dimensional families of linear spaces whose focal scheme on the general space has given dimension, and how to compute the corresponding Bertini s series. Statement a) has been proved by Corrado Segre (see [12] and [4]; p.385, lemma 2.2); statement b) has been proved by myself (see [9]; Lemma 4.10). 6

a) Let Σ be a scroll of type (1, k) and let f be the dimension of the space of foci on its general fibre (for clarity I shall denote sometimes this number by f Σ ). For the general k dimensional linear space G of the family there exist l + 1 points P 0,..., P l, k f curves Q 1 (t),..., Q k f (t), and positive integers a 1,..., a k f with l = k a i such that i) For all sufficiently small t, the linear space < P 0,..., P l, Q 1 (t), Q 1(t),..., Q (a1 1) 1 (t),..., Q k f (t), Q k f (t),..., Q (a k f 1) k f (t) > which I shall denote by G(t), belongs to Σ. ii) G(0) = G iii) The space of foci of Σ over G(t) is < P 0,..., P l, Q 1 (t), Q 1(t),..., Q (a1 2) 1 (t),..., Q k f (t), Q k f (t),..., Q (a k f 2) k f (t) > (where, if an index becomes negative, the corresponding term is dropped Viceversa, for each f with 1 f k 1, it is possible to construct, as above, a scroll Σ of type (1, k) for which the dimension of the focal scheme on the general element is f. I shall refer to this construction as the Segre construction. b) Let Σ G(k, N) be a scroll of type (1, k) and let f be the dimension of the space of foci over its general fibre. i) If f = 1, then Bert(V (Σ)) = (k 1)/(1 t). In particular δ(v (Σ)) = k 1 and dim(b 1 (V (Σ))) = k + 1. ii) If 0 f k 2, then Bert(V (Σ)) = (k 1) + ((k 2 f)/(1 t)). In particular δ(v (Σ)) = k 1 and dim(b 1 (V (Σ))) = k f. iii) If f = k 1, then Bert(V (Σ)) = k. In particular δ(v (Σ)) = k and dim(b 1 (V (Σ))) = 1. 3 Bounds on the dimension of the first Bertini s image for varieties of defect 1 In [9] I set some bounds for the dimension of the first Bertini s image of a variety. The goal of this section is to improve those bounds for varieties of defect 1. 7

3.1 Some recalls Let me recall the results proved in [9] (Corollary 7.7, and Proposition 7.9). Theorem 3.1.1 Let X P N, be an n-dimensional variety with defect δ < n. Then: i) n δ dim(b 1 (X)) (δ + 1)(n δ) δ. ii) If dim(b 1 (X)) = n δ, then either X is a scroll in δ-spaces, or it contains a P δ+1 through its general point. iii) Let δ > 1. If then (δ + 1)(n δ) 2δ + 1 dim(b 1 (X)) (δ + 1)(n δ) δ, δ + 1 n δ + 2, and X is a scroll of type (1, n 1). Moreover, if n = δ + 1, then there exists an integer k with 0 k δ, such that X is a cone with a (δ k 1)- dimensional vertex, projecting the scroll of osculating k-spaces to a curve. Another elementary but basic fact about contact spaces is contained in the following lemma. Lemma 3.1.2 Let X P N be a variety having defect δ. The family of the δ-spaces of contact passing through a general point of X is irreducible. Proof. The family of tangent hyperplanes to X at a general point p X is the family of hyperplanes of P N containing T X,p, hence it is represented in P(V ) by the linear space (T X,p ) defined in section 1. The family of the P δ s of contact through the point p in G(δ(X), P N ) is the image under the Bertini map of a dense subset of (T X,p ), hence it is irreducible. q.e.d. 3.2 Bounds on the dimension of the Bertini s image of a variety Lemma 3.2.1 Let X be a variety containing a family F of P k s such that a general point of X is contained in some element of F. Let α be a general element of F, p 0,..., p k be general points of α, and let m be a point of α, smooth on X. Then, T X,m < T X,p0,..., T X,pk >. 8

Proof. Let q be a general point of T X,m and let π q be the linear space spanned by q and α. I shall show that in each T X,pi there exists a k + 1-dimensional linear space π q,i such that π q < π q,0,..., π q,k >. (6) By letting q vary in T X,m, π q describes a family of subspaces which covers T X,m and π q,i describes a family of subspaces of T X,pi. Therefore, by (6) it follows that T X,m = q TX,m π q q TX,m < π q,0,..., π q,k > < T X,p0,..., T X,pk >. Let ˇα be the (n k 1) dimensional linear space representing all P k+1 s through α. To prove (6) it is enough to prove the existence of a projective transformation Φ : α ˇα such that Φ(m) = π q and Φ(p i ) T X,pi for i = 0,..., k. Let γ be a curve contained in X, containing m, and such that the linear space spanned by α and by the tangent line to γ at m is π q. I want to deform α along γ. Let F γ F G(k, P(V )) be the family of P k s of F intersecting γ and let ζ be a curve in F γ, passing through α. ζ can be thought of as a curve in G(k, P(V )), hence its Zariski tangent space at α can be identified to an element Φ P(Θ G(k,P (V ),α ). But it is well known that there exists a natural identification between the Zariski tangent space Θ G(k,P (V ),P (W ) and Hom(W, V/W ) (see for example [12] or [4]), hence Φ can be identified to a projective transformation Φ : α ˇα. Moreover, Φ(m) = π q by construction, and obviously Φ(p i ) T X,pi. This completes the proof. q.e.d. Corollary 3.2.2 Let X P N be a variety with defect δ, let π be a general δ-space of contact and let p 0,..., p δ be a set of points which span π. Let l = dim < T X,p0,..., T X,pδ >. Then dim(b 1 (X)) = l δ. Proof. Let d be the dimension of the general fibre of the Gauss map G of X. By the identification of the Bertini map with the Gauss map of dual variety given in section 1, dimb 1 (X) = dimx d. On the other side the fibre of G over < p 0,..., p δ > is the family of hyperplanes whose contact locus is π =< p 0,..., p δ >, i.e. the set of hyperplanes containing T X,m for every m π. By lemma 3.2.1 this set is the linear space < T X,p0,..., T X,pk >, 9

therefore d = N 1 l, hence the result. q.e.d. Let X P N be a variety with defect δ > 0. I shall denote by Σ p B 1 (X) the family of contact δ-spaces containing p and by γ p = V (Σ p ) the cone in P N described by these spaces. Obviously, dim(γ p ) = dim(σ p ) + δ. Lemma 3.2.3 Let x be a general point of X. Then dim(σ x ) = l n, where, dim(x) = n and l is defined in Corollary 3.2.2. Proof. Let us consider the incidence variety and the projection mappings I = {(α, x) B 1 (X) X x α} p : I B 1 (X), q : I X. Let α B 1 (X) be a general point. Since and also we get dim(i) = dim(b 1 (X)) + dim(p 1 (α)) = dim(b 1 (X)) + δ dim(i) = dim(x) + dim(q 1 (x)) = dim(x) + dim(σ x ), dim(σ x ) = dim(b 1 (X)) + δ n. and by corollary 3.2.2 we get the result. q.e.d. Lemma 3.2.4 Let X P N be a variety with defect δ > 0 and let dimb 1 (X) = k. i) If X is the general projection of X to P k+δ+1, then dim(b 1 (X )) = dim(b 1 (X)). ii) If H is a general hyperplane and if X H is the hyperplene section of X with H, then def(x H ) = def(x) 1. Proof. i) Let α be the general P δ of contact. The family of hyperplanes of P N whose contact locus is α has dimension N 1 k δ (the dimension of the fibre over α of the Bertini map). Let us project X from a general point p P N, and let X be the projection. The projection of a P δ of contact for X is of contact for X if and only if at least one hyperplane whose contact locus is that δ-space, 10

contains p. This happens for the general P δ of contact if N k δ 1 1, hence the result. ii) The general hyperplane K tangent to X H at a point p can be lifted to a hyperplane K tangent to X at p, in a unique way, by setting K =< K, T X,p >. Let C K be the contact space corresponding to K. Then, C K = C K H, which, for general H, has dimension one less than C K. q.e.d. Lemma 3.2.5 Let X P N be a variety with defect 1 and let γ x be the cone of contact lines through x. Then dimγ x dimt X,p T X,q where p, q are general points of a contact line through x. Proof. For p (γ x ) reg, dim(γ x ) = dim(t γx,p). Since γ x is a cone through x, T γx,p = T γx,q if < p, q > is a contact line containing x. Obviously T γx,p T X,p, hence T γx,p = T γx,q T q,x. Therefore T γx,p T X,p T X,q, hence the result. q.e.d. Theorem 3.2.6 Let X be an n-dimensional variety with defect one, then n 1 dimb 1 (X) 3(n 1)/2. Proof. The lower bound is part of 3.1.1 i). For the upper bound, let l = dim < T X,p, T X,q >. Then l = 2dim(T X,p ) dim(t X,p T X,q ) = 2n dim(t X,p T X,q ). By corollary 3.2.2, lemma 3.2.5 and the fact that dim(x) = dim(b 1 (X)) dim(γ x ) + 2, we get dim(b 1 (X)) = l 1 = 2n dim(t X,p T X,q ) 1 2n dim(γ x ) 1 = = 2n + n dim(b 1 (X)) 2 1 = 3n 3 dim(b 1 (X)), hence the result. q.e.d. 3.3 Varieties of defect 1 with Bertini s image of extremal dimension In general, if def(x) = δ and dim(b 1 (X)) = n δ, then the Gauss map has δ-dimensional general fibre and the family of contact spaces coincides with the family of fibres of the Gauss map. In fact, the tangent space to X at a smooth point p is the intersection of all hyperplanes which are tangent to X at p, therefore, if all tangent hyperplanes to X at p are tangent to X along the same contact space C, then the tangent space itself is constant along C. Hence the structure of varieties X for which dim(b 1 (X)) = n δ can be analyzed further by using a theorem of classification of varieties whose Gauss map has δ-dimensional general fibre. Theorems of this type are classical (see 11

for example [[12]; sect. 22, p. 108 and following sections] or [15]), and have been reproved for example in [[4]; (2.21) p.391 and (2.27) p. 392)]. I shall give a refinement of these theorems for 3-dimensional varieties in theorem 4.3.4 Something can also be said about the structure of varieties which reach the upper bound in theorem 3.2.6. First of all let me remark that there are simple examples for which that bound is reached, i.e. P m P m+1 P 2m+2 (see [[9]; proposition 6.4]). An example of an even dimensional variety which reach the bound is the one described in 6.5. Let Σ be an irreducible family of lines which cover a variety X P N. By ω r I shall denote the linear space spanned by the family of tangent spaces to X at the points of a general r Σ, which are smooth on X. Lemma 3.3.1 Let X P N be a k-dimensional variety with defect 1. Then dim(b 1 (X)) = dim(ω r ) 1. Proof. By lemma 3.2.1, ω r =< T X,p, T X,q >, with < p, q >= r, hence by corollary 3.2.2, dim(b 1 (X)) = dim(ω r ) 1. Lemma 3.3.2 Let Σ be a d-dimensional family of lines, let x be a general point of V (Σ), let k = dim(v (Σ)), let γ x = V (Σ x ) be the cone of lines of Σ through x, and let r be a general element of Σ. Then, dim(ω r ) 3k d 2. Moreover, if dim(ω r ) = 3k d 2, then for p and q general points of r, T γp,r = T γq,r, where T γp,r denotes the tangent space to γ p along the line r. Proof. Let s = d k + 1, then, dim(σ x ) = s. Therefore ω r is spanned by the tangent space to k 1 independent deformations of r in Σ (defined at the end of subsection 2.3). Given a general p r, we can choose s of these deformations in Σ p ; the tangent space to each of these is a P 2. The tangent spaces to the others are either a P 3 or a P 2. Therefore, dim(ω r ) 1 + s + 2(k 1 s) = 3k d 2. Moreover, if dim(ω r ) = 3k d 2 and if T γp,r T γq,r, there would exist a deformation of r in Σ q which is not contained in T γp,r, therefore contradiction. q.e.d. dim(ω r ) 1 + s + 1 + 2(k 2 s), Theorem 3.3.3 Let X P N be a 2m + 1-dimensional variety with defect 1 and such that dim(b 1 (X)) = 3m. If r is a general contact line and p and q are two general points of r, then T γp,r = T γq,r. In particular, if γ p = γ q, then X is an (m, m + 1) scroll. 12

Proof. By lemma 3.3.1, dim(ω r ) = 3m + 1, hence the result follows by lemma 3.3.2. q.e.d. Remark. If X is as in theorem 3.3.3, and m = 2, then X is an (m, m + 1) scroll (even without assuming γ p = γ q ). This might be true in general. 4 Classification of Bertini s series of varieties of dimension less than or equal to 3 In this section I shall classify Bertini s series of varieties of dimension less than or equal to 3. 4.1 Varieties of dimension less than or equal to two Let us begin with the classification of Bertini s series of varieties of dimension less than or equal to 2. 1. Varieties of dimension 1. The only irreducible curve with nonzero Bertini s series is P 1, linearly embedded in some P N, for which Bert(X) = 1. 2. Varieties of dimension 2. Also the classification of surfaces with nonzero Bertini s series is trivial. A surface may have defect 1 or 2 and we divide the discussion accordingly. Defect 2: The only irreducible surface with defect 2 is X = P 2 linearily embedded in some P N, for which Bert(X) = 2. Defect 1: If the defect is one, then the family of contact lines must be one dimensional (otherwise X would be a plane), hence the contact lines are the fibres of the Gauss map (see remark in subsection 3.3); therefore, either X is a cone or it is the tangent developable to a curve. In both cases, B 1 (X) is a curve, which cannot be a line because X is not a plane. Therefore B 1 (X) has no defect, and Bert(X) = 1. 4.2 Varieties of dimension 3 A complete classification of the possible Bertini s series of 3-dimensional varieties and a first rough classification of varieties with non zero Bertini s series (i.e. positive defect) can be given with small effort (theorem 4.2.2 below) by combining some results of [9] with some well known classical facts. In subsection 4.3 I shall refine some of the results of this subsection. 13

First of all, let me recall an important construction for building (singular) varieties with defect. Let A 1,..., A s P N be s projective nondegenerate varieties and let S(A 1,... A s ) be the variety obtained by projecting in P N the closure of the incidence variety I = {a 1,..., a s, t t < a 1,..., a s >, a i a j } A 1... A s P N. Obviously, dims(a 1,..., A s ) dima 1 +... + dima s + s 1. Proposition 4.2.1 If A 1,..., A s P N are s projective nondegenerate varieties, then def(s(a 1,... A s )) s 1. Moreover, if equality holds, then B 1 (S(A 1,..., A s )) is birationally equivalent to the family of s 1-dimensional linear subspaces intersecting the varieties A i. In particular, if A 1 =... = A s = A, then the first Bertini s image is birationally equivalent to the s-th symmetric product of A. Proof. Let a i A i be a general point, let S(a 1,... a s ) be the (s 1)-dimensional linear space spanned by a 1,..., a s, and let b S(a 1,..., a s ) be a general point of S(a 1,..., a s ). Then, by Terracini s lemma, (see [16] or [3]) T b S(A 1,..., A s ) =< T a1 A 1,..., T as A s >, hence T b S(A 1,..., A s ) is constant along < a 1,..., a s >, and the result follows. q.e.d. Theorem 4.2.2 Let X P N be a three dimensional irreducible variety, let δ = def(x), and let b = dim(b 1 (X)). If δ > 0, then one of the following cases holds. i) If δ = 3, then X is a three dimensional linear subspace of P N, b = 0, and Bert(X) = 3. ii) If δ = 2, then X is a scroll V (Σ) of type (1, 2), b = 1, f Σ Bert(X) = 2. = 1 and iii) If δ = 1 and b = 3, then X is a scroll V (Σ) of type (1, 2), f Σ = 1 and Bert(X) = 1/(1 t). iv) If δ = 1, b = 2 and B 1 (X) is ruled, then X is a scroll V (Σ) of type (1, 2) (whose Gauss map has 1-dimensional general fibre), f Σ = 0, and Bert(X) = 1. v) If δ = 1, b = 2 and B 1 (X) is not ruled, then the Gauss map of X has 1-dimensional general fibre and Bert(X) = 1. Proof. 14

δ = 3 The general point of X is contained in a P 3 of contact, which must coincide with X by irreducibility. This is case i). δ = 2 Through the general point of X there is a P 2 of contact, (unique by Lemma 3.1.2), which endows X with the structure of a scroll of type (1,2), hence the result follows by 2.4, b). This is case ii). δ = 1 I shall divide the case in two further cases according to the possible values of b = dimb 1 (X). By theorem 3.2.6 one has 2 b 3. b=3 X is a scroll in planes over a curve. In fact, by a theorem of Severi (see [10]) a 3-fold containing a 3-dimensional family of lines either is a scroll or it is a quadric. But for a 3-dimensional quadric with defect 1, b = 2. Moreover, for a scroll of type (1, 2), b = 3 if and only if the general fibre has no foci by 2.4, b). This is case iii). b=2 In this case the contact lines are the fibres of the Gauss map by 3.3. For the defect of B 1 (X), let us observe that if B 1 (X) is ruled, then X is a scroll of type (1, 2) whose general fibre contains a focus by 2.4. The focal variety F (defined in 2.1) cannot be a point, because, if it were, X would be a cone and then def(x) would be greater or equal to 2, against the hypothesis. Therefore F is a curve. In these hypothesis, there exists a family Σ of planes which are tangent to but not osculating to F, and X = V (Σ) by 2.4 b). Therefore B 1 (X) has no defect and Bert(X) = 1, see again 2.4 b). This is case iv). If B 1 (X) is not ruled, then B 1 (X) has no defect, and again, Bert(X) = 1. This is case v). Note that in this case X is not a scroll in P 2 s 4.3 Refinements of the classification Theorem For refining the analysis of case v) of theorem 4.2.2, I shall look at the focal variety F introduced in 2.1 and I shall improve a result proved by Corrado Segre [[12]; sect. 22] about the classification of varieties with degenerate Gauss map (see also [[4]; (2.21) p. 391 and (2.27) p. 392]). This will be theorem 4.3.4. For that I need to prove some preliminary results (theorems 4.3.1, 4.3.2 and 4.3.3). The first result is classical (see [12] and [14]). I shall give a proof for the reader s convenience. Theorem 4.3.1 Let Σ be k P 1 P N and let F be its focal variety. Then, if there exists a component F F such that dimf = k, the general line r Σ is tangent to F. Proof. The proof consists of an easy local computation. It is always possible to find a local parametric representation a N 1 (u 1,..., u k ), a N (u 1,..., u k ) 15

for Σ around r such that i) a N (u 1,..., u k ) is a local equation for F ; ii) a focal deformation of r with focus a N is (1, 0,..., 0). One verifies, by using the notations introduced in 2.3, that i) and ii) imply M(1, 0,..., 0)(0, 1) t = 0 i.e. (a N ) u1 < a N 1, a N >. (7) Note that a N and (a N ) u1 are distinct in general, since they span the tangent space to F toghether with (a N ) u2,..., (a N ) uk 2, and the tangent space has dimension k 1 by hypothesis, hence (7) implies that a N 1 < a N, (a N ) u1 >, whence the claim. q.e.d. Theorem 4.3.2 Let Σ be a two dimensional family of lines in P N intersecting a curve C. Let r be a general line of Σ and let a C be a point at which r intersects C. Then: i) the focal scheme of Σ over r contains a; ii) if the focal scheme on r is nonreduced (i.e. it coincides with 2a), then r is contained in a tangent plane to C at a. Proof. Let a N 1 (u 1 ) be a local parametric representation for C such that a N 1 = a. For P in C, let γ p be the cone of lines of Σ containing P. Let q 0,..., q N 2, q N 1 be points of P N spanning an (N 1)-dimensional linear subspace π P N not containing a N 1 (u 1 ) for general u 1. The intersection γ an 1 (u 1 ) π is a curve. We call it Γ(u 1 ). Let a N (u 1, u 2 ) = K 0 (u 1, u 2 )q 0 +... + K N 1 (u 1, u 2 )q N 1. (8) A local parametric representation for Σ is given by a N 1 (u 1 ), a N (u 1, u 2 ). Let us consider the projective reference system q 0,..., q N 2, a N 1, a N. We have and (a N 1 ) u1 = N 2 s=0 β s 01q s + β N 1 01 a N 1 + β N 01a N, (a N 1 ) u2 = 0, 16

hence β s 02 = 0 for s = 1,..., N, and (a N ) u2 = (K 0 ) u2 q 0 +... + (K N 1 ) u2 q N 1. Therefore, β i 12 = (K i ) u2 β N 12K i 0 i N 2.... β N 1 12 = 0 The focal matrix is F (λ 0, λ 1 ) = β N 12 = (K N 1 ) u2 /K N 1 β 0 01λ 0 + β 0 11λ 1 β 0 12λ 1.. β01 N 2 λ 0 + β11 N 2 λ 1 β12 N 2 λ 1. (9) Since rk(f (λ 0, 0)) = 1, then the point a is a focal point. This proves part i). To prove part ii) we observe that by (9) the focal scheme is non reduced i.e. its equation is λ 2 1 = 0, if and only if rank β 0 01 β 0 12.. β01 N 2 β12 N 2 1 that is if and only if the following relation holds ( β01 rank 0... β01 N 2 ) 1 K N 1 (K 0 ) u2 (K N 1 ) u2 K 0... K N 1 (K N 2 ) u2 (K N 1 ) u2 K N 2 (10) Condition (10) is equivalent to the fact that Γ(u 1 ) is a line containing the point of intersection of the tangent line to C at a N 1 (u 1 ), which we denote by R. To prove the equivalence, hence the theorem, it is enough to compute the coordinates of R and of the tangent vectors to Γ(u 1 ) in the open affine subset U n 1 of π where the n 1-th homogeneous coordinate is non zero and to check that condition (10) means that all tangent vectors to Γ(u 1 ) contain R. q.e.d. Let Σ be a two dimensional family of lines in P N such that a component of the focal variety is a surface F. We proved in theorem 4.3.1 that the general line of Σ is tangent to F. If the focal scheme on the general line is non reduced we can say more. Theorem 4.3.3 For Σ as above the focal scheme on the general line is non reduced if and only if Σ is the family of tangent lines to a family of asymptotic curves of F. 17

Proof. Let a(u 1, u 2 ) be a local parametrization for F. We can choose a local parametric representation of Σ around r of the form Let us define and let us consider a reference system a(u 1, u 2 ), a u1 (u 1, u 2 ). q N 2 = a u2, a N 1 = a, a N = a u1, q 0,..., q N 2, a N 1, a N. We have a u1 = a N, hence β01 s = 0 if s N and β01 N = 1. a u2 = q N 2, hence β02 s = 0 if s N 2 and β02 N 2 = 1. Therefore, the focal matrix is β11λ 0 1 β12λ 0 1 F (λ 0, λ 1 ) =... β11 N 2 λ 1 λ 0 + β12 N 2 λ 1 Hence the focal scheme is nonreduced if and only if it has equation λ 2 1 = 0, i.e. ( β 0 rank 11... β11 N 2 ) 1, 0... 1 and this is equivalent to a u1u 1 < a u1, a u2, a >. If a u1u 1 < a, a u1 >, then F is a ruled surface, but then the family Σ is 1 dimensional. Therefore we must have < a, a u1, a u1u 1 >=< a u1, a u2, a >, hence, the u 1 -lines are asymptotic lines. q.e.d. By using the above results we can get some more information about three dimensional varieties whose Gauss map has 1-dimensional general fibre, then about cases iv and v) of theorem 4.2.2. Theorem 4.3.4 Let X be an irreducible 3-dimensional subvariety of P N such that the general fibre of the Gauss map is 1-dimensional, and let Σ be the family of 1-dimensional fibres of the Gauss map. Then one of the following cases holds. i) There exist two irreducible surfaces S 1 and S 2 such that the general line in Σ is tangent to both S 1 and S 2. 18

ii) There exists an irreducible surface S and an irreducible curve C such that the general line in Σ is tangent to S and intersects C. iii) There exist two irreducible curves C 1 and C 2 such that Σ = S(C 1, C 2 ). (S(C 1, C 2 ) has be defined in subsection 4.2). iv) There exists an irreducible surface S such that the general line in Σ is bitangent to S. v) There exists an irreducible curve C such that Σ = S(C, C). vi) There exists a surface S such that Σ is the family of tangent lines to a family of asymptotic curves of S. vii) There exists a curve C and a scroll Σ of planes, tangent but not osculating C, such that the general line in Σ intersects C and for the general P C, γ P is a plane of the family Σ, where γ P is the cone spanned by the lines of the family Σ containing P. viii) The variety X is a cone. Proof. Let r be a general line of the family Σ, and let F r be the focal scheme on r. Since the Gauss map is constant along r, the first order deformations of r in Σ are contained in the tangent space to X along r, which is a P 3. By choosing a suitable reference system for writing the equations of the focal scheme, either the focal scheme has degree 2, or it is the entire line by theorem 2.3.1. The second possibility does not occur by theorem 2.1.1. Then the focal variety F has at most two components since it has at most two components on the general line. Case a) F has two components F 1 and F 2. Then, on the general line r Σ, F r is a reduced 2 points scheme F r = (F 1 ) r + (F 2 ) r. We can assume that dimf 1 dimf 2. By considering all numerically possible values for (dimf 1, dimf 2 ) we have the following cases: (2,2) the general line in Σ is tangent to both F 1 and F 2 by theorem 4.3.1. This is case i). (2,1) the general line in Σ is tangent to F 1 by theorem 4.3.1 and intersects F 2. This is case ii). (2,0) is impossible, otherwise all tangent planes to the surface F 1 pass through a fixed point. (1,1) gives rise to case iii). (1,0) and (0,0) are impossible because dimσ = 2. Case b) F is irreducible and, on the general line r Σ, F r is a reduced 2 points scheme. By considering all numerically possible values for (dimf ) we have either case iv) or case v). 19

Case c) F is irreducible and, on the general line r Σ, F r is a nonreduced 1 point scheme. By considering all numerically possible values for (dimf ) we have: (2); the general line r is tangent to F along an asymptotic line by theorem 4.7. This is case vi). (1); for the general point P F, γ P is a tangent plane to F, not osculating F, by theorem 4.3.2. This is case vii). (0); Σ is a family of lines containing the point F, and we get case viii). q.e.d. 5 The second fundamental form of a variety In this paragraph I shall briefly recall the definition of the second fundamental form of a subvariety of P N and its main properties, in order to prove a result which is needed for the classification of Bertini s series of 4 dimensional varieties (theorem 5.2.2). For details about the second fundamental form of a variety, see [4] and [[7]; cap I, par. 2]. 5.1 Generalities Let X P N be a k-dimensional variety, and let p X be a general point of X, let y = (y 0 :... : y N ) be homogeneous coordinates in P N, and let α be a hyperplane, with homogeneous equation ξ y = 0. We consider local homogeneous parametric equations y = y(x 1,..., x k ) for X such that y(0) = p. The hyperplane section α X has equation: ξ y(x 1,..., x k ) = 0. (11) The hyperplane α is tangent to X at p, if and only if all terms of order one vanish in the Taylor expansion of (11) in a neighborhood of 0. Let P p be the set of hyperplanes tangent to X at p and let M p be the set of hyperplanes tangent to X at p for which all second order terms in the Taylor expansion at the origin vanish. Let m = dim(m p ). If α P p \M p, then the quadratic term in the Taylor expansion of (11) at 0 defines a quadric in P(Θ X,p ), where Θ X,p is the Zariski tangent space to X at p. This quadric is the projectivization of the tangent cone to α X at p. In this way we get a linear map π Mp : P p P N k m 2 P(Sym 2 (Θ X,p)). The linear system of quadrics obtained in this way is called the second fundamental form of X at p, and is usually denoted by II p. Remark. Let X P N and let p be a general point of X. Let T (2) X,p be the second osculating space to X at p, defined as follows. Let y = y(x 1,..., x k ) be a local parametrization of X centered at p, as above. Then T (2) X,p is the linear 20

space spanned by p and by all partial derivatives of degree less than or equal to 2 of y(x), evaluated at zero. The definition is easily proved to be independent on the parametrization. Then, since M p is the space of hyperplanes containing T (2) X,p, m = N 1 dim(t (2) X,p ), hence dim(ii p ) = dim(t (2) X,p ) k 1. In lemma 5.1.1 I shall collect some facts concerning the second fundamental form of a variety which I shall use for proving theorem 5.2.2. Proofs can be found in [4] and in [7]. Lemma 5.1.1 a) Let p be a general point of X P N. The points contained in the base locus of the linear system of quadrics of the second fundamental form of X at p correspond to lines in P N having multiplicity of intersection at least 3 with X at p. b) The Gauss map of X has m-dimensional fibres if and only if, at the general point p X, all quadrics of II p (X) are singular along a fixed P m 1. Moreover, any such P m 1 gives the direction of a P m on X which is the fibre of the Gauss map G of X over G(p). c) X has defect δ if and only if, at the general point p X, each quadric in II p (X) has a P δ 1 of singular points. Moreover, each of these P δ 1 gives the directions a family of P δ s on X which are the contact loci for the hyperplanes tangent to X at p. 5.2 Applications Lemma 5.2.1 Let Σ be a positive dimensional linear system of surfaces in P 3 and let γ be a curve contained in the base locus of Σ. Let us suppose that the general surface of Σ is nonsingular at the general point of γ but it has at least a singular point, variable on γ when the surface varies in Σ. Then, all surfaces of Σ are tangent along γ, that is γ has multiplicity at least two in the intersection of all surfaces of Σ. Proof. Let π : P P 3 be the blow up of P 3 along γ and let E = P(N γ P 3) be the exceptional divisor of the blow up. Let Σ be the linear system { X = π X E} X Σ and let Σ be the linear system cut by Σ on E, i.e. Σ = { X E} X Σ. 21

Let X Σ be a surface having only a finite number of singularities on γ, and let them be {x 1,..., x r }. The divisor X = X E can be written as X = D + r n i E i, where, E i is the fibre of E over x i, n i is a positive integer and D is the unisecant of E representing, over the general point of γ, the tangent plane to X at x. To prove the lemma we need only to show that D remains fixed when X Σ varies. Under the hypothesis that the general surface X Σ has at least one singular point variable on γ, it follows that the linear system Σ in E consists of reducible curves. By discarding the fixed components we get a new linear system Σ. Each curve in Σ contains the fibre E i over the singular point of the surface X which moves on γ. If the general curve of Σ is irreducible, then D is a part of the fixed component and the lemma is proved. Otherwise, by Bertini s theorem, each curve in Σ would be composed with a pencil. Assume by contradiction that D is not fixed. Then the pencil should contain the general fibre and D, which is a unisecant of the fibre: contradiction. q.e.d. Theorem 5.2.2 Let X P N be a 4-dimensional variety with defect 1 and dim(b 1 (X)) = 4. Then, X is a scroll of type (2, 2). Proof. Let II p (X) be the second fundamental form of X in P(Θ X,p ) = P 3. Then II p (X) is a linear system of rank 3 quadrics (by Lemma 5.1.1 c)), of positive dimension, otherwise the Gauss map would be degenerate and dim(b 1 (X)) = 3. The vertex of the general qudric in II p (X) describes one component of the base locus of the linear system II p (X) (by Lemma 5.1.1). The base locus has degree less than or equal to four. Moreover, the general vertex of these quadrics gives the direction of a line in P N contained in the variety X, which is a contact line for a hyperplane. The multiplicity of our component of the base locus of the second fundamental form is at least 2 by lemma 5.2.1. Therefore its degree is at most 2, but it cannot be 2 otherwise each cone of the linear system II p would have a double point on it. Hence for the general x X, the contact lines containing it fill a plane, and X contains a plane through its general point. On the other side, X does not contain a P 3 through its general point or the base locus of its second fundamental form would be a plane and the defect of X would be 2. Therefore X is a scroll in planes over a surface. q.e.d. i=1 22

6 Classification of Bertini s series of varieties of dimension 4 In theorem 6.1.1 I shall prove a rough classification theorem for varieties of dimension 4 with non zero Bertini s series (i.e. positive defect). Like the corresponding result for 3 dimensional varieties, i.e. theorem 4.2.2, some cases of 6.1.1, namely cases vi)-ix) deserve more attention. In the rest of this section I shall give examples of any of these cases. The analysis can be further deepened and I plan to do this in a forthcoming paper. 6.1 General results Theorem 6.1.1 Let X P N be a 4-dimensional irreducible variety, and let δ = def(x) and b = dim(b 1 (X)). If δ > 0, then one of the following cases holds. i) If δ = 4, then X is a 4-dimensional linear subspace of P N, b = 0, and Bert(X) = 4. ii) If δ = 3, then X is a scroll V (Σ) of type (1, 3), b = 1, f Σ Bert(X) = 3. = 2 and iii) If δ = 2 and b = 4, then X is a scroll V (Σ) of type (1, 3), f Σ = 1 and Bert(X) = 1/(1 t). iv) If δ = 2, and b = 3, then X is a scroll V (Σ) of type (1, 3), f Σ = 0 and Bert(X) = 2 + 1/(1 t). v) If δ = 2, b = 2, and B 1 (X) is a scroll of type (1,1), then the general contact plane is a fibre of the Gauss map, X is a scroll V (Σ) of type (1, 3), f Σ = 1 and Bert(X) = 2. vi) If δ = 2, b = 2, and B 1 (X) is not a scroll of type (1,1), then the general contact plane is a fibre of the Gauss map, X is a scroll of type (2, 2) and Bert(X) = 2. vii) If δ = 1, b = 3, and B 1 (X) is a scroll of type (2,1), then X is a scroll of type (2, 2) and the general contact line is a fibre of the Gauss map. viii) If δ = 1, b = 3, and B 1 (X) is not a scroll of type (2,1), then X is a scroll of type (3, 1), the general contact line is a fibre of the Gauss map and Bert(X) = 1. ix) If δ = 1 and b = 4, then X is a scroll V (Σ) of type (2, 2) such that the general line in the general plane of the family Σ is a contact line. Proof. 23

δ = 4 The general point of X is contained in a P 4 of contact, wich must coincide with X by irreducibility. This is case i). δ = 3 The general point of X is contained in a P 3 of contact, which is unique by Lemma 3.1.2. Therefore B 1 (X) endows X with the structure of a scroll of type (1, 3). We get ii) by 2.4 b). δ = 2 I shall divide the case in three subcases according to the possible values of b = dimb 1 (X). By theorem 3.1.1 i); 2 b 4. b=4 We get iii) by theorem 3.1.1, iii) and by 2.4 b). b=3 We get iv) by theorem 3.1.1, iii) and by 2.4 b). b=2 By theorem 3.1.1, ii) and by 2.4 b), we get either v) or vi). In both cases the general contact plane is a fibre of the Gauss map. δ = 1 I shall divide the case in two subcases, according to the possible values for b = dimb 1 (X). By theorem 3.2.6; 3 b 4. b=3 If B 1 (X) is not a scroll, then it has no defect and we get vii). If B 1 (X) is a scroll, then the contact lines through a general point span a well defined plane which cannot sit in a 3-dimensional plane contained in X because of 2.4 b) (the defect is one). Hence we get viii). In both cases the general contact line is a fibre of the Gauss map. b=4 We get ix) by theorem 5.2.2. q.e.d. Remarks. Let X be a 4 dimensional variety with non zero Bertini s series, i.e. positive defect. We say that X has b-type j if X satisfies case j of theorem 6.1.1. Theorem 2.4, part a), provides us with an explicit procedure for building all 4 dimensional varieties of b-types ii), iii), iv) and v) (Segre s construction), therefore for these types we have a complete classification. The classification problem for 4 dimensional varieties with non zero Bertini s series would be completely solved if one could find explicit constructions for building all 4 dimensional varieties of the remaining b-types. In the rest of this section I shall only give some examples for each occuring type. Four dimensional varieties of b-type vi), vii) and ix) are covered by a 2 dimensional family of planes. Varieties which are covered by a two dimensional family of planes have in general (in a sense which is not necessary to specify here, but see 7) defect zero. The condition of having positive defect is of first order infinitesimal nature, and we have already introduced two tools for dealing with first order infinitesimal questions related to families of linear spaces, namely the theory of first order foci and the theory of fundamental forms. In section 7 I shall discuss another tool, namely the Segre cone at a general element of a family of linear spaces and I shall use it for investigating some properties of two dimensional families of planes with defect 1 (see proposition 7.3.5). 24

As we shall see, our examples of varieties of b-type vi) are particularizations of examples of varieties of b-type vii), hence our list of examples begins with four dimensional varieties of b-type vii). 6.2 Varieties of b-type vii) 1. Cones Let Y P N be a scroll of type (2, 1) with defect zero, let P P N be a general point and let X be the cone over Y with vertex P. Then X is a scroll of type (2, 2) and b-type vii). 2. Tangent planes to a surface Let X P N be a surface and let Σ be the closure of the family of tangent planes to smooth points of X. I shall investigate the b-type of V (Σ). The following definition is useful. A surface X P N satisfies k independent Laplace equations when the dimension of the second osculating space at its general point is 5 k. This concept is classical and has been considered by many authors Obviously, 0 k 3. Moreover we have: i) k = 3 if and only if X is a plane; ii) k = 2 if and only if either X P 3, or X is a developable ruled surface. (See [[15]; p.571]). Let X P N be a surface, let n 5 and let X(u, v) be a local parametric representation of X. Then, X(u, v), X u (u, v), X v (u, v) is a local parametric representation of the family Σ of tangent planes to X. This family is twodimensional if and only if X is not developable. We shall consider now the Gauss map of V (Σ). The tangent space to V (Σ) at the point λx +µx u +νx v is spanned by the points X, X u, X v, λx u +µx u,u +νx u,v λx v + µx u,v + νx v,v hence it coincides with < X, X u, X v, µx u,u + νx u,v, µx u,v + νx v,v >. Therefore it is clear that the Gauss map of V (Σ) is constant along the line spanned by X and < µx u + νx v >. Moreover is easily seen that the condition that fibres of the Gauss map are strictly greater than these lines is that the second osculating space has dimension less than or equal to 4, in which case the Gauss map is constant along each tangent plane. Putting all this toghether we get that V (Σ) has b-type vii) if and only if X does not satisfy any Laplace equation and has b-type vi) if and only if X satisfies one independent Laplace equation. 3. Tangent planes to a curve Let γ be a curve in P N, N 5. For each smooth point p γ, let Σ p be a one dimensional family of planes containing the tangent line to γ at p and let Σ be the closure of the union of such families. I shall investigate the b-type of V (Σ). Let X(t) be a local parametric 25