The geometry of the flex locus of a hypersurface

We give a formula in terms of multidimensional resultants for an equation for the flex locus of a projective hypersurface, generalizing a classical result of Salmon for surfaces. Using this formula, we compute the dimension of this flex locus, and an upper bound for the degree of its defining equations. We also show that, when the hypersurface is generic, this bound is reached, and that the generic flex line is unique and has the expected order of contact with the hypersurface.


Introduction
A point of a projective variety is a flex point if there is a line with order of contact with the variety at this point higher than expected. It is a generalization of the notion of inflexion point of a curve. The study of the flex locus of curves and surfaces is a classical subject of geometry from the XIXth century, treated by Monge, Salmon and Cayley, among others. Currently, there is an increasing interest in this object in low dimensions, mainly due to its applications in incidence geometry [Tao14,Kat14,GK15,Kol15,EH16,SS18].
In this text, we study the geometry of the flex locus of a hypersurface of a projective space of arbitrary dimension. Before explaining our results, we introduce some notation and summarize the previous. Let K be an algebraically closed field of characteristic zero, P n the projective space over K of dimension n ≥ 1, and V a hypersurface of P n of degree d ≥ 1. A point p ∈ V is a flex point if there is a line with order of contact at least n + 1 with the hypersurface V at the point p, and any such line is called a flex line (Definition 3.3). The flex locus of V is the set of all the flex points of V .
An important result in this context is the so-called Monge-Salmon-Cayley theorem for surfaces in P 3 , see for instance [Tao14,Kol15], generalized by Landsberg to the higher dimensional case [Lan99,Theorem 3]. It states that if the hypersurface V is irreducible, then it is ruled if and only if all of its points are flexes.
A hypersurface of degree less than n is necessarily ruled (Proposition 3.6) and its flex locus is the whole hypersurface. Hence, one restricts the study of the flex locus to the case d ≥ n.
For a plane curve C ⊂ P 2 of degree d ≥ 2, a point p ∈ C is an inflexion point if and only if the determinant of the Hessian matrix of the defining polynomial of C vanishes at p. This implies that the flex locus of C is defined by a polynomial of degree 3d − 6. Hence if C contains no line, then it has at most 3d 2 − 6d inflexion points, by Bézout theorem.
For a surface S ⊂ P 3 of degree d ≥ 3, an old result of Salmon states that there is a homogeneous polynomial in K[x 0 , x 1 , x 2 , x 3 ] of degree 11d − 24 defining its flex locus [Sal65, Article 588, pages 277-278], see also [EH16,§11.2.1]. If S has no ruled component, then this result together with the Monge-Salmon-Cayley theorem and Bézout's theorem imply that the flex locus is a curve of S of degree at most 11d 2 −24d.
We first address the problem of computing the dimension, and the degree of both the defining equations and the flex locus. Let x = {x 0 , . . . , x n } be a set of n + 1 variables and f V ∈ K[x] a squarefree homogeneous polynomial defining V . Let t be another variable and y = {y 0 , . . . , y n } a further set of n+1 variables. Then we consider the family of bihomogeneous polynomials f V,k , k = 0, . . . , d, in K[x, y] determined by the expansion Our first main result gives an equation for the flex locus of V in terms of multivariate resultants.
defining the flex locus of V . It is uniquely determined modulo f V by the condition for any linear form ∈ K[x], where Res y denotes the resultant of n + 1 homogeneous polynomials in the variables y.
This result recovers the previous degree computations for the polynomial defining the flex locus of a plane curve or of a surface in P 3 . It also allows us to give a scheme structure to the flex locus: we define the flex scheme Flex(V ) as the subscheme of P n defined by the homogeneous polynomials f V and ρ V (Definition 3.11). This scheme does not depend on the choice f V , unique up to a nonzero scalar factor, nor on that of ρ V , unique modulo f V . Thus, the flex locus of V is the reduced scheme associated to Flex(V ).
Giving a closed form for a canonical representative for ρ V modulo f V seems to be a challenge on its own. In the case of curves, such a representative is given by the determinant of the Hessian matrix of f V (Example 3.12). For n = 3, Salmon also obtained a representative of this polynomial as a determinantal closed formula in terms of covariants, based on an approach by Clebsch [Sal65, Articles 589 to 597]. It would be interesting to generalize these formulae to higher dimensions.
The next corollary is a direct consequence of Theorem 1.1 and Landsberg's theorem generalizing the Monge-Salmon Cayley theorem [Lan99, Theorem 3].
Corollary 1.2. If V has no ruled irreducible components, then Flex(V ) is a complete intersection subscheme of P n of dimension n − 2 and of degree In particular, the flex locus of V is set-theoretically defined by equations of degree at most max(d, d n k=1 n! k − (n + 1)!), and its degree, as an algebraic set, is at most d 2 n k=1 n! k − d(n + 1)!.
Set L V for the union of lines contained in V . When d = n, a flex line of V at a point p ∈ V has order of contact at least n + 1 at this point, and so it is necessarily contained in V by Bézout theorem. Hence in this case, L V coincides with the flex locus of V . Corollary 1.3. Let V be a hypersurface of P n of degree n without ruled irreducible component. Then L V is a ruled subvariety of V of dimension n − 2 and of degree at most Our second main result ensures that the bound for the degree of the flex locus is sharp, and that other expected properties hold true in the generic case.
Theorem 1.4. Let V be a generic hypersurface of P n of degree d ≥ n. Then (1) Flex(V ) is a reduced subscheme (that is, a subvariety) of V of dimension n − 2; (2) for a generic flex point p of V , there is a unique flex line passing through it.
If d = n, then this line is contained in V , whereas if d > n, then its order of contact with V at p is exactly n + 1.
For a cubic surface S in P 3 , Salmon's degree bound is 11 · 3 − 24 = 9. If S is smooth, it contains 27 lines and their union is the complete intersection of S with a surface of degree 9. The next result gives an analogous result for generic hypersurfaces of P n of degree n. It is a direct consequence of Theorem 1.4 and Corollary 1.3.
Corollary 1.5. Let V be a generic hypersurface of P n of degree n. Then L V is a ruled subvariety of V of dimension n − 2 of degree equal to Salmon's theorem for surfaces has been revisited several times. In particular, in the recent book [EH16], the authors reprove it by performing suitable computations in the Chow ring of a Grassmaniann.
Our proof of this result and of the general version in Theorem 1.1 is elementary, and closer to Salmon's approach [Sal65, Articles 473 and 588, pages 94-95 and 277-278], see Remark 3.13. It proceeds by identifying lines with points of P n × P n outside the diagonal. Although this seems less natural from the point of view of intersection theory, it nevertheless allows us to find explicit equations for the flex locus using resultants. The proof of Theorem 1.4 is also elementary and based on the properties of resultants.
The paper is organized in the following way. In Section 2 we review the definition and properties of multidimensional resultants that will be used in the sequel. The proof of Theorem 1.1 is given in Section 3, whereas in Section 4 and Section 5 we show that the flex subscheme is generically reduced and that the generic flex line is unique and has the expected order of contact, thus proving Theorem 1.4.
Acknowledgements. We thank Marc Chardin, Martí Lahoz, Juan Carlos Naranjo and Patrice Philippon for helpful discussions. Part of this work was done while the authors met at the Universitat de Barcelona, the Université de Nice -Sophia Antipolis and the Université de Caen. We thank these institutions for their hospitality.
Busé and Weimann were partially supported by the CNRS research project PICS 6381 "Diophantine geometry and computer algebra". D'Andrea and Sombra were partially supported by the MINECO research project MTM2015-65361-P, and by the "María de Maeztu" program for units of excellence in R&D MDM-2014-0445.

Preliminaries on resultants
The resultant of a family of homogeneous multivariate polynomials plays a central role throughout this text. Therefore, in this section we briefly review this notion and some of its basic properties. We refer to [CLO05,Jou91,GKZ94] for the proofs and more details.
We denote by N the set of nonnegative integers and by K an algebraically closed field of characteristic zero. Boldface symbols indicate finite sets or sequences, where the type and number should be clear from the context. For instance, for n ∈ N we denote by y the set of variables {y 0 , . . . , y n }, so that K[y] = K[y 0 , . . . , y n ].
Let d = (d 0 , . . . , d n ) ∈ N n+1 . For i = 0, . . . , n, we consider the general homogeneous polynomial of degree d i in the variables y given by the sum being over the vectors of a = (a 0 , . . . , a n ) ∈ N n+1 of length |a| = n j=0 a j = d i , and where each c i,a is a variable and y a stands for the monomial n j=0 y a j j . For each i, set c i = {c i,a | a ∈ N n+1 , |a| = d i } for the set of d i +n n variables corresponding to the coefficients of F i , and A = Z[c 0 , . . . , c n ] for the universal ring of coefficients. As usual, given P ∈ A and a system of homogeneous polynomials g i ∈ K[y] of degree d i , i = 0, . . . , n, we write P (g 0 , . . . , g n ) ∈ K for the evaluation of P in the coefficients of the g i 's.
Denote by I and by m the ideals of A[y] respectively defined by F 0 , . . . , F n and by y 0 , . . . , y n . The elimination ideal of the system It is a principal ideal, and the resultant of F , denoted by Res d , is defined as its unique generator satisfying the additional condition The resultant also verifies the following formula for the descent of dimension [Jou91, Lemme 4.8.9 and §5.7].
The resultant satisfies the Poisson formula that we state below, see [Jou91, Proposition 2.7] or [CLO05, Theorem 3.4, Chapter 3] for its proof.
Proposition 2.2. Let g 0 , g 0 ∈ K[y] be homogeneous polynomials of degree d 0 , and g 1 , . . . , g n ∈ K[y] homogeneous polynomials of respective degrees d 1 , . . . , d n with a finite number of common zeros in P n . For each such common zero η ∈ P n , let m η denote its multiplicity. Then both products being over the set of common zeros of g 1 , . . . , g n in P n .
A fundamental property of resultants is that their vanishing characterizes the systems of n + 1 homogeneous polynomials in n + 1 variables that are degenerate, in the sense that their zero set in P n is nonempty. Precisely, a system of homogeneous polynomials g 0 , . . . , g n ∈ K[y] of respective degrees d 0 , . . . , d n , has a common zero in P n if and only if Res d (g 0 , . . . , , g n ) = 0.
The following result gives a criterion to decide if a such degenerate system has a unique zero and, if it does, allows to compute it, see [Jou91, Lemma 4.6.1] or [JKSS04, Corollary 4.7] for its proof.
Then the zero set of g 0 , . . . , g n in P n consists of a single point η, and for i = 0, . . . , n, there is an equality of projective points

The equation of the flex locus
In this section, we obtain an explicit equation for the flex locus of a projective hypersurface by means of resultants. Using this equation, we define the flex scheme and we compute its dimension, the degree of its defining equations and its degree, thus giving the proof of Theorem 1.1. Definition 3.1. Let V be a subvariety of P n and p a point V . For a line L of P n passing by p, its order of contact with V at p is defined as where O L,p is the local ring of L at p, I V the ideal sheaf of V , and ι : L → P n the inclusion map.
The order of contact of a line is either a positive integer or +∞. We have that ord p (V, L) = 1 if and only if L intersects V transversally at p, and ord p (V, L) = +∞ if and only if L is contained in V .
For the rest of this section, we assume that V is a (non necessarily irreducible) hypersurface of degree d ≥ 1. Fix then a defining polynomial f V of V , that is, a homogeneous polynomial in K[x] = K[x 0 , . . . , x n ] of minimal degree such that V coincides with Z(f V ), the set of zeros of f V in P n . Such a polynomial is squarefree, and unique up to a nonzero scalar factor.
The next lemma translates the notion of order of contact with the hypersurface V into algebraic terms. Given a variable t, we denote by val t the t-adic valuation in the local ring Lemma 3.2. Let p ∈ V and L a line of P n passing by p. Let ϕ : A 1 → P n be an affine map parameterizing a neighborhood of p in L, and such that ϕ(0) = p. Write ϕ = ( 0 , . . . , n ) with i ∈ K[t] an affine polynomial, i = 0, . . . , n. Then Proof. Up to a reordering of the homogeneous coordinates of P n , we can suppose that where the second equality follows from the definition of the order of contact, and the fact that f V is a local equation for the germ of hypersurface (V, p) and ϕ a parametrization of the germ of line (L, p).
The order of osculation of V at p is defined as where the supremum is taken over the lines L of P n passing by p. The point p is a flex point of V whenever µ p (V ) ≥ n + 1. A line L with order of contact with V at p at least n + 1 is called a flex line.
Consider again the group of variables y = {y 0 , . . . , y n } and a further variable t, and let f V,k , k = 0, . . . , d, be the family of polynomials in K[x, y] determined by the expansion For k = 0, . . . , d, In particular, f V,k is bihomogeneous of bidegree (d − k, k). For a point p ∈ P n and each k ∈ N, consider the subvariety of P n defined as . The next lemma shows that the order of osculation of V at p can be read from the dimensions of these subvarieties. (1) For each k ∈ N, the subvariety Z k p ⊂ P n is a cone centered at p, union of the lines having order of contact with V at p greater than k.
(2) The order of osculation of V at p is the least k ∈ N such that Z k p = {p}.
Proof. Fix k ∈ N and choose a representative p ∈ K n+1 \ {0} of the point p ∈ P n . We have that f V (p + tp) = (1 + t) d f V (p) and so, for all j ∈ N, Hence p ∈ Z k p . Let q ∈ P n be a point different from p and L the line passing by p and q. This line is parametrized by the affine map ϕ : A 1 → P n defined by ϕ(t) = p + q t for any choice of representatives p, q ∈ K n+1 \ {0} of p and q. Lemma 3.2 combined with the expansion (3.1) implies that the condition q ∈ Z k p is equivalent to ord p (V, L) > k. Hence q lies in Z k p if and only if the line L is contained in this subvariety and has order of contact with V at p greater than k, which proves (1).
By definition, the order of osculation of V at p is the least k ∈ N such that there is no line L with ord p (V, L) > k. Hence (2) is a consequence of (1).
The following corollary follows directly from Lemma 3.4 and the definition of flex points.  Proof. For k ∈ N, the subvariety Z k p is defined by k equations. If this subvariety consists of the single point p, then this number of equations k has to be at least n, by Krull's Hauptidealsatz. Lemma 3.4(2) then gives the lower bound µ p (V ) ≥ n.
On the other hand, if k > d then Z k p = Z n p because f V,j = 0 for all j > d. Hence the Z k p 's form a sequence of subvarieties that is decreasing with respect to the inclusion, and constant for k ≥ d. By Lemma 3.4(2), if Z d p = {p} then µ p (V ) ≤ d. Else, by Lemma 3.4(1), each line contained in Z d p has an order of contact that is arbitrarily large. By Lemma 3.2, such a line is necessarily contained in V .
To conclude, we observe that the last statement is a direct consequence of the first one. where Res y (1,...,n,e) denotes the resultant of n + 1 homogeneous polynomials in the variables y of respective degrees 1, . . . , n, e. Proposition 3.8. Let g ∈ K[x] be a homogeneous polynomial of degree e ≥ 1. Then R V,g defines the flex locus of V in the open subset P n \ Z(g).
Proof. Let p ∈ V such that g(p) = 0. If R V,g (p) = 0 then Z(g) intersects Z n p , by the vanishing property of the resultant. Since p / ∈ Z(g), this implies that Z n p = {p}. By Corollary 3.5, p is a flex point. Conversely, suppose that p is a flex point. Since g is not a constant, Z(g) is a hypersurface and, by Corollary 3.5, dim(Z n p ) ≥ 1. Hence Z(g) does intersect Z n p and so R V,g (p) = 0, as stated. The polynomial R V,g gives an equation for the flex locus of V outside the hypersurface Z(g), but might vanish at points in Z(g) that are not flexes. The next result, corresponding to Theorem 1.1 in the introduction, shows that this equation can be replaced by another one defining the flex locus of V in the whole of the projective space.
To prove it, we need the following auxiliary result. Proof. Let p ∈ V and p ∈ K n+1 \ {0} a representative of this point. If p is not a flex, then Z n p = {p} by Corollary 3.5. By Bézout's theorem, the intersection multiplicity of f V,1 (p, y), . . . , f V,n (p, y) at p is n! and hence, by the Poisson formula (Proposition 2.2), On the other hand, if p is flex then Z n p has positive dimension, again by Corollary 3.5. This implies that the system f V,1 (p, y), . . . , f V,n (p, y), G(y) has a common zero and so R V,g (p) = 0 and, similarly R V,h (p) = 0. Hence (3.4) reduces to 0 = 0 in this case. Thus the equality (3.4) holds for every point of V , which implies the statement.
Proof of Theorem 3.9. Let u = {u 0 , . . . , u n } and v = {v 0 , . . . , v n } be two sets of n + 1 variables and consider the linear forms By Lemma 3.10, for every choice of α, β ∈ K n+1 \ {0}, We deduce that there is a trihomogeneous polynomial s ∈ K[u, v, x] such that v, x], and hence the syzygy (3.5) is necessarily a Koszul syzygy. Hence there are trihomogeneous polynomials The resultant Res y (1,...,n,1) is a multihomogeneous polynomial and, for i = 0, . . . , n − 1, its degree in the set of variables c i corresponding to the coefficients of the ith polynomial is n!/(i + 1). Hence n! k − (n + 1)!, as stated. The uniqueness of the polynomial ρ V satisfying (3.3) follows by considering any linear form that is not a zero divisor modulo f V , completing the proof. This scheme does not depend on the choice of f V , unique up to a nonzero scalar factor, nor on that of ρ V , unique modulo f V . By Theorem 3.9, its support | Flex(V )|, that is, its set of closed points, coincides with the flex locus of V .
Example 3.12. Let C be a plane curve of degree d ≥ 2, and f C ∈ K[x 0 , x 1 , x 2 ] its defining polynomial. A computation using the Euler identities shows that, for any linear form , (3.6) − (d − 1) 2 Res y (1,2,1) (f C,1 (x, y), f C,2 (x, y), (y)) ≡ 2 det(H(f C )) mod f C , where H(f C ) stands for the Hessian matrix of f C . Thus we recover from Theorem 3.9 the well-known fact that a point p ∈ C is an inflexion point if and only the determinant of the Hessian matrix of f C vanishes at p, see for instance [BK86, §7.3, Theorem 1].
For a surface S in P 3 , Theorem 3.9 shows that the flex locus of S is defined by an equation of degree recovering the result of Salmon.
Remark 3.13. In the book [Sal65], Salmon studied the flex locus of the surface S by means of elimination theory. His Article 473 in pages 94-95 of loc. cit. gives a general method to compute, for three surfaces depending on parameters and satisfying a certain intersection theoretic condition, the degree of the condition so that these surfaces contain a common line. His Article 588 in pages 277-278 of loc. cit. applies this degree computation to the three surfaces that arise in the study of the flex locus. In our notation, these three surfaces are those defining the variety Z 3 p for a point p ∈ S.

The flex subscheme of a generic hypersurface
In this section, we show that for a generic hypersurface of P n of degree d ≥ n, the bounds for the flex locus in Corollary 1.2 are sharp, or equivalently that the flex scheme is reduced, and hence that it is equal to the flex locus. The next result corresponds to Theorem 1.4(1) in the introduction.  (1, d).
We first prove the existence of a universal polynomial Φ d in K[c, x] with the property that, for any hypersurface V of P n of degree d, its flex polynomial ρ V can be obtained as the evaluation of Φ d at the coefficients of a defining polynomial of V .
n! k − (n + 1)! such that, for any squarefree homogeneous polynomial f ∈ K[x] of degree d, It is uniquely determined modulo F in the ring K[c, x] by the condition that, for any linear form ∈ K[x], Proof. Adapting the proof of Theorem 3.9 to the present situation, we can show the existence of a bihomogeneous polynomial Φ d ∈ K[c, x] satisfying the congruence (4.5) for any linear form ∈ K[x]. The formulae (4.3) for the degrees of Φ d in the variables c and x follow from this congruence and the corresponding formulae for R F, in (4.2). For a squarefree homogeneous polynomial f ∈ K[x] of degree d, the congruence (4.5) can be evaluated into the coefficients of f , specializing to The equality (4.4) then follows from the unicity of ρ Z(f ) modulo f .  Proof. The first statement follows from the formula (4.6) and the fact that the polynomials F k (x, 0, y ), k = 1, . . . , n, do not depend on c d,0,...,0 and so neither does R, which gives the first statement.
To prove the second one, set again R = R F,y 0 and consider a factorization x], we can assume that its factors are also of this kind. By Lemma 4.4, R(1, 0, . . . , 0) is an irreducible polynomial in K[c] and so one of these factors, say Q 1 , has degree 0 in the variables c or equivalently, does not depend on the coefficients of F . For each choice of p ∈ P n \ Z(x 0 ), we can construct a squarefree homogeneous polynomial f of degree d such that p is not a flex point of Z(f ), and a linear form such that (p) = 0. Proposition 3.8 then implies that R(p) = 0 and, a fortiori, Q 1 (p) = 0. Hence Q 1 is a unit of K[c, x] x 0 and R is irreducible, concluding the proof.
Lemma 4.6. The ideal (F, Φ d ) ⊂ K[c, x] is of height 2, and x 0 is not a zero divisor modulo this ideal.
Proof. Set again R = R F,y 0 for short. By Lemma 4.5, this polynomial does not depend on the variable c d,0,...,0 . Hence, it is coprime with F , as F is irreducible. By Proposition 4.2, R ≡ x n! 0 Φ d mod F , and so Φ d is also coprime with F , giving the first statement. For the second statement, set F = F (0, x 1 , . . . , x n ) and Φ d = Φ d (0, x 1 , . . . , x n ), so that Again by Proposition 4.2, With the same arguments as for the previous case, we deduce that F and Φ d are coprime. Hence x 0 , F, P is a regular sequence in K[c, x].
Since F, Φ d is a regular sequence in K[c, x] and this ring is Cohen-Macaulay, the associated primes of the ideal (F, Φ d ) are of height 2. Since x 0 , F, Φ d is also a regular sequence, x 0 does not lie in any of these associated primes and so this variable is not a zero divisor modulo (F, Φ d ), as stated.
Proof. By Lemma 4.6, x 0 is not a zero divisor modulo (F, Φ d ) and so the morphism is an inclusion. Hence, it is enough to prove that the ideal (F, Φ d ) ⊂ K[c, x] x 0 is prime. Thanks to (4.5) applied with = x 0 , we obtain an isomorphism and we are reduced to show that (F, R F,y 0 ) ⊂ K[c, x] x 0 is prime.
Proof of Theorem 4.1. Setting N = d+n n − 1, let Y be the subscheme of P N × P n defined by F and Φ d . By Lemmas 4.6 and 4.7, this is an irreducible variety of dimension N + n − 2. Let π : Y −→ P N the map induced by the projection onto the second factor.
For a generic choice of α ∈ P N , the homogeneous polynomial F (α, x) ∈ K[x] is squarefree and, by Proposition 4.2 and Theorem 3.9, its fiber π −1 (α) identifies with the flex locus of the hypersurface of P n defined by this polynomial. The same result implies that the dimension of this flex locus is either n − 1 or n − 2. Since Y has dimension N + n − 2, the theorem of dimension of fibers implies that π −1 (α) has dimension n − 2.
Finally, the fact that Y is a variety and Bertini's theorem [Jou83, Théorème 6.3(3)] imply that this fiber is reduced, completing the proof.

Generic flex points
For a squarefree homogeneous polynomial f ∈ K[x] of degree d ≥ n and a flex point p of the hypersurface Z(f ), we consider the following properties: (1) there is a unique flex line of Z(f ) at p; (2) for a flex line L of Z(f ) at p, if d = n, then L is contained in Z(f ) whereas if d > n, then the order of contact of L with Z(f ) at p is equal to n + 1.
In this section we prove the next result, corresponding to Theorem 1.4(2) stated in the introduction.
Theorem 5.1. Let f ∈ K[x] be a generic homogeneous polynomial of degree d ≥ n, and p a generic point of Flex(Z(f )). Then (f, p) satisfies the conditions (1) and (2).
We begin with some notation and preliminary results. For d ≥ 0, set N = d+n n − 1 and let P N be the projective space of nonzero homogeneous forms of degree d modulo scalar factors. For k = 0, . . . , d, we introduce the incidence subvariety with Z(x 0 ) the hyperplane at infinity of P n and F 0 , . . . , F k as in (4.1).
Lemma 5.2. The subvariety Γ k is irreducible and has dimension N + 2n − k.
Proof. Consider the surjective map pr 1 : Γ k → (P n \ Z(x 0 )) × Z(x 0 ) induced by the projection onto the last two factors. To study the fibers of this map over a point (p, q), we can reduce to the case p = (1 : 0 : · · · : 0), by applying a suitable linear change of coordinates.
For a point q ∈ Z(x 0 ), the identities in (4.7) imply that F j ((1, 0, . . . , 0), q), j = 0, . . . , k, are nonzero linear forms in the variables c depending on disjoint subsets of variables, and so they are independent. Hence the fiber pr −1 1 ((1, 0, . . . , 0), q) is a linear space of dimension N − k, and a similar statement holds for any pair of points (p, q). Thus Γ k is a geometric vector bundle of dimension N − k − 1 over the base space (P n \ Z(x 0 )) × Z(x 0 ). Since this base is irreducible and has dimension 2n − 1, the subvariety is also irreducible and has dimension N + 2n − k, as claimed.
For the special case k = n, the subvariety Γ n consists of the triples (f, p, q) where f is a homogeneous polynomial of degree d, p ∈ P n \ Z(x 0 ) is a flex point of the hypersurface Z(f ), and q ∈ Z(x 0 ) determines a flex line passing through p. Let Ω ⊂ P N × (P n \ Z(x 0 )) denotes the set of pairs (f, p) where p ∈ P n \ Z(x 0 ) is a flex point of Z(f ), and (5.1) π : Γ n −→ Ω the map induced by the projection of P N × (P n \ Z(x 0 )) × Z(x 0 ) onto its first two factors.
Proof. Since π is the restriction to the irreducible subvariety Γ n of the proper map P N × (P n \ Z(x 0 )) × Z(x 0 ) −→ P N × (P n \ Z(x 0 )), its image Ω is also an irreducible subvariety. Indeed, by Proposition 3.8 and Proposition 4.2, it is the subvariety of P N × (P n \ Z(x 0 )) defined by the polynomials F and R F,y 0 .
The inversion property of the resultant (Proposition 2.3), implies that the map π is invertible on the open subset of points (f, p) ∈ Ω where (5.2) ∂ Res y (1,...,n,1) ∂c i 0 ,a 0 (f 1 (p, y), . . . , f n (p, y), y 0 ) = 0 for a representative p ∈ K n+1 \ {0} of p and a pair of indices 0 ≤ i 0 ≤ n − 1 and a 0 ∈ N n+1 with |a 0 | = i 0 + 1. We next want to prove that this open subset is nonempty. To this end, it is enough to show that there is a point (f, p 0 ) in Ω with p 0 = (1 : 0 · · · : 0) ∈ P n \ Z(x 0 ) satisfying at least one of the inequations (5.2). We have that F (1, 0, . . . , 0) = c d,0,...,0 and, by Lemma 4.5, the polynomial R F,y 0 does not depend on this variable. Thus In both cases, there is a dense open subset T of P N such that, for each f ∈ T , we have that f is squarefree and there is a dense open subset U f of the flex locus of Z(f ) such that for each p ∈ U f , the pair (f, p) satisfies the conditions (1) and (2), completing the proof.