THE QUADRO-QUADRIC CREMONA TRANSFORMATIONS OF P 4 AND P 5

Transcription

1 THE QUADRO-QUADRIC CREMONA TRANSFORMATIONS OF P 4 AND P 5 ANDREA BRUNO AND ALESSANDRO VERRA 1. Introduction A Cremona transformation is a birational automorphism of the projective space P r over a field k. Cremona transformations bear the name of Luigi Cremona, who first investigated them in XIX century achieving pioneering results. Any Cremona transformation f is defined by a unique r-dimensional linear system Φ O P r(m) without fixed components. The same is true for f 1, which is defined by some Φ O P r(n) without fixed components. In this case f is said to be of type (m, n). In the language of classical algebraic geometry Cremona transformations of type (2, 2) are also called quadro-quadric transformations. One uses the words quadro-cubic transformation, cubo-cubic transformation and so on in an analogous way. A linear system without fixed components defining a Cremona transformation is said to be homaloidal (see [H] or [D] for generalities on Cremona transformations). The general classification of Cremona transformations is a very difficult problem. Due to the lack of knowledge on the Cremona group Cr(P r ) of birational automorphisms of P r it seems presently out of reach. The study of special classes of Cremona transformations appears on the other hand useful and often reveals some beautiful, unexpected geometry. To a birational map f : P r P r we can associate the Hilbert scheme of (the closure of) its graph Γ P r P r. In its first instance the classification of Cremona transformations of type (m, n) is the description of the Hilbert schemes of their graphs and of the irreducible components of these Hilbert schemes whose general member is such a graph. In this paper we focus on the case m = n = 2, assuming that k is the complex field. We will therefore discuss quadro-quadric transformations. Quadro-quadric transformations of P 2 are well known with the simpler name of quadratic transformations of P 2. Together with linear maps they generate Cr(P 2 ), (see [D]). The Hilbert scheme of their graphs is irreducible and birational to the Hilbert scheme of three points in P 2. We will see in a moment that there is a unique family of quadro-quadric transformations also in the case of P 3. The research was supported by the research program PRIN : Geometria delle varietá algebriche e dei loro spazi di moduli. 1

2 2 ANDREA BRUNO AND ALESSANDRO VERRA The classification of the irreducible families of Cremona transformations of P 3 of type (2, n) was given by Cremona himself in [C]. On the other hand only recently a stratification of the corresponding Hilbert schemes, together with a precise description of the transformations parametrized by the different strata, was obtained by Pan, Ronga and Vust, ([PRV]). In Cremona s list there is only one family of quadro-quadric transformations of P 3. In P 4 we have for the first time more than one family of Cremona transformations of type (2, 2). In this paper we prove the following Theorem 1.1. Let f : P 4 P 4 be a Cremona transformation of type (2, 2) and let B be the indeterminacy scheme of f. Then B is in the Hilbert scheme of one of the following subschemes of P 4 : a disjoint union of a quadric surface and a point, a union of a plane P and two disjoint lines intersecting P, C L, where L is a line with a double structure embedded in a hyperplane H, C is a smooth conic, H is tangent to C and C L is one point. The proof relies on an inductive procedure which in principle could work for any quadratic transformation f : P r P r, defined by a homaloidal linear system of quadrics Φ. Let Q Φ be general, then Q is integral and f(q) = P is a hyperplane. Consider a point o Q which is in the biregular locus of f and consider the birational map σ : Q P r 1, defined by the projection of center o. If f is of type (2, n), the composition σ f 1 : P P r 1 is a Cremona transformation of type (n, n ) with n 4. Knowing the classification of these transformations of P r 1 is therefore useful to afford the classification of quadro-quadric transformations of P r. This is what we do for r = 4 and n = 2. After completing the proof of this Theorem we found an apparently neglected paper of Semple [S], in which a classification of quadratic transformations of P 4 is given. On the basis of such a list, with the method above described, we prove: Theorem 1.2. Let f : P 5 P 5 be a Cremona transformation of type (2, 2) and let B be the indeterminacy scheme of f. Then B is in the Hilbert scheme of one of the following subschemes of P 5 : a disjoint union of a quadric surface and a point, a union of a 3-dimensional linear space P, a plane σ, intersecting P along a line, and a line l intersecting Π and disjoint from σ, C L, where L is a plane with a double structure embedded in a hyperplane H, C is a smooth conic, H is tangent to C and C L is one point. B is the Veronese surface.

3 THE QUADRO-QUADRIC CREMONA TRANSFORMATIONS OF P 4 AND P 5 3 Not so much seems to be known for higher values of r, see however [RS]. Nevertheless a beautiful exception deserves to be mentioned: the classification of quadro-quadric transformations with smooth and connected base locus, due to Ein and Shepherd-Barron in [ES]: there are four cases, for each of them the base locus is one of the four Severi varieties. In this paper we find (see proposition 6.2) three families of quadro-quadric transformations of P r, for each r, generalising those in P 4 and in P 5 ; furthermore, Proposition 6.2 and Conjecture 6.3 lead us to conjecture that there are only two families of quadro-quadric Cremona transformations in P r, for r 3, whose base locus contains a codimension 2 component. Luc Pirio and Francesco Russo announced recently that they showed that quadro-quadric Cremona transformations are classified by Cubic Jordan algebras; as a Corollary they find that quadro-quadric Cremona Transformations are all symmetrical, as remarked here in Remark 6.1 only a-posteriori. Moreover, their result carries the possibility to give an algebraic classification of quadro-quadric transformations at least for low r. 2. The inductive method We will fix in what follows a quadratic transformation f in P r, given by a linear system Φ = PV O P r(2) with base locus B. Also f 1 is a birational map, induced by a linear system Φ = PV O P r(n) with base locus B. It will also be useful to fix a basis q 0,..., q r of V H 0 (O P r(2)). Lemma 2.1. Let b B be a base point of Φ; then b is smooth for the general quadric Q Φ. In particular, the general quadric Q Φ is integral. Proof. Assume that b Sing Q for each Q Φ and fix a general point x P r B. Since f is birational the base locus of the linear system Φ x := {Q Φ / x Q} is B {x}. On the other hand each Q Φ is a cone of vertex b, therefore the line < bx > is contained in each Q Φ x and hence in the base locus of Φ x. Since < bx > is not in B this is a contradiction. If the general quadric Q Φ is not integral, then its vertex Λ is contained in B by Bertini s Theorem. But then each point b Λ B is singular for any Q Φ and this is a contradiction. Let o P r be a point where f is biregular, so that f(o) is a point and f(o) / B. If P := f(q) then P is a hyperplane through f(o). Since f is biregular at o, the map Q P establishes a projective isomorphism between the linear system of quadrics Q Φ passing through o and the linear system of hyperplanes through f(o). Furthermore, from Bertini theorems it follows that a general hyperplane P P r

4 4 ANDREA BRUNO AND ALESSANDRO VERRA passing through f(o) intersects properly each schematic component of B. Let σ : Q P r 1 be the stereographic projection of center o. Let then be given a general point o in the biregular locus of f and a general quadric Q containing o, such that P = f(q) is a general hyperplane containing f(o). Consider the induced rational map h := σ f 1 : P P r 1. Proposition 2.2. With the above notations, h is a Cremona transformation on P of type (n, n ) with n 4. If B h is the base locus of h we have B h = (B P ) {f(o)}, where B and P intersect transversely. Proof. The map h is birational by construction; moreover, h is defined by the restriction to P of the sublinear system of Φ of quadrics through f(o). We have already remarked that by generality of P the intersection B P is transverse. The base locus B h surely contains B P and f(o). By generality of P and biregularity of f 1 around f(o) it follows that B h = (B P ) {f(o)}. We will show that h 1 = f σ 1 is given by a linear system of hypersurfaces of degree m 4. Fix projective coordinates (t 0 : : t r 1 ) on P r 1 and (x 0 : : x r ) on P r. Then σ 1 is defined by the equations x 0 = p 0 (t 0..., t r 1 ),..., x r = p r (t 0,..., t r 1 ) where p 0,..., p r are quadratic forms. On the other hand f is defined by the quadratic forms q 0,..., q r in x 0,..., x r and we can assume that q r is the equation of Q. Therefore f σ 1 is defined by the quartic forms s 0 = q 0 (p 0,..., p r ),..., s r 1 = q r 1 (p 0,..., p r ). s 0,..., s r 1 span a (r 1)-dimensional linear system Ψ O P r 1(4) which defines h 1 and possibly has some fixed components. We can add to this even further information on B. Let in fact o P r and consider the polar linear map p o : Φ O P r(1) which associates to Q the polar hyperplane of Q with respect to o. Such a map is in general a rational map and is not defined at Q if and only if o Sing Q. Lemma 2.3. A point o P r is a point where f is biregular if and only if p o is an isomorphism

5 THE QUADRO-QUADRIC CREMONA TRANSFORMATIONS OF P 4 AND P 5 5 Proof. For a point o P r, we have that o Sing Q for some Q Φ if and only if o is a base point of Φ or if f is ramified at o. More precisely the fundamental locus of f is defined by the determinant of the matrix of linear forms ( q i ) x j which, evaluated at o, is the polar map p o. It then follows that p o is a linear isomorphism if and only if o is a biregular point of f. If o is as always a point in the biregular locus of f, in particular o is not in B and hence, by Bertini theorems, a general hyperplane T P r passing through o intersects properly each schematic component of B. Since the polar map p o is an isomorphism there exists a unique quadric Q Φ such that T is the tangent hyperplane to Q at o and we can assume that Q is such that P = f(q) is transverse to B. If σ : Q P r 1 is the stereographic projection of center o, we recall that σ 1 blows up the quadric q := Q T P r 1 and then blows down the strict transform by this blowing up of the hyperplane t := T P r 1. Let U = Q (Q T ) and V = P r 1 q; then the map σ : U V is biregular and we can conclude that Lemma 2.4. σ : B U σ(b) V is biregular. In particular, the base locus of h 1 = f σ 1, besides fixed divisors, is the union B h 1 = q σ(b) V. 3. Quadro-quadric transformations in P 4 Let f be a quadro-quadric transformation in P 4, given by a linear system Φ = PV O P 4(2) with base locus B. Let f 1 be its inverse, induced by a linear system Φ = PV O P 4(2) with base locus B. From the results of previous section, we can find a point o in the biregular locus of f and a hyperplane P containing f(o), such that the composition of f and of the projection from o induce a Cremona transformation h := σ f 1 : P P 3 of type (2, n ) with base locus B h. Therefore h belongs to one of the three families of quadratic Cremona transformations h : P P 3

6 6 ANDREA BRUNO AND ALESSANDRO VERRA classified by Cremona. The families are the following: type (2, 2): a general B is the disjoint union of a smooth conic and a point, type (2, 3): a general B is the disjoint union of a line and three distinct points, type (2, 4): a general Φ is the linear system of quadrics through three distinct points and tangent to a given plane at a fourth point. Since we have enough information on the base locus of B h, we can deduce properties of B from the formula B P = B h \ {f(o)} and the fact that P and B are transverse. Moreover, Proposition 2.2 can also be applied to f 1, which is a quadratic transformation, so that we dispose of informations about a transverse hyperplane section of B. Notice that by our choice, f(o) is an isolated point in B h, so that h cannot be too special as a Cremona transformation (we refer to the list at page 1157 of [PRV]). Lemma 3.1. If f is a quadro-quadric transformation, the base locus of f either contains a quadric surface, or contains a plane or doesn t contain two-dimensional components. With respect to these possibilities, f and f 1 belong to the same family of quadratic transformations. Proof. Let as always B be the indeterminacy scheme of f 1 and let B h be the indeterminacy scheme of h; from Proposition 2.2 we have: B h \ {f(o)} = B P, with P and B transverse. If h is of type (2, 2), B h \ {f(o)} is a conic and it follows that B is the union of a quadric surface S and a zero-dimensional scheme Z. If h is of type (2, 3), B h \ {f(o)} is the union of a line and two points and it follows that B is the union of a plane Π, a conic C and a zero-dimensional scheme Z. If h is of type (2, 4), B h \ {f(o)} = B P = {o 1, o 2, o} where o 1, o 2 are smooth points and o is defined as follows: if x, y, z, t are coordinates in P 3 and x = 0 is the equation of the hyperplane tangent to all quadrics of Φ h at the point x = y = z = 0, o is defined by the ideal (x, y 2, yz, z 2 ). We deduce from this that, up to a projectivity, B contains a double line L of ideal (x, y 2, yz, z 2 ), a conic C and a zero-dimensional scheme Z. Since also f 1 is a quadratic transformation, we can apply as above Proposition 2.2 to f 1 in order to deduce that also B, the base locus of f, satisfies only one of the above three possibilities. After this remark, we go back to the cases above:

7 THE QUADRO-QUADRIC CREMONA TRANSFORMATIONS OF P 4 AND P 5 7 If h is of type (2, 2) we know that h 1 is the composition of the blowup σ 1 : P Q and of the restriction of f to Q; since h 1 is given by a linear system of quadrics we must have that B contains a quadric surface which is a base component for h 1. The conclusion is that B and B both contain a quadric surface as a component and that f and f 1 are of the same type. If h is of type (2, 3) we know that h 1 is the composition of the blow-up σ 1 : P Q and of the restriction of f to Q; since h 1 is given by a linear system of cubics we must have that B contains a plane which is a base component for h 1. The conclusion is that B and B both contain a plane as a component and that f and f 1 are of the same type. Since the above reasonings are symmetric, we deduce that also in last case, i.e. when h is a space Cremona transformation of type (2, 4), f and f 1 are of the same type and do not have two-dimensional components. We will use Lemma 3.1 and Lemma 2.4 in order to classify quadro-quadric transformations. Since the classification relies by construction on properties of the base locus of quadratic space transformations and of their inverses, we will briefly recall, from [C] and [PVR], such properties. Let h : P P 3 be a quadratic transformation of type (2, n ), let B h be the base locus Φ the linear system inducing h, B h the base locus of h 1, Φ h the linear system inducing h 1. Then: type (2, 2): a general B h is the disjoint union of a smooth conic and a point, and the same holds for B h ; type (2, 3): a general B h is the disjoint union of a line and three distinct points; a general Φ h is the linear system of cubic surfaces passing doubly through a line L and simply through three lines l 1, l 2, l 3 ; type (2, 4): a general Φ h is the linear system of quadrics through three distinct points and tangent to a given plane at a fourth point; a general Φ h is the linear system of Steiner surfaces passing doubly through 3 edges l 1, l 2, l 3 of the same vertex of a tetrahedron and simply through a smooth conic c. Once we have the above informations on the base loci of quadro-quadric transformations we are able to prove: Theorem 3.2. Let f : P 4 P 4 be a Cremona transformation of type (2, 2) and let B be the indeterminacy scheme of f. Then B is in the Hilbert scheme of one of the following subschemes of P 4 : a disjoint union of a quadric surface and a point, a union of a plane Π and two disjoint lines intersecting Π, C L, where L is a line with a double structure embedded in a hyperplane H, C is a smooth conic, H is tangent to C and C L is one point. Proof. With all the notations we have at hand, we start from h, which is a quadratic space Cremona transformations. We have three possibilities:

8 8 ANDREA BRUNO AND ALESSANDRO VERRA 1) h is of type (2,2). From Lemma 3.1, we know that B = S Z, and B = S Z, where S and S are quadric surfaces and Z and Z are zero-dimensional schemes of the same length. To understand Z we consider the inverse map h 1 : P 3 P : the base locus is the union of a conic q and of σ(z) which consists then of one point. Hence, from Lemma 2.4, Z is a point and f and f 1 are elementary quadratic transformations of P 4. 2) h is of type (2, 3). From Lemma 3.1, we know that B = Π C Z, and B = Π C Z, where Π and Π are planes, C and C are conics and Z and Z are zero-dimensional schemes of the same length. We first notice that, since all quadrics in Φ contain the plane Π, they are singular with vertex in Π. The general element of the system defining h 1 is the projection of a smooth cubic scroll in P 4, so that we can assume that Q has rank 4. In particular, the intersection T o Q is the union of two planes Π 1 Π 2, intersecting at the line joining o and the vertex of Q. The base locus of h 1 is the union of a line L, which is singular for the general element of Φ h and three lines l 1 l 2 l 3, or a limit of such a scheme. If A is any cubic scroll in Q, we have that Π 1 A = a conic, Π 2 A = a line. Their projection via σ are respectively L and l 1. From Lemma 2.4, it follows that σ(c Z) = l 2 l 3. It then follows, from Lemma 2.4, that C = l 2 l 3 and that Z =. The lines l 2 and l 3 are skew and Π intersects them because the intersection of Π with any cubic scroll A is a conic which is a bisection of A. 3) h is of type (2, 4). From Lemma 3.1, we know that B = L C Z, and B = L C Z, where L and L are double lines of ideal (x, y 2, yz, z 2 ) in planes H and H, C and C are conics and Z and Z zero-dimensional schemes of the same length. Let us consider now the inverse map h 1 : this is defined by the linear system of Steiner surfaces passing doubly through 3 edges l 1, l 2, l 3 of the same vertex of a tetrahedron and simply through a smooth conic c, or a degeneration of this. From Lemma 2.4, σ(b) q = l 1 l 2 l 3 c. In particular Z = Z =. We will show that q c and this will prove that σ(b), and hence B, is the union of a double line L and a smooth conic C. Suppose to the contrary that q = c and let F be a general surface of the system defining h 1 ; the surface F is obtained by projection of the Veronese surface in P 5 from a line and σ 1 (F ) is a complete intersection of two quadrics, so that it has been projected from a point in the secant variety of the Veronese surface. If q = c, σ 1 (F ) has three singular lines, from Lemma 2.4, and this is impossible.

9 THE QUADRO-QUADRIC CREMONA TRANSFORMATIONS OF P 4 AND P 5 9 We will next show that L C = 2 and that C (L) red = 1, i.e that H is tangent to C and the tangent vector to C (L) red = 1 is normal to (L) red. Since h 0 (P 4, I L (2)) = 8, quadrics containing the scheme L must cut on C a linear system of dimension at most 2; from standard exact sequences, using the fact that a transverse section of B imposes independent conditions on quadrics of P 4, we see that such a restriction must in fact be a complete linear system of dimension 2 on C. It then follows that h 0 (C, I L/C (2)) = 3, so that C L = 2 as schemes; this can happen only if C (L) red {1, 2}. If C (L) red = 2 and we cut with a general hyperplane section B we get in the base locus of the inverse of a (2, 4) Cremona of P 3 points o, o 1, o 2 on the same line, which does not hold (see the list at page 1157 of [PRV]). We then have C (L) red = 1 and the only possibility is that quadrics containing L are tangent to C at the point C H. Conversely, assume that f is defined by one of the three linear systems above. Let us fix projective coordinates < x, y, z, t, w > in P 4. We will show that such linear systems are homaloidal. If h is of type (2, 2) it is well known that f is birational. In order to describe a homaloidal system of this type it is enough to give to the point in B coordinates [0 : 0 : 0 : 0 : 1] and to the quadric in B equations w = Q(x, y, z, t) = 0. A basis for Φ is then given by < wx, wy, wz, wt, Q >. If h is of type (2, 3), the linear system of quadrics through a plane Π maps P 4 to a quintic scroll in P 8 and if we project this from two lines, the composition is birational. If Π has equations t = w = 0 and lines have equations t = x = y = 0, w = x = z = 0 a basis for Φ is given by < xt, xw, yw, tw, tz >. If h is of type (2, 4), the linear system of quadrics which are tangent to a given hyperplane along a line maps P 4 to a quartic scroll in P 7, which is the cone with vertex a line over the Veronese surface: if L has equations x = y 2 = yz = z 2 the linear system H 0 (P 4, I L (2)) has basis < x 2, xy, xz, y 2, yz, z 2, xt, xw >. The image of the conic C via this linear system does not intersect the vertex of the cone and if we project this from a conic, the composition is birational. According to Del Pezzo ([DP], see also [S]) canonical equations are given in this case as follows: if the conic has equations z = t = y 2 xw = 0, a basis for Φ is given by < y 2 xw, yz, xz, xt,, z 2 >. 4. Semple s list of quadratic transformations of P 4 Semple, in [S], classifies quadratic transformations of P 4 ; we don t have space to reproduce a complete account of his result. Semple s Theorem starts with a classification of all surfaces which can be the intersection of two quadrics of a homaloidal system; he then proceeds by considering the fact that a homaloidal system induces on such a surface a birational morphism to the plane.

10 10 ANDREA BRUNO AND ALESSANDRO VERRA Since we are interested in classifying quadro-quadric transformations of P 5, from Proposition 2.2, we will restrict our attention to describe, from Semple s list, those quadratic transformations of P 4 which are of type (2, n), with n 4, and contain an isolated point in their base locus. Theorem 4.1. Let f be a general quadratic transformation of type (2, n) in P 4. Suppose that n 4, that B contains an isolated point. Then f belongs to one of the following families: (i) type (2, 2): a general B is the disjoint union of a smooth codimension 2 quadric and a point, (ii) type (2, 3): a general B is the union of a plane π, a line l with π l and two distinct points, (iii) type (2, 4): a general Φ is the linear system of quadrics through three distinct points and tangent to a hyperplane h along a line l h, (iv) type (2, 4): a general B is the disjoint union of a smooth rational quartic curve and a point, (v) type (2, 4): a general B is the disjoint union of a plane π and four distinct points, Actually, in Semple s list, at page 370, also appear the following cases: (vi) type (2, 4): a general Φ is the linear system of quadrics which touch a plane σ along a line l σ, are tangent to a given hyperplane H at o l, contain a conic c touching H at o and a point P, (vii) type (2, 4): a general Φ is the linear system of quadrics which touch a plane σ along a line l σ, are tangent to hyperplanes H 1, H 2 respectively at o 1, o 2 l, contain two lines o i l i H i for i = 1, 2 and a point P, (viii) type (2, 4): a general Φ is the linear system of quadrics which touch a plane σ along a line l σ, are tangent to hyperplanes H 1, H 2 respectively at o 1, o 2 l, contain two lines o 1 l 1, l 2 H 1 and a point P. but it easy to recognise that all these cases appear as degenrate cases of (iv), when a rational normal quartic degenerates to the union of a double line and a conic or two skew lines or two concurrent lines; hence such transformations are not general and don t appear in the list above. In what follows we will describe these transformations and their inverses. First of all, if f is as in (i), (ii), or (iii), the description of f is classical and in fact we can definitely say more: Proposition 4.2. The following quadratic transformations are defined in P r for each r 3: (i) type (2, 2): a general B is the disjoint union of a smooth codimension 2 quadric and a point; a general B belongs to the same class of B; (ii) type (2, 3): a general B is the union of a codimension 2 linear space π, a codimension 3 linear space l such that < π, l > is a hyperplane,

11 THE QUADRO-QUADRIC CREMONA TRANSFORMATIONS OF P 4 AND P 5 11 and two distinct points p 1, p 2 P r ; a general Φ f is the linear system of cubic hypersurfaces passing doubly through a codimension 2 linear space L and simply through two codimension 2 linear spaces l 1, l 2 and a line l 3 with l 3 L ; (iii) type (2, 4): a general Φ f is the linear system of quadrics through three distinct points p 1, p 2, p 3 P r and tangent to a given hyperplane h along a codimension 3 linear space l; a general Φ f is the linear system of quartic hypersurfaces which are cones with vertex a point, passing doubly through 3 codimension 2 linear spaces l 1, l 2, l 3, with multiplicity 3 at a common codimension 3 linear intersection m and simply through a smooth conic c; the conic c intersects each l i in a point. Proof. Case (i) corresponds to elementary quadratic transformations: the linear system Ψ of quadrics containing a codimension 2 quadric embeds P r into a quadric hypersurface and then projection from a point gives the elementary transformation; its inverse factors through the inverse of stereographic projection (see for instance Proposition 2.2) and hence has in its base locus a codimension 2 quadric. Consider now case (ii): the linear system Ψ of quadrics through a codimension 2 linear space π embeds P r into a scroll of minimal degree r + 1 in P 2r which is a fibration onto P 1 with fibers linear spaces of dimension r 1: if< π, l > is a hyperplane, the linear space l is sent by Ψ to a linear space contained in a fibre of the scroll, so that projection from such a linear space is birational to a cubic in P r+2 ; projection from two points gives a birational morphism; from this description, a general hyperplane misses the two points so that its image is a cubic hypersurface and this shows that the inverse system is given by cubics. In order to identify B we simply notice what follows: first of all Φ contracts conics passing through p 1, p 2 and intersecting π and l; such conics are parametrised twice by < l, p 1, p 2 > so that they correspond to a double codimension two linear space L B. Also, each codimension two linear space containing l and intersecting π in codimension one gets contracted by Φ and the family of such linear spaces corresponds to a line l 3 B ; notice that l 3 and L intersect at a point corresponding to < l, p 1, p 2 >. Finally, lines through p 1 or p 2 intersecting π are contracted and correspond to two codimension 2 linear spaces l 1, l 2 in B. This exausts B. Finally, in case (iii), consider the hyperplane h with equation X = 0 and the codimension 3 linear space l h, with equations X = Y = Z = 0. If a quadric Q is tangent to h along l, its equation is given by XL + F (Y, Z) = 0 where L is a linear form and F is a homogeneous quadratic polynomial in Y and Z. It follows from this description that the linear system Ψ of quadrics tangent to h along l embeds P r into the cone over the Veronese surface in P r+3 so that the linear projection from three points not belonging to the vertex is P r and Φ f is homaloidal. Since a general

12 12 ANDREA BRUNO AND ALESSANDRO VERRA hyperplane does not intersect the three points, the inverse system is given by quartic hypersurfaces. Again it is quite simple to determine the base locus B. First of all, an isolated conic c is in B : it corresponds to the pencil of hyperplanes in h containing l and c is a conic because each quadric in Φ f contains two such hyperplanes. Three codimension two linear spaces l 1, l 2, l 3 appear doubly in B : they correspond to conics through two of the three points p 1, p 2, p 3 and tangent to l. Each such family of conics fills a hyperplane < l, p i, p j > so that each l i is intersected by c at a point. The intersection l 1 l 2 l 3 is a codimension three linear space which appears with multiplicity three in B : in fact for any point p l any line through p and normal to h is contracted because a quadric in Φ which contains it is singular at p; since the three lines < p, p i, p j > are normal to h there exists a codimension three linear space m B which lies in the intersection l 1 l 2 l 3 with multiplicity three. hypersurfaces in Φ f are cones for the following reason: for any p l the lines < p, p i > for i = 1, 2, 3 and any other line through p are the preimage of a line through a point of m, so that any hypersurface in Φ f is met by a line through a point in m in no variable point. All other quadratic transformation in Theorem 4.1 seem to be specific of P 4. In particular if f is of type (iv) it is fairly known and for further discussion we don t need to describe its inverse: Ψ is given by the linear system of quartic hypersurfaces through a projected Veronese surface and a double quadric. If f is of type (v)(page 375 of [S]), f is defined on the blow-up of P 4 at a plane π and its image factors through a quintic fourfold of minimal degree in P 8. The morphism contracts the P 3 spanned by the four points to a plane and the span of π and each of the four points to planes a i. The map f 1 is then given by quartics and B is the union of a triple plane π and four simple planes a i in such a way that each a i intersects π in a line. Let 5. Quadro-quadric transformations in P 5 f : P 5 P 5 be a (2,2) Cremona transformation. We choose a general Q Φ, with P = f(q), a general point o Q, let σ : Q P 4 be the stereographic projection and we obtain a birational map h := σ f 1 : P P 4 of type (2, n ), with n 4 and base locus B h. From Proposition 2.2 we obtain that B h = (B P ) {f(o)}, where B and P intersect transversely. In particular we obtain that B h contains an isolated point in its base locus.

13 THE QUADRO-QUADRIC CREMONA TRANSFORMATIONS OF P 4 AND P 5 13 It then follows that h is as in Theorem 4.1.Furthermore, if B h 1 is the base locus of h 1, we obtain from Lemma 2.4 that there exists a quadric surface q P 4 such that B h 1 \ q is birational to B. First of all, we prove: Lemma 5.1. If f is a quadro-quadric transformation of P 5, if Q is a general quadric in Φ, o Q is general and σ : Q P 4 is the stereographic projection, if h = σ f 1 is of type (2, 2), then f and h are elementary transformations. Proof. From Proposition 2.2, if h is obtained from f, B h contains an isolated point. If we check the list of quadro-quadric transformations in P 4 in Theorem 3.2, we see that the only case in which B h contains an isolated point is the one of elementary transformations. Hence h is an elementary transformation and, argueing as in the proof of Theorem 3.2 it is easy to deduce that f is an elementary quadratic transformation. We are ready to prove the main theorem of this section: Theorem 5.2. Let f : P 5 P 5 be a Cremona transformation of type (2, 2) and let B be the indeterminacy scheme of f. Then B is in the Hilbert scheme of one of the following subschemes of P 5 : a disjoint union of a quadric threefold and a point, a union of a 3-dimensional linear space P, a plane σ, intersecting P along a line, and a line l intersecting Π and disjoint from σ, C L, where L is a plane with a double structure embedded in a hyperplane H, C is a smooth conic, H is tangent to C and C L is one point. the Veronese surface. Proof. With all previous notations, if f is a Cremona quadro-quadric transformation in P 5, we associate to f a quadratic transformation h of type (2, n) from the list in Theorem 4.1. After Lemma 5.1 it suffices to consider the case in which n 3. Suppose that h is of type (2, 3). We look up to Semple s list in Theorem 4.1 for an h of type (2, 3) containing an isolated point in its base locus. We find only one type of Cremona transformation of this type: h is given by the system of quadrics containing a plane π, a line l intersecting π at a point, and two isolated points. Moreover, we notice that, since h is of type (2, 3), B contains a three-dimensional linear space Π. We now apply Lemma 2.4: we know that B h 1 = q σ(b). Furthermore, from Proposition 4.2, the birational map h 1 is given by the linear system of cubic hypersurfaces which have a double plane σ, contain simply two planes σ 1, σ 2 intersecting σ in lines and a line l intersecting σ. Since all quadrics in Φ contain Π, they are singular with vertex in Π. In particular, the intersection T o Q is singular along the line joining o and

14 14 ANDREA BRUNO AND ALESSANDRO VERRA the vertex of Q, so that is the union of two spaces Π 1 Π 2, intersecting at such line. If A is any cubic residual to Π in Q, we have that Π 1 A = a quadric threefold, Π 2 A = a linear space. Their projection via σ must be respectively σ and σ 1. From Lemma 2.4, it follows that σ(b \ T o ) = σ 2 l. From this it follows that B = Π σ l, so that in particular f is symmetrical. We will now consider the case in which h is of type (2, 4). First of all, if h is of type (iv) it is well known that f is the symmetrical quadro-quadric transformation of P 5 induced by the quadrics containing the Veronese surface. Notice that if h is as in (vi), (vii), (viii) in the list after Theorem 4.1, the Veronese surface degenerates to a double plane and a quadric or a couple of planes. A second important remark is that h cannot be of type (v) because in this case B would have a three-dimensional linear space as a component and then the (2, n) transformation associated to f 1 would be of type (2, 3); but we have already classified this case and this is symmetrical, i.e. f and f 1 belong to the same family. Let then assume that h is as in (iii), i.e. that B = Π c Z, where Π is a double structure on a plane (Π) red contained in a hyperplane H, c is a conic and Z is zero-dimensional. In this case, from Proposition 4.2, the birational transformation h 1 is given by the system of quartics with three double planes σ 1, σ 2, σ 3 intersecting at a line l of triple points, passing through a conic intersecting each plane at a point. From Lemma 2.4 we then have: σ(b) q = σ 1 σ 2 σ 3 c. For dimension reasons we must have q = σ 1 σ 2 so that σ(b) is the union of a double structure on a plane and a conic intersecting at one point this plane: it is evident that also B is like this and that f and f 1 belong to the same family. We will give canonical equations in the second and third case in next section, while it is well known that in the first and fourth case we actually have quadro-quadric Cremona transformations. 6. Further remarks We collect in this section remarks and observations stemming out from previous results, which might share some light on the classification of quadroquadric transformations in P r for r 6.

15 THE QUADRO-QUADRIC CREMONA TRANSFORMATIONS OF P 4 AND P 5 15 Remark 6.1. If f is a quadro-quadric Cremona transformation in P r for r 5, then f and f 1 belong to the same family. It is quite natural to conjecture that this what happens in general. Unfortunately our remark is only a consequence of our classification Theorems. As a matter of fact, we notice: Proposition 6.1. Suppose that f is a quadro-quadric symmetrical transformation of P r. Then the induced quadratic transformation h of P r 1 has the following property: the base locus of h is the union of a transverse intersection of B and an isolated point, while the base locus of h 1 is supported on the union of a codimension two quadric and a general projection of B. Also, besides elementary transformations, we find two infinite series of quadro-quadric Cremona transformations in P r : Proposition 6.2. In P r we have the following classes of quadro quadric transformations, corresponding to B P r being in one of the following families: a disjoint union of a quadric variety of codimension 2 and a point, B = T σ l where T, σ, l are linear of dimensions resp. r 2, r 3, 1, σ and l are disjoint and each intersects T in dimension resp. n 4 and zero, C T, where T is a codimension 3 linear space with a double structure in a hyperplane H, C is a smooth conic, H is tangent to C and C T red is one point. These quadro-quadric transformations are symmetrical and induce in P r 1 the quadratic transformations listed in Proposition 4.2. Proof. The first case in this list is the one of elementary transformations, which is known. The second case induces a quadratic transformations as in second case of Proposition 4.2. If B is as described, quadrics through T embed P r in a scroll of minimal degree in such a way that projection from the images of σ and l are birational. In last case we just give canonical equations for f: if r 5, P r has coordinates < x, y, z, t, w, x 1,..., x r 4 >, T has equations x = y 2, yz, z 2 and C has equations z = t = y 2 zw = x 1 =... = x r 4, the linear system Φ is given by < y 2 xw, yz, xz, xt, z 2, xx 1,..., xx r 4 >. Proposition 6.3. With the above notations, suppose that f : P r P r is a quadro-quadric transformation. Then B contains a quadric Q of dimension r 2 if and only if h is of type (2, 2) if and only if f is elementary and B = Q p for a point p P r. Proof. The composition h is of type (2, 2) if and only if there exists a codimension 2 quadric Q B: this follows from the description of h 1 and the description of σ 1. We must show that if h is of type (2, 2) then f is elementary. But h is of type (2, 2) if and only if Q B; by a dimension count

16 16 ANDREA BRUNO AND ALESSANDRO VERRA it easily follows that there must exist a point p P n such that B = Q p so that f is elementary. Conjecture 6.4 Analogously, we propose the following: With the above notations, suppose that f : P r P r is a quadro-quadric transformation. Then B contains a codimension 2 linear space Π if and only if h is of type (2, 3) if and only if f is the second type of quadro-quadric transformation in Proposition 6.3. References [C] [DP] [D] [ES] [H] [PRV] [RS] [S] L. Cremona Sulle transformazioni razionali nello spazio. Annali di Mat. ser. II, V ( ), P. Del Pezzo Una trasformazione cremoniana fra spazi a quattro dimensioni Rend. R. Acc. delle Scienze Fisiche e Mat. di Napoli, (3), 2, (1896), I. Dolgachev Topics in Classical Algebraic Geometry, part I in preparation. L. Ein, N. Shepherd-Barron Some special Cremona transformations. Amer. J. Math. 111 (1989), no. 5, H.P. Hudson Cremona transformation in Plane and Space Cambridge University Press, I. Pan, F. Ronga, T. Vust Transformations birationnelles quadratiques de l espace projectif complexe trois dimensions. Ann. Inst. Fourier (Grenoble) 51 (2001), no. 5, F. Russo, A. Simis On birational maps and Jacobian matrices. Compositio Math. 126 (2001), no. 3, J. G. Semple Cremona Transformations of Space of Four Dimensions by Means of Quadrics, and the Reverse Transformations. Phil. Trans. R. Soc. Lond. A 228 (1929), Universita di Roma 3, Dipartimento di Matematica, L.go S. Leonardo Murialdo, Roma; bruno@mat.uniroma3.it Universita di Roma 3, Dipartimento di Matematica, L.go S. Leonardo Murialdo, Roma; verra@mat.uniroma3.it