The XJC-correspondence · 2015-07-20 · 4 Pirio and Russo, The XJC-correspondence Example 2.1. We have the following examples. (1)Up to projective equivalence, there exists a unique

J. reine angew. Math., Ahead of Print Journal für die reine und angewandte MathematikDOI 10.1515/crelle-2014-0052 © De Gruyter 2014

The XJC-correspondenceBy Luc Pirio at Rennes and Francesco Russo at Catania

Abstract. For any n � 3, we prove that there are equivalences between

� irreducible n-dimensional non-degenerate complex projective varieties X � P2nC1

different from rational normal scrolls and 3-covered by cubic curves, up to projectiveequivalence,

� n-dimensional complex Jordan algebras J of rank 3, up to isotopy,

� quadro-quadric Cremona transformations C W Pn�1 Ü Pn�1 of the complex projectivespace of dimension n � 1, up to linear equivalence.

These three equivalences form what we call the XJC-correspondence.We also provide some applications to the classification of particular types of varieties in

the class defined above and of quadro-quadric Cremona transformations.

Introduction

In this paper we shall work over the field of complex numbers C.We consider the following sets: (1) n-dimensional complex Jordan algebras of rank 3,

modulo isotopy; (2) irreducible n-dimensional non-degenerate projective varietiesX � P2nC1

that are not scrolls such that through three general points passes a twisted cubic curve containedin it, modulo projective equivalence; (3) quadro-quadric Cremona transformations in Pn�1,modulo linear equivalence. The main result of the paper shows that the previous three sets arein bijection and that, when it is defined, the composition of two such bijections is the identitymap as soon as the source and the target space of the composition coincide, see Theorem 4.1and the related diagram in Section 4.

This correspondence, which we call the XJC-correspondence, is based on the followingresults having their own interest: every quadro-quadric Cremona transformation of Pn�1

is linearly equivalent to an involution which is the adjoint of a rank 3 Jordan algebra ofdimension n (Theorem 3.4); every irreducible n-dimensional variety Xn � P2nC1 whichis 3-covered by twisted cubics and different from a rational normal scroll is projectively

Both authors partially supported by P.R.A. of the University of Catania (Italy) and by G.R.I.F.G.A. duringthe preparation of the paper. The first author was also supported by the C.N.R.S. The second author is a member ofthe G.N.S.A.G.A. and he was also partially supported by P.R.I.N. “Geometria delle varietà algebriche”.

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2 Pirio and Russo, The XJC-correspondence

equivalent to a twisted cubic over a rank 3 complex Jordan algebra (Theorem 3.7). Some specialversions of the XJC-correspondence are the following: cartesian products of varieties 3-coveredby twisted cubics correspond to direct product Jordan algebras of rank 3 and to the so-calledelementary quadratic transformations (Proposition 4.2); smooth varieties 3-covered by twistedcubics, modulo projective equivalence, are in bijection with semi-simple rank 3 Jordan alge-bras, modulo isotopy, and with semi-special quadro-quadric Cremona transformations, up tolinear equivalence (Theorem 4.3).

Probably some of these last special versions of the XJC-correspondence have beenestablished before although we are not aware of any reference different from [19]. As faras we know, the equivalence of the previous sets in full generality was conjectured for thefirst time in the final remarks of [19]. The XJC-correspondence yields unexpected relationsbetween different mathematical objects which a posteriori appear to be only three incarnationsin different worlds of the same thing.

Quadro-quadric Cremona transformations can be considered as the simplest examples ofbirational maps of a projective space different from linear automorphisms. In the plane thesetransformations are completely classified and together with projective automorphisms generatethe whole group of birational maps of P2. In low dimension they were studied classically by theItalian school, see for example [8] and the references therein, and soon later by Semple [24].These results were reconsidered recently in [18], where the classification in P3 originallyoutlined in [8] is completed, see also [2]. The XJC-correspondence applied in low dimensionsallowed us to obtain in an uniform way the complete classification of quadro-quadric Cremonatransformations in PN for N � 5, which was unknown for N D 4 and N D 5, see [20].Moreover in [20], we constructed via the XJC-correspondence new series of examples ofquadro-quadric Cremona transformations.

In [10] it is proved that there are only four examples of quadro-quadric Cremona trans-formations with smooth irreducible base locus. These four examples are related to the so-calledSeveri varieties and are linked to the four simple complex Jordan algebras of hermitian 3 � 3matrices with coefficients in the complexification of one of the four real division algebras R, C,H and O, see [10,28], and also [5,12] and Corollary 4.6 here. As an application we also provein Corollary 4.6 that every homaloidal polynomial f of degree 3 defining a quadro-quadricCremona transformation whose ramification locus scheme is cut out by f is, modulo linearequivalence, the norm of a rank 3 semi-simple Jordan algebra, providing a new short proofof [12, Theorem 3.10] and of [5, Theorem 2, Corollary 4]. Finally in [21] we defined thenotions of semi-simple and radical part of a quadro-quadric Cremona transformation or of anextremal variety 3-covered by twisted cubics yielding some structure theorems for these objectsobtained by reinterpreting in these two worlds the theory of the solvability of the radical ofa Jordan algebra.

The paper is organized as follows. In Section 1 we introduce some notations which are notstandard. In Section 2 we define precisely the objects which we study giving some examples:the X -world consisting of extremal varieties 3-covered by twisted cubics; the C -worldconsisting of quadro-quadric Cremona transformations and the J -world consisting of rank 3complex Jordan algebras. Moreover the natural equivalence relations: projective equivalence,linear equivalence, respectively isotopy are introduced as well as the notion of cubic Jordanpair. In Section 3 we define the correspondences between the three sets modulo equivalences;first from the J -world to the X - and C -worlds, then from the C -world to the X -world. Weprove the equivalence between the C - and J -worlds in Theorem 3.4 while the equivalence



Pirio and Russo, The XJC-correspondence 3

between the X - and J -worlds is proved in Theorem 3.7. The XJC-correspondence and its par-ticular forms recalled above are stated in Section 4, where some applications are included, seefor example Corollary 4.6.

1. Notation

For every integer N � 0, PN denotes the complex projective space of dimension N .If V is a complex vector space of finite dimension and if A � V is a subset, then hAi

denotes the smallest linear subspace of V containing A, analogous notions being definedin P .V /. The projective equivalence class of x 2 V n¹0º is the element Œx� 2 P .V /. Let P1; P2be two projective subspaces in PN . When P1 \ P2 D ;, we define their direct sum as

P1 ˚ P2 D hP1; P2i � PN :

We shall consider (irreducible) algebraic varieties defined over the complex field. If Xis an irreducible algebraic variety and if n D dim.X/, we shall write X D Xn or simply Xn.We denote by ŒX� the projective equivalence class of an irreducible projective varietyX � PN .We shall indicate by .X/m the cartesian product of m copies of X . We denote by TxX theembedded projective tangent space to X � PN at a smooth point x of X while TX;x indicatesthe abstract tangent space to X at x.

An irreducible quadric hypersurface in P rC1 is denoted byQr while v3.P1/� P3 standsfor the twisted cubic curve with parametrization P1 3 Œs W t �! Œs3 W s2t W st2 W t3� 2 P3.

2. The objects

2.1. The X -world: Varieties Xn.3; 3/. An irreducible projective variety X in PN issaid to be 3-rationally connected by cubic curves (3-RC by cubics for short) if for a general3-uplet of points .xi /3iD1 2 .X/

3, there exists an irreducible rational cubic curve included inXpassing through x1; x2 and x3.

If X D Xn � PN is 3-RC by cubics and of dimension n, then by projecting X froma general projective tangent space TxX , we get an irreducible variety Y n�ı � PN�n�1, ı � 0,such that through two general points passes a line contained in Y n�ı .

This immediately implies Y D Pn�ı so that

(2.1) dimhXi � 2nC 1 � ı � 2nC 1;

see also [22, Section 1.2] for more general results and formulations.We will say that a varietyXn � PN 3-RC by cubics is extremal ifN D dimhXi D 2nC1.

In what follows, we shall use the notation X D Xn.3; 3/ when X � P2nC1 is an extremalvariety which is 3-RC by cubics.

Thus for X D Xn.3; 3/ � P2nC1 and for two general points x1; x2 2 X , we have

P2nC1 D hXi D Tx1X ˚ Tx2X:

Moreover, there exists a unique cubic curve passing through three general points which is easilyseen to be a twisted cubic, see [19, 22] for details and more general results.




Example 2.1. We have the following examples.

(1) Up to projective equivalence, there exists a unique 3-RC curveX1.3; 3/: the twisted cubiccurve v3.P1/ � P3.

(2) Let Q D Qn�1 be an irreducible hyperquadric in Pn. It is well known that Q is 3-RCby conics and since P1 is 3-covered by lines(!), it immediately follows that the Segreproduct Seg.P1 �Q/ � P2nC1 is 3-RC by cubics so that Seg.P1 �Q/ D Xn.3; 3/.These examples produce a family of Xn.3; 3/, with n � 2.

(3) Let .…i /3iD1 be a 3-uplet of elements of the grassmannian variety

G.2; 5/ D G3.C6/ � P19;

Plücker embedded. If the elements …i are general, one can find a basis .ui /6iD1 of C6

such that

…1 D u1 ^ u2 ^ u3;

…2 D u4 ^ u5 ^ u6

and …3 D .u1 C u4/ ^ .u2 C u5/ ^ .u3 C u6/:

Then s 7! .u1Csu4/^.u2Csu5/^.u3Csu6/ extends to a morphism ' W P1 ! G3.C6/

such that '.0/ D …1; '.1/ D …2 and '.1/ D …3. The curve '.P1/ � G.2; 5/ � P19

in the Plücker embedding is a twisted cubic, showing that G.2; 5/ D X9.3; 3/.

(4) The n-dimensional rational normal scrolls S1:::13 � P2nC1, n� 1, and S1:::122 � P2nC1,n � 2, are classical examples of Xn.3; 3/, which we shall call degenerated examples.

We shall denote by Xn.3; 3/ the set of irreducible non-degenerate varietiesXn � P2nC1

which are 3-RC by twisted cubics and which are not degenerated in the above sense, i.e. that aredifferent from S1:::13 or from S1:::122. The description of the projective equivalence classes ofelements in Xn.3; 3/ is a natural geometrical problem already considered in [19], see also [22]for general classification results of this kind. Indeed, this problem naturally appears when tryingto solve the question of which maximal rank webs are algebraic, a central problem in webgeometry, see [23].

2.2. The C -world: Cremona transformations of bidegree .2 ; 2/. Consider a rationalmap f W Pn�1 Ü Pn�1. There exists a unique integer d � 1 and elements fi 2 jOPn�1.d/j,i D 1; : : : ; n, with gcd.f1; : : : ; fn/ D 1 such that

f .x/ D Œf1.x/ W � � � W fn.x/�

for x 2 Pn�1 outside the base locus scheme B D V.f1; : : : ; fn/ � Pn�1 of f . By definition,the degree of f is deg.f / D d . We will denote byF W Cn ! Cn the homogeneous affine poly-nomial map defined by F.x/ D .f1.x/; : : : ; fn.x// for x 2 Cn. Note that the projectivizationof F is of course the rational map f and that F depends on f only up to multiplication bya nonzero constant.

A rational map f W Pn�1 Ü Pn�1 is birational (or is a Cremona transformation) if itadmits a rational inverse f �1 W Pn�1 Ü Pn�1. In this case, one defines the bidegree of f asbideg.f / D .degf; degf �1/. In this paper we will mainly consider quadro-quadric Cremonatransformations, i.e. Cremona transformations of bidegree .2; 2/. The set of such birationalmaps of Pn�1 will be indicated by Bir2;2.Pn�1/.





(1) The standard involution of Pn�1 is the birational map

Œx1 W x2 W � � � W xn� 7! Œx2x3 : : : xn W x1x3 : : : xn W � � � W x1x2 : : : xn�1�:

It has bidegree .n � 1; n � 1/ and it is an involution, that is f D f �1 or equivalentlyf ı f is equal to the identity of Pn�1 as a rational map.

(2) Let us assume that x 7! .`0.x/; : : : ; `n.x// is a linear automorphism of Cn. Then forany nonzero linear form ` W Cn ! C, the map x 7! Œ`.x/`0.x/ W � � � W `.x/`n.x/� is a bi-rational map. With the previous definitions it is a birational map of bidegree .1; 1/ but weshall consider such a map as a fake quadro-quadric Cremona transformation.

(3) Let Qn�1 � Pn be an irreducible hyperquadric. Given p 2 Qreg, the projection from p

induces a birational map �p W Q Ü Pn�1. For p; p0 2 Qreg with p0 62 TpQ, the com-position �p0 ı ��1p W Pn�1 Ü Pn�1 is a birational map of bidegree .2; 2/, called anelementary quadratic transformation.

(4) Let J be a finite-dimensional power-associative algebra. The inversion x Ü x�1

induces a birational involution j W P .J / Ü P .J /. If J has rank r , see Section 2.3 forthe definitions, then j is of bidegree .r � 1; r � 1/.

In the next lines we shall indicate by V the vector space Cn.Let f1; : : : ; fn and g1; : : : ; gn be quadratic forms on V defining two affine polynomial

maps F D .f1; : : : ; fn/ W V ! V and G D .g1; : : : ; gn/ W V ! V . Let f W Pn�1 Ü Pn�1

respectively g W Pn�1 Ü Pn�1 be the induced rational maps. Then g D f �1 as rationalmaps if and only if there are two homogeneous cubic forms N;M 2 Sym3.V �/ such that, forevery x; y 2 V ,

(2.2) G.F.x// D N.x/x and F.G.y// DM.y/y:

In the previous case, one easily verifies that for every x; y 2 V , we also have

(2.3) M.F.x// D N.x/2 and N.G.y// DM.y/2:

Two Cremona transformations f; Qf W Pn�1 Ü Pn�1 are said to be linearly equivalent(or just equivalent for short) if there are projective transformations `1; `2 W Pn�1 ! Pn�1 suchthat Qf D `1ıf ı`2. This is an equivalence relation on Bir2;2.Pn�1/ and in the sequel we shallinvestigate the quotient set Bir2;2.Pn�1/=¹linear equivalenceº and its various incarnations.

If f 2 Bir2;2.Pn�1/, we will denote by Œf � its linear equivalence class.

2.3. The J -world: Jordan algebras and Jordan pairs of degree 3. By definition,a Jordan algebra is a commutative complex algebra J with a unity e such that the Jordanidentity

(2.4) x2.xy/ D x.x2y/

holds for every x; y 2 J (see [14, 17]). Here we shall also assume that J is finite-dimensional.It is well known that a Jordan algebra is power-associative: for every x 2 J , the sub-algebra hxiof J spanned by x and the unity e is associative. By definition, the rank rk.J/ of J is thecomplex dimension of hxi for x a general element in J , i.e. for x in a Zariski-dense open subsetof J . Note that such a general element x is invertible, i.e. there exists a unique x�1 2 hxi suchthat xx�1 D x�1x D e.





(1) Let A be a non-necessarily commutative associative algebra with a unity. Denote by AC

the vector space A with the symmetrized product a � a0 D 12.aa0 C a0a/. Then AC is

a Jordan algebra. Note that AC D A if A is commutative.

(2) Let q W W ! C be a quadratic form on the vector spaceW . For .�;w/; .�0; w0/ 2 C˚W ,the product

.�;w/ � .�0; w0/ D .��0 � q.w;w0/; �w0 C �0w/

induces a structure of rank 2 Jordan algebra on C ˚W with unity e D .1; 0/.

(3) Let A be the complexification of one of the four Hurwitz algebras R;C;H or O anddenote by Herm3.A/ the algebra of Hermitian 3 � 3 matrices with coefficients in A:

Herm3.A/ D

8̂<̂:0B@r1 x3 x2

x3 r2 x1

x2 x1 r3

1CA W r1; r2; r3 2 C; x1; x2; x3 2 A

9>=>; :Then the symmetrized productM �N D 1

2.MN CNM/ induces on Herm3.A/ a struc-

ture of rank 3 Jordan algebra.

A Jordan algebra of rank 1 is isomorphic to C (with the standard multiplicativeproduct). It is a classical result that any rank 2 Jordan algebra is isomorphic to an algebraas in Example 2.3 (2). In this paper, we will mainly consider Jordan algebras of rank 3. Theseare the simplest Jordan algebras which have not been yet classified in arbitrary dimension.

Let J be a rank 3 Jordan algebra. The general theory specializes in this case and ensuresthe existence of a linear form T W J !C (the generic trace), of a quadratic form S 2 Sym2.J�/and of a cubic form N 2 Sym3.J�/ (the generic norm) such that

x3 � T .x/x2 C S.x/x �N.x/e D 0 for every x 2 J .(2.5)

Moreover, x is invertible in J if and only if N.x/ ¤ 0 and in this case x�1 D N.x/�1x#,where x# stands for the adjoint of x defined by x# D x2�T .x/xCS.x/e. The adjoint satisfiesthe identity

.x#/# D N.x/x:


(1) The algebraM3.C/ of 3 � 3matrices with complex entries is associative. ThenM3.C/C

is a rank 3 Jordan algebra. If M 2M3.C/, the generic trace of M is the usual trace, thenorm is the determinant of M and the adjoint is the classical one, that is the transpose ofthe cofactor matrix of M .

(2) Let C ˚W be a rank 2 Jordan algebra as defined in Example 2.3 (2). For x D .�;w/with� 2 C and w 2 W , one has a trace T .x/ D 2� and a quadric norm N.x/ D �2 C q.w/

such that x2 � T .x/x CN.x/e D 0 for every x. Then one defines the adjoint of xby x# D .�;�w/. In the rank 2 case, one has .x#/# D x.




(3) Let A be as in Example 2.3 (3). Since A is the complexification of a Hurwitz’s alge-bra, it comes with a non-degenerate quadratic form k � k2 W A ! C that is multiplicative.If h � ; � i stands for its polarization, then the generic norm on Herm3.A/ is given by

N

0B@r1 x3 x2

x3 r2 x1

x2 x1 r3

1CA D r1r2r3 C 2hx1x2; x3i � r1kx1k2 � r2kx2k2 � r3kx3k2(2.6)

for every r1; r2; r3 2 C, x1; x2; x3 2 A.

(4) Let J be a power-associative algebra. Again in this case one can define the notions ofrank, adjoint x#, norm N.x/ and trace and the theory is completely analogous to the pre-vious one. Let r D rk.J / � 2. The adjoint satisfies the identity .x#

�#D N.x/r�2x thus

its projectivization is a birational involution of bidegree .r � 1; r � 1/ of P .J /, see alsoExample 2.2 (4).

The inverse map x 7! x�1 D N.x/�1x# on J naturally induces a birational involu-tione| W P .J �C/ Ü P .J �C/ of bidegree .r; r/, defined bye|.Œx; r�/ D Œrx#; N.x/�.Such maps were classically investigated by N. Spampinato and C. Carbonaro Marletta,see [3, 4, 26], producing examples of interesting Cremona involutions in higher dimen-sional projective spaces. It is easy to see that setting eJ D J �C, then for .x; r/ 2 eJone has .x; r/# D .rx#; N.x// so that the map e| is the adjoint map of the algebra eJ .A Cremona transformation of bidegree .r; r/ will be called of Spampinato type if it islinearly equivalent to the adjoint of a direct product J �C where J is a power-associativealgebra of rank r .

The previous construction will be used in [20] to produce some interesting Cremonainvolutions and to describe some known examples in a different way. In Section 3.1 and inSection 3.3.1 maps of this type will naturally appear in relation to tangential projectionsof twisted cubics over rank 3 Jordan algebras, respectively extremal varieties 3-coveredby twisted cubics.

The set of complex Jordan algebras of dimension n will be denoted by Jordann whileJordann3 will indicate the subset formed by the elements having rank equal to 3. Here we willfocus on the description of Jordann3 up to a certain equivalence relation that we now introduce.

2.3.1. Isotopy. Let J be a Jordan algebra. By definition, the quadratic operator associ-ated to an element x 2 J is the endomorphism Ux D 2Lx ı Lx � Lx2 of J where Lx standsfor the multiplication by x in J . If u 2 J is invertible, one defines the u-isotope J .u/ of J asthe algebra structure on J induced by the product �.u/ defined by

x �.u/ y D1

2Ux;y.u/;

where as usual Ux;y D UxCy � Ux � Uy is the linearization of the quadratic representationP W V ! End.V / of J given by x 7! P.x/ D Ux (the name is justified by the fact that P isa homogenous polynomial map of degree 2). Then u�1 is a unity for the new product �.u/ andmoreover J .u/ is a Jordan algebra, the u-isotope of J . Let us recall that x 2 J is invertible ifand only if Ux is invertible; moreover x�1 D U�1x .x/ and Ux�1 D U

�1x in this case.

Two Jordan algebras J and J 0 are called isotopic if J 0 is isomorphic to an isotope J .u/

of J . One immediately proves that the rank is invariant by isotopy. The norm N .u/.x/ and the




adjoint x#.u/ of an element x 2 J .u/ are expressed in terms of the normN.x/ and the adjoint x#

in the algebra J in the following way, see [17, Section II.7.4]:

(2.7) N .u/.x/ D N.u/N.x/ and x#.u/D N.u/�1Uu#.x#/:

If J is a Jordan algebra, then we shall denote by ŒJ � its isotopy class.Of course, isotopy defines an equivalence relation on Jordann and hence on Jordann3

since the rank is isotopy-invariant. In this paper, we are interested in the description of thequotient set Jordann3=¹isotopyº.

The concept of ‘Jordan pair’ is a useful notion in order to deal with Jordan algebras up toisotopy. We introduce it in the next section. This notion will be used later in Section 3.3.2.

2.3.2. Jordan pairs. By definition (see [15]), a Jordan pair is a pair V D .V C; V �/ ofcomplex vector spaces together with quadratic maps (for � D ˙)

Q� W V � ! Hom.V �� ; V � /

satisfying the following relations for every .x; y/ 2 V � � V �� :

D�x;yQ�x D Q

�xD��y;x; D�Q�x .y/;y

D D�x;Q��x .y/; Q�Q�x .y/D Q�xQ

��y Q�x ;

where D�x;y 2 End.V � / is defined by

D�x;y.z/ D Q�xCz.y/ �Q

�x .y/ �Q

�z .y/

for every z 2 V � .


(1) Let J be a Jordan algebra. Then V D .J ; J/ with quadratic operators Q˙x D Ux forevery x 2 J is a Jordan pair. By definition, it is the Jordan pair associated to J ;

(2) Given integers p; q > 0, the pair V D .Mp;q.C/;Mq;p.C// together with the quadraticoperators defined by Q�x .y/ D x � y � x (where � designates the usual matrix product) isa Jordan pair.

By definition, a morphism between two Jordan pairs .V C; V �/ and .VC; V�/ with

respective associated quadratic operators Q� and Q�

is a pair h D .hC; h�/ of linear mapsh� W V � ! V

�such that, for all � D ˙ and every .x; y/ 2 V � � V �� , one has

h� .Q�x .y// D Q�

h� .x/.h�� .y//:

Isomorphisms and automorphisms of Jordan pairs are defined in the obvious way.An element u 2 V �� is said to be invertible ifQ��u is invertible (as a linear map from V �

into V �� ). In this case, one verifies that the product

x � x0 WD1

2Q�x;x0.u/ D

1

2.Q�xCx0.u/ �Q

�x .u/ �Q

�x0.u//

induces on V � a Jordan algebra structure with unit .Q��u /�1.u/ 2 V � . This Jordan algebra isdenoted by V �u . Then it can be proved that V is isomorphic to the Jordan pair associated to V �u .This gives an equivalence between Jordan algebras up to isotopies and Jordan pairs admittinginvertible elements up to isomorphisms, see [15].




3. Equivalences

In this section, we establish some equivalences between the three mathematical worldsintroduced above.

3.1. Starting from the J -world. Let J be a Jordan algebra of dimension n and ofrank 3. Following Freudenthal in [13], one defines the twisted cubic over J as the Zariskiclosure XJ of the image of the affine embedding

�J W J ! P .C ˚ J ˚ J ˚C/;

x 7! Œ1 W x W x#W N.x/�:

It is known that XJ � P2nC1 belongs to the class Xn.3; 3/, see for example [19, Sec-tion 4.3]. We shall provide other proofs of this fact below, see Proposition 3.3.

Let J .u/ be the u-isotope of J relatively to an invertible element u 2 J . Let ù be thelinear automorphism of P .C ˚ J ˚ J ˚C/ D P2nC1 defined by

ù.Œs W X W Y W t �/ D Œs W X W N.u/�1Uu#.Y / W N.u/t �:

It follows from (2.7) that, as affine maps from J D J .u/ to P2nC1, one has �J .u/ D ù ı �J .Hence the projective varieties XJ and XJ .u/ are projectively equivalent. Therefore the associ-ation J ! XJ factorizes and induces a well-defined application

Jordann3=¹isotopyº ! Xn.3; 3/=¹projective equivalenceº;(3.1)

ŒJ � 7! ŒXJ �:

Similarly, since x#.u/ D N.u/�1Uu#.x#/, the linear equivalence class of the birationalmap #J W Œx� Ü Œx#� of Pn�1 does not depend on J but only on its isotopy class. Hence wealso get a well-defined map

Jordann3=¹isotopyº ! Bir2;2.Pn�1/=¹linear equivalenceº;

ŒJ � 7! Œ#J �:

Remark 3.1. The tools used above to construct the ‘twisted cubics over Jordan algebrasof rank 3’ are the adjoint x# and the norm N.x/. These notions have been introduced forevery unital power-associative algebra so that one can ask if it were possible to define ‘twistedcubics over commutative unital power-associative algebras of rank 3’. Since a commutativepower-associative algebra of rank 3 with unity is necessarily a Jordan algebra of the same rank,according to [11, Corollary 13], this generalization would not produce new examples.

In the same vein, one could define a map associating to a rank 3 power-associative algebrawith unity the quadro-quadro Cremona transformation given by the linear equivalence classof its adjoint. As we shall see in Theorem 3.4 below, this generalization is useless since therestriction of this map to Jordan algebras of rank 3 will be surjective. Moreover, by applyingTheorem 3.4 to the adjoint of a commutative power-associative algebra of rank 3 with unityone could deduce a new proof of [11, Corollary 13] mentioned above.

One verifies easily that1J D Œ0 W 0 W 0 W 1� is a smooth point of XJ and that the homog-enization of �J is the inverse of the birational map �1J W XJ Ü Pn given by the restrictionto X of the projection from T1JXJ , see for example [19, Section 4]. It is also immediate




to verify that 0J D �J .0/ D Œ1 W 0 W 0 W 0� 2 XJ is also a smooth point and that T0JXJ isthe closure of the locus of points of the form Œ1 W x W 0 W 0� with x 2 J . Thus the birationalmap W P .J˚C/Ü P .J˚C/ given by .Œx W x0�/D Œx0x# WN.x/� is a birational involutionof type .3; 3/ of Spampinato type (see Example 2.4 (4)) and it is clearly the composition of thehomogenization of �J with the (restriction to XJ of the) linear projection �0J from T0JXJ ,that is D �1J ı �

�10J

as rational maps. We shall return to this in Section 3.3.2.

3.2. Starting from the C -world. Let f 2 Bir2;2.Pn�1/, let g 2 Bir2;2.Pn�1/ be itsinverse, let F;G W Cn ! Cn be some associated quadratic lifts and let N;M be the associatedcubic forms, see Section 2.2.

3.2.1. From the C -world to the X -world. Let us consider the affine embedding

�f W Cn! P .C ˚Cn

˚Cn˚C/ D P2nC1;(3.2)

x 7! Œ1 W x W F.x/ W N.x/�:

The Zariski-closure Xf of its image is a non-degenerate irreducible n-dimensional sub-variety of P2nC1 containing 0f D �f .0/ D Œ1 W 0 W 0 W 0�.

In order to prove that Xf is 3-covered by twisted cubics, we shall use in different waysthe following crucial result whose incarnations in the three worlds we defined till now will bethe starting points of the bridges connecting these apparently different universes.

Let the notation be as above.

Lemma 3.2. There exists a bilinear form BF W Cn �Cn ! C such that

dNx D BF .F.x/; dx/

for every x 2 Cn.

Proof. In coordinates, the relation G.F.x// D N.x/x translates into

(3.3) gi .f1.x/; : : : ; fn.x// D xiN.x/; i D 1; : : : ; n:

Let I D hf1; : : : ; fni � CŒx1; : : : ; xn� D S DLd�0 Sd and let I D

Ld�0 Id , where

for every integer d � 0, Sd and Id stand for the homogeneous components of degree dof S and I respectively. Let us recall that the biggest homogeneous ideal of S defining thescheme B D V.I / is the saturated ideal

I satD

Md�0

I satd D

Md�0

H 0.IB.d//:

It follows from (3.3) that

(3.4) xiN.x/ 2 .I2/2� I4 for every i D 1; : : : ; n:

By derivation of (3.3) with respect to xj for j distinct from i , we deduce that xi .@N=@xj / 2 I3yielding

(3.5) x2i@N

@xj2 I4 for every i; j D 1; : : : ; n; i ¤ j:




By derivation of relation (3.3) with respect to xi we obtain N.x/C xi .@N=@xi / 2 I3for every i D 1; : : : ; n. Multiplying by xi and using (3.4) we deduce x2i .@N=@xi / 2 I4 forevery i . Combined with (3.5), this shows that x2i .@N=@xj / 2 I4 for every i; j D 1; : : : ; n.Then by definition

(3.6)@N

@xi2 I sat

2 for every i D 1; : : : ; n.

Since I sat2 D H

0.IB.2// D span¹f1; : : : ; fnº, the last equality being an immediate conse-quence of the birationality of f , there exist constants bij 2 C such that @N=@xi D

PnjD1 bijfj

for every i . Then letting BF .x; y/ DPni;jD1 bijxiyj , we have dNx D BF .F.x/; dx/ for

every x 2 Cn.

We now provide below an elementary but computational proof that Xf 2 Xn.3; 3/. Foran algebro-geometric proof with the computations hidden in some well-known facts of thetheory of schemes, see the end of the proof of [19, Corollary 5.3].

Proposition 3.3. Let the notation be as above. The variety Xf belongs to Xn.3; 3/:Xf is non-degenerate in P2nC1, is 3-RC by twisted cubics and is different from a rationalnormal scroll.

First proof. For a; b 2 Cn with M.b/ ¤ 0, the rational map

a;b W t 7!G.aC tb/

M.aC tb/

is well-defined and it follows from (2.2) and (2.3) that for t generic, one has

�f . a;b.t// D

�1 W

G.aC t b/

M.aC t b/WF.G.aC t b//

M.aC t b/2WN.G.aC t b//

M.aC t b/3

�D�M.aC t b/ W G.aC t b/ W aC t b W 1

�:

Thus �f ı a;b.P1/ is a twisted cubic curve passing through 0f D �f . a;b.1// andmoreover Xf is 2-covered by the family of these twisted cubics passing through 0f : for.p; p0/ 2 .Xf /

2 general, there exist a; b 2 Cn such that Im.�f ı a;b/ is a twisted cubicincluded in Xf that passes through the points p; p0 and 0f .

Now let x? 2 Cn be such that N.x?/ ¤ 0, let �x? be the translation by x? in Cn and letus consider the linear automorphism of P2nC1 defined by

`x?.!/ D�s W x C s x? W y C dFx?.x/C sF.x?/ W t C BF .y; x?/C dNx?.x/C sN.x?/

�for ! D Œs W x W y W t � 2 P .C ˚Cn ˚Cn ˚C/ D P2nC1, where BF stands for the bilinearform given by Lemma 3.2. One verifies immediately that

`x? ı �f D �f ı �x? :

This shows that the pair .Xf ; �f .x?// is projectively equivalent to .Xf ; 0f /, hence Xfis also 2-covered by twisted cubics passing through �f .x?/. Since this holds for any x? 2 Cn

such that N.x?/ ¤ 0, this implies that Xf D Xn.3; 3/. The variety Xf is not a rational normalscroll since the linear system of quadrics defining the so-called second fundamental form ata general point has no fixed component since it is naturally identified with the linear systemdefining f , see [19, Section 5] for definitions and details.




One immediately verifies that the projective equivalence class of Xf does not dependon f but only on its linear equivalence class. Hence there exists a well-defined map

Bir2;2.Pn�1/=¹linear equivalenceº ! Xn.3; 3/=¹projective equivalenceº

Œf � 7! ŒXf �:

3.2.2. From the C -world to the J -world. Let us now explain how we can asso-ciate in a direct and algebraic way a rank 3 Jordan algebra to a Cremona transformationf 2 Bir2;2.Pn�1/. Assume that Pn�1 D P .V / for an n-dimensional vector space V . On theopen set defined by N.x/ ¤ 0 we define

jf .x/ DF.x/

N.x/:

Then jf W V Ü V is a birational map which is homogeneous of degree �1. Following [16],we say that the map jf W V Ü V is an inversion and that the elements x 2 V with N.x/ ¤ 0are invertible. For x 2 V invertible, one sets

Pf .x/ D �d.jf /�1x :

We defined in this way a rational map Pf W V Ü End.V / which is homogeneous of degree 2.Similarly one defines jg W V Ü V and Pg W V Ü End.V /.

Theorem 3.4. Let the notation be as above. For every linear equivalence class Œf �with f 2 Bir2;2.Pn�1/, there exists a Jordan algebra Jf of rank 3 such that Œ#Jf � D Œf �.

In particular every quadro-quadric Cremona transformation is linearly equivalent to aninvolution which is the adjoint of a rank 3 Jordan algebra.

Proof. Replacing F by Pf .e/ ı F if necessary, we can assume that there exists aninvertible element e 2 V such that Pf .e/ D IdV . Euler’s formula and the homogeneity of jfimply Pf .x/.jf .x// D x for every invertible x so that, without loss of generality, we canalso assume jf .e/ D e. Similarly, one sets jg.y/ D G.y/=M.y/ for y such that M.y/ ¤ 0.Taking the exterior derivative of the relation jg ı jf .x/ D x, we deducePf .x/ D �d.jg/jf .x/for any invertible x. The differentials dGy and dMy are homogeneous of degree 1, respectivelyof degree 2, in y. Hence the substitution y D jf .x/ D N.x/�1F.x/ in

d.jg/y DM.y/�1dGy �M.y/

�2G.y/dMy

yields

Pf .x/ D �dGF.x/ C xN.x/�1dMF.x/ (by (2.2) and (2.3))

D �dGF.x/ C xN.x/�1BG.G.F.x//; dx/ (by Lemma 3.2)

D �dGF.x/ C x BG.x; dx/:

Thus the rational map Pf W V Ü End.V / extends to a polynomial quadratic affine morphismPf W V ! End.V /. Therefore

Pf .x; y/ D Pf .x C y/ � Pf .x/ � Pf .y/ 2 End.V /




is bilinear in x and y and the results of [16] (in particular Theorem 4.4 and Remark 4.5 therein)ensure that the product �f on V defined by

x �f y D1

2Pf .x; y/.e/

satisfies the Jordan identity (2.4), admits e as a unital element and induces on V a structureof Jordan algebra denoted by Jf . If x 2 V is an invertible element, the inverse of x for thisproduct is given by x�1 D jf .x/, hence the adjoint of x is x# D F.x/, yielding rk.Jf / D 3,see Example 2.2 (4).

It can be verified that the isotopy equivalence class of Jf depends only on the linearequivalence class of f yielding a well-defined map

Bir2;2.Pn�1/=¹linear equivalenceº ! Jordann3=¹isotopyº

Œf � 7! ŒJf �:

Remark 3.5. In the previous proof we chose a point e such that Pf .e/ D IdV , which isnot natural from an intrinsic point of view. More generally one can consider the source spaceand the target space of f as distinct n-dimensional projective spaces associated to two vectorspaces VF ; VG of dimension n. We can consider f as a birational map f W P .VF / Ü P .VG/with inverse g W P .VG/ Ü P .VF /. Reasoning as in the proof of Theorem 3.4, one provesthat �d.jf /�1x (resp. �d.jg/�1y ) depends quadratically on x 2 VF (resp. on y 2 VG), defininga quadratic map Pf W VF ! End.VG ; VF / (resp. Pg W VG ! End.VF ; VG/). Then .VF ; VG/together with the pair of quadratic operators .Pf ; Pg/ is a Jordan pair.

3.3. Starting from the X -world. In this subsection, we describe how to associate toan X 2 Xn.3; 3/ an equivalence class of quadro-quadric Cremona transformations of Pn�1

and also how to produce directly (that is not through the previous construction) a rank 3 Jordanalgebra J of dimension n, defined modulo isotopy, such thatX is projectively equivalent toXJ .

Let X � P2nC1 be an element of Xn.3; 3/. Let x 2 Xreg be a general point such that Xis 2-RC by a family†x � Hilb3tC1.X; x/ of twisted cubics included inX and passing throughthe point x. Denote by �x W X Ü Pn the restriction toX of the tangential projection with cen-ter TxX � P2nC1. It is known that �x is birational, cf. [22, Proposition 2.6]. Let ˇx W eX ! X

be the blow-up of X at x and let Ex D ˇ�1x .x/ be the associated exceptional divisor. Let 'X;xbe the restriction to Ex of the lift of �x to eX :

(3.7) 'X;x D .�x ı ˇx/jEx W Ex Ü Pn:

In [19, Section 5], we proved that:

(a) 'X;x is birational onto its image which is a hyperplane Hx � Pn,

(b) 'X;x is induced by the second fundamental form jIIX;xj � jOEx .2/j,

(c) as a scheme, the base locus scheme of 'X;x coincides with the Hilbert scheme of linespassing through x and contained in X ,

(d) .'X;x/�1 W Hx Ü Ex is also induced by a linear system of hyperquadrics in Hx ,

(e) 'X;x is a fake quadro-quadric transformation if and only if X � P2nC1 is a rationalnormal scroll.




From (c) and (d) one could deduce another proof of Lemma 3.2, see [19], or equivalentlyone can say that (c) is the incarnation in the X -world of the result proved in Lemma 3.2,see [19, Theorem 5.2] for details.

Remark 3.6. The map v 7! Qv considered in Section 3.3.2 below is an alternativegeometrical definition of (a quadratic lift of) 'X;x which is more intrinsic than the previousone since it does not depend on the embedding of X in the projective space P2nC1.

3.3.1. From the X -world to the C -world. From now on we shall assume that X isin the class Xn.3; 3/ so that X is not a rational normal scroll. The results listed above implythat, after identifying Ex and Hx with Pn�1, the map 'X;x is a Cremona transformation ofbidegree .2; 2/ of Pn�1. Moreover, in [19, Theorem 5.2] it is proved that X is projectivelyequivalent to the variety X'X;x associated to 'X;x via the construction in Section 3.2. We leaveit to the reader to verify that the linear equivalence class of 'X;x does not depend on x but onlyon the projective equivalence class of X . Therefore we have a well-defined application

Xn.3; 3/=¹projective equivalenceº ! Bir2;2.Pn�1/=¹linear equivalenceº;

ŒX� 7! Œ'X;x�:

The results of the previous sections show that this map is a bijection.

3.3.2. From the X -world to the J -world. The results of [19, Section 5] recalled aboveand Theorem 3.4 immediately imply that anyX 2 Xn.3; 3/ is of Jordan type, that is there existsa rank 3 Jordan algebra J such that X is projectively equivalent to XJ .

Theorem 3.7. If X D Xn.3; 3/ � P2nC1 is not a rational normal scroll, then thereexists a rank 3 Jordan algebra JX such that X is projectively equivalent to XJX .

This result, conjectured firstly in [19, Section 5], is not proved directly but via C -world.It would be interesting to have a direct proof.

Nevertheless, there is a direct way of recovering geometrically the underlying structureof Jordan algebra from X . We provide below a geometric and intrinsic equivalence fromthe X -world to the J -world without any reference to the C -world. Since there is no realdifficulty here, we leave the interested readers fill up the details.

LetX be as in the statement of the theorem. Let xC; x� denote two general points ofXreg

such that X is 1-RC by the family †xCx� of twisted cubics included in X passing through xC

and x�. One hasP2nC1 D TxCX ˚ Tx�X:

For � D ˙, let �� D �x� be the restriction to X of the tangential projection withcenter Tx�X onto the projective tangent space Tx��X at the other point. This map is definedat x�� and by definition x�� D �� .x�� /.

Define V � as the complex tangent space TX;x� . For v 2 V � generic, there exists a uniquetwisted cubic curve Cv included in X , joining x� to x�� and having Œv� as tangent directionat x� . More precisely, there exists a unique projective isomorphism ˛v W P1 ! Cv such that˛v.0 W 1/ D x

� , ˛v.1 W 0/ D x�� and d˛v.s W 1/=dsjsD0 D v. The map

v 7! exp.v/ WD ˛v.1 W 1/




can be extended to the whole V � since, after some natural identifications, it is nothing else butthe affine embedding � � defined in (3.2), where � is the inverse of the quadro-quadraticbirational map 'X;x�� associated to �� through formula (3.7) above. We thus defined geo-metrically an exponential map

exp W V � ! X

whose image is denoted by X� . Being an affine embedding, its differential

d expv W TV � ;v ! TX;exp.v/

is an isomorphism for every v 2 V � . Using the linear structure of V � , one can (canonically)identify TV � ;v with V � itself obtaining a linear isomorphism ı�v W V

� ! TX;exp.v/. For v gen-eral, one has exp.v/ 2 X�� , thus there exists a unique Qv 2 V �� such that exp.v/ D exp. Qv/(moreover Qv D d˛v.1 W t /=dt jtD0). Thus we can define a linear isomorphism by setting

Q�v D �.ı�v /�1ı ı��Qv W V �� ! V � :

The linear map Q�v depends quadratically on v 2 V � and this association extends to thewhole V � yielding a quadratic polynomial map

Q� W V � ! Hom.V �� ; V � /:

Using the fact that X is of Jordan type, it is then easy to verify that

(a) the quadratic maps Q˙ just defined induce a structure of Jordan pair on .V C; V �/admitting invertible elements,

(b) for u 2 V � invertible, the Jordan algebra V Cu has rank 3 and X is projectively equivalentto X

VCu

.

Using the fact that the group of projective automorphisms of X acts transitively on3-uples of general points of X (cf. [19, Proposition 4.7]), one verifies that the isotopy classof V Cu does not depend on the points xC; x� and u but only on the projective equivalence classof X . Therefore we have a well-defined application

Xn.3; 3/=¹projective equivalenceº ! Jordann3=¹isotopyº

ŒX� 7! ŒV Cu �:

To establish that this map is bijective and that is the inverse map of (3.1) does not present anyreal difficulty and is left to the reader.

Remark 3.8. (1) It would be interesting and more satisfying to prove the two state-ments (a) and (b) above in a direct geometric way without using the fact that the consideredvariety X 2 Xn.3; 3/ is actually of Jordan type.

(2) The previous arguments show that the Jordan avatar of the geometrical data formedby X 2 Xn.3; 3/ together with two general points xC; x� on it is the Jordan pair .V C; V �/.Similarly, the geometrical object corresponding precisely to a rank 3 Jordan algebra J is notreally XJ but rather the geometrical data formed by XJ together with three general pointson it. These two remarks lead to the following heuristic question: what are the Jordan-theoreticcounterparts of the data of X alone, or of a pair .X; x/ where x is a general point on X?

Some of the notions introduced in [1] seem to be relevant to study this question.




To end this section, we shall briefly outline another geometrical but extrinsic way ofrecovering the algebra JX naturally associated to X 2 Xn.3; 3/. Let the notation be as inSection 3.3, let x1; x2 2 X be two general points and let W Pn Ü Pn be the birationalmap �x1 ı �

�1x2

, see also end of Section 3.1. From the results in [19, Section 5] recalled above,it is not difficult to see that the birational map , which is clearly of bidegree .3; 3/, is ofSpampinato type. Indeed, arguing as in the proof of Theorem 3.4, one proves that is lin-early equivalent to the involution of a rank 4 Jordan algebra eJX , which is clearly isomorphicto JX �C. We leave it to the reader to check the details of the proof of this claim.

In conclusion, from an extrinsic geometrical point of view, the transition from X -worldto the C - and J -worlds is completely determined by general tangential projections.

4. Statement of the main theorem and of a general principle

The constructions of the previous sections are all represented in the diagram below, whichwe will call the ‘XJC-diagram’:

Xn.3; 3/=¹projective equivalenceº

ŒX� 7! ŒVC

X;u�

##

ŒX� 7!Œ'X;x �

**

\\ŒXf�

7!Œf�

Jordann3=¹isotopyºŒXJ � 7!ŒJ �oo

Bir2;2.Pn�1/=¹linear equivalenceº

Œf� 7!ŒJf�

RR

��

Œ#J�

7!ŒJ�

Then the main result of this paper can be formulated in concise terms by referring to thisdiagram.

Theorem 4.1. The above diagram is commutative, all maps appearing in it are bijec-tions and, when it is defined, the composition of two such bijections is the identity map if thesource and the target space of the composition coincide.

Once the maps in the XJC-diagram have been introduced, the proof of the previoustheorem reduces to straightforward verifications left to the reader.




Theorem 4.1 says in some sense that (up to certain well-understood equivalence relations)there are correspondences between the objects of these three distinct worlds. The X -world isa world of particular projective algebraic varieties sharing deep geometrical properties and itcan be considered as a ‘geometrical world’. The J -world is a world of particular algebras sothat it is an ‘algebraic world’, while we consider the C -world to be of another nature, whichwe will call ‘cremonian’.

A consequence of the preceding main theorem is the following general principle:

XJC-principle. Any notion, construction or result concerning one of the X -, J - orC -world admits a counterpart in the other two worlds.

The XJC-principle is not a mathematical result in the classical sense and it has to beconsidered as a kind of meta-theorem. In the sequel we present some applications regardingclassification results for particular classes of objects in the different worlds. Other applicationsare obtained in [20] and in [21].

4.1. A first occurrence of the XJC-principle. Assume that X , J and f are corre-sponding objects.

Proposition 4.2. The following assertions are equivalent:

(i) The variety X is a cartesian product.

(ii) The algebra J is a direct product.

(iii) The Cremona map f is an elementary quadratic transformation, see Example 2.2.

Moreover, the objects satisfying these properties are respectively:

(1) the Segre embeddings Seg.P1 �Qn�1/,

(2) the direct products C � J 0 where J 0 is a Jordan algebra of rank 2,

(3) the elementary quadratic transformations.

Proof. Clearly (iii) implies (ii) and (i). If (i) holds and if X D X1 �X2 � P2nC1, thenwe can suppose that through three general points ofX1 � P2nC1 passes a line and that throughthree general points of X2 � P2nC1 passes a conic. Then X1 is a line and X2 is a quadrichypersurface in its linear span. Thus X is projectively equivalent to the Segre embeddingof P1 �Qn�1. The other implications/conclusions easily follow.

4.2. A second occurrence of the XJC-principle. In this subsection, we relate thesmoothness property in the X -world to an algebraic one in the J -world and to another onein the C -world. We introduce these properties.

By definition, the radical R of a Jordan algebra J , indicated by Rad.J/, is defined as thebiggest solvable ideal of J (see also Proposition 4.4 below for a characterization of the radicalwhen J has rank 3). Then J is said to be semi-simple if Rad.J/ D 0. In this case, a classicalresult of the theory asserts that J is isomorphic to a finite direct product J1 � � � � � Jm wherethe Jk are simple Jordan algebras, that is Jordan algebras without any non-trivial ideal.

Following [10,25], a Cremona transformation f W Pn�1 Ü Pn�1 is called semi-specialif the base locus scheme of f is smooth. A Cremona transformation is said to be special




if the base locus scheme is smooth and irreducible. Thus special Cremona transformationsf W Pn�1 Ü Pn�1 can be solved, as rational maps, by a single blow-up along an irreduciblesmooth variety, while semi-special Cremona transformations can be solved by blowing-upsmooth irreducible subvarieties of Pn�1, that is there are no “infinitely near base points”.In conclusion the semi-special Cremona transformations are the simplest objects from the pointof view of Hironaka’s resolution of rational maps.

Assume that X , J and f are corresponding objects.

Theorem 4.3. The following assertions are equivalent:

(i) The variety X is smooth.

(ii) The algebra J is semi-simple.

(iii) The Cremona transformation f is semi-special.

Moreover, the classification of the objects satisfying these properties is given in the table belowand f is semi-special but not special if and only if it is an elementary quadratic transformationassociated to a smooth quadric.

Semi-simple rank 3 Jordanalgebra

Smooth variety Xn � P2nC1, 3-RCby cubics, not of Castelnuovo type

Special Cremonatransformation

direct product C � Jwith J simple of rank 2

Segre embedding Seg.P1 �Qn�1/with Qn�1 smooth hyperquadric

elementaryquadratic

Herm3.R/ 6-dimensional Lagrangian Grassman-nian LG3.C6/ � P13

Œx� Ü Œx#�

Herm3.C / 9-dimensional Grassmannian mani-fold G3.C6/ � P19

Œx� Ü Œx#�

Herm3.H / 15-dimensional orthogonal Grassman-nian OG6.C12/ � P31

Œx� Ü Œx#�

Herm3.O/ 27-dimensional E7-variety in P55 Œx� Ü Œx#�

Table 1. Classification of complex semi-simple Jordan algebras of rank 3.

Proof. Semi-special Cremona transformations are classified and they correspond tosemi-simple Jordan algebras of rank 3, see for example [19, Proposition 5.6], showing theequivalence between (ii) and (iii). It is known that the twisted cubics associated to semi-simpleJordan algebras are smooth and they are described in the table above. We proved the remainingimplications in [19, Theorem 5.7].

4.3. The radical and the ramification scheme of a quadro-quadric Cremona trans-formation. We will use the following fact which shows that the radical can be determinedfrom the generic norm.

Proposition 4.4 ([27, 0.15 and 9.10]). For any Jordan algebra J , one has

Rad.J/ D ¹x 2 J W N.x C y/ D N.y/ for every y 2 Jº:




Let f be a quadro-quadric Cremona transformation of Pn�1 D P .V / with base locusscheme Bf � Pn�1. The cubic hypersurface V.N.x// � Pn�1, whereN.x/ is the cubic formappearing in (2.2), is the ramification scheme of f . This terminology is justified by the factthat the locus of points where the differential of the birational map f is not of maximal rank isexactly V.N.x//red, see (2.2) and also [6, Section 1.3].

The radical of f is the set Rf of points of multiplicity 3 of V.N.x// and it has a naturalscheme structure given by

Rf D V.d2Nx/ � Pn�1:

The support of Rf , if not empty, is then a linear subspace of Pn�1, contained in V.N.x//,and it is the vertex of the cone V.N.x// � Pn�1 by Proposition 4.4. We remark that Rf canhave any dimension between �1 and n � 2 (with the usual convention that the empty set isa subspace of dimension �1). The case Rf D ; corresponds to the semi-simple case and, atthe opposite side, Rf is a hyperplane if and only if N is the cubic power of a linear form.

4.4. Applications to homaloidal cubic polynomials. As an application we deduce twoclassification results. Let us recall that a homogeneous polynomial P 2 CŒx1; : : : ; xn� is calledhomaloidal if the associated polar map

P 0 D

�@P

@x1W � � � W

@P

@xn

�W Pn�1 Ü Pn�1

is birational. Let notation be as in Section 4.3, and set

BP D BP 0 D V

�@P

@x1; : : : ;

@P

@xn

�� Pn�1:

Assuming that P 0 has bidegree .2; 2/, we know that there exists a cubic form N such thatV.N / � Pn�1 is the ramification scheme of P 0. Let us remark that since P 0 is birational, thepartial derivatives of P are linearly independent so that V.P / � Pn�1 is not a cone. In particu-lar if P has degree three, then it is necessarily a reduced polynomial. We first classify reduciblehomaloidal polynomials of degree three defining quadro-quadric Cremona transformations.

Corollary 4.5. Let P be a reducible cubic homaloidal polynomial in n � 3 variablessuch that P 0 is a quadro-quadric Cremona transformation whose ramification scheme is V.N /.Then one of the following holds:

� one has V.P / D V.N / and P is linearly equivalent to the norm of a semi-simple (butnot simple) rank 3 complex Jordan algebra of the first line in Table 1 above.

� one has V.P / ¤ V.N / and V.P / is the union of a smooth hyperquadric in Pn�1 witha tangent hyperplane; in some coordinates, one has P.x/ D x1.x22C� � �Cx

2n�1�x1xn/

and N.x/ D x31 .

Proof. If P is the product of three distinct linear forms, then necessarily n D 3 andclearly we are in the first case.

Suppose that P is the product of a linear form ` with a quadratic form q. Without lossof generality we can assume ` D x1. Let Q D V.q/ � Pn�1. By Leibniz’s rule of derivationof a product the restriction of P 0 to Q is the Gauss map of Q, i.e. the map associating toeach smooth point of Q the point in the dual space representing its tangent hyperplane. Thus




the quadric hypersurface Q is contracted by P 0 if and only if Q is a cone. There is an inclu-sion of schemes V.x1/ \Q � BP from which it follows that V.x1/ � V.N /. Thus V.x1/ iscontracted by P 0 and x1 is an irreducible factor of N.x/.

If the hyperquadric Q is also contracted by P 0, then it is necessarily a cone with vertexa point and we can suppose, modulo constants, N.x/ D x1q.x/ D P.x/. Since the hypersur-face V.N / D V.P / � Pn�1 is not a cone, the rank 3 Jordan algebra JP 0 has trivial radical byProposition 4.4, hence is semi-simple. Since N D P is reducible, JP 0 is not simple, hence weare in the case corresponding to the first line of Table 1.

Finally, if Q is not contracted by P 0, then it is a smooth hyperquadric. In this case thehyperplane V.x1/ � Pn�1 is necessarily tangent to Q at a point (otherwise BP would bea degenerated smooth quadric Qn�3 � Pn�2, which is impossible) and we are in the secondcase. In the isotopy class ŒJP 0 � we can choose a representative such that

x#D .x21 ;�x1x2; : : : ;�x1xn�1; x

22 C � � � C x

2n�1 � x1xn/ and N.x/ D x31 :

Following [9], we will say that a homogeneous polynomial P 2 CŒx1; : : : ; xn� suchthat det.Hess.lnP // ¤ 0 is EKP-homaloidal if its multiplicative Legendre transform P� isagain a polynomial function. In this case P� is a homogeneous polynomial function too anddeg.P / D deg.P�/, see also [12] where this condition was investigated. By the preliminaryresults of [12], an EKP-homaloidal polynomial is homaloidal and, after having identified Cn

with its bidual, we get

(4.1)P 0�P�ıP 0

PD IdCn :

Therefore such an EKP-homaloidal polynomial of degree d defines a Cremona transfor-mation of type .d � 1; d � 1/. If moreover d D 3, it follows from (4.1) (combined with (2.3))that we have V.N / D V.P /, that is V.P / is the ramification locus scheme of P 0. Conversely,as we shall see in the proof of Corollary 4.6 below, if P is a homaloidal cubic polynomial suchthat V.N / D V.P /, then it is EKP-homaloidal. Note that the EKP condition defined above isnot satisfied by the reducible polynomials of the type described in the second case of Corol-lary 4.5 where V.N / is a cubic hypersurface supported on the tangent hyperplane.

Corollary 4.6. Let P be a homogeneous polynomial in n � 3 variables. The followingassertions are equivalent:

(1) P is a cubic EKP-homaloidal polynomial.

(2) P is homaloidal, P 0 has bidegree .2; 2/ and V.P / D V.N /.

(3) P is the norm of a semi-simple rank 3 complex Jordan algebra.

When these assertions are verified, P is linearly equivalent to one of the norms of the algebrasin Table 1.

Proof. We have seen before that (1) implies (2). Assume that the latter is satisfied by P .Since P 0 has bidegree .2; 2/, the JC-correspondence ensures that (modulo composition bylinear automorphisms), one can assume that P 0 is nothing but the adjoint map of a rank 3complex Jordan algebra denoted by JP . Since P is homaloidal, V.N / D V.P / � Pn�1 is nota cone. Proposition 4.4 ensures that the radical of JP is trivial. Thus JP is semi-simple and theconclusion follows from the classification recalled in the Table contained in Theorem 4.3.




That (3) implies (1) has been remarked in [12] (see also [5, 9]). It follows from the moregeneral fact that the relative invariant of a prehomogeneous vector space is an EKP-homaloidalpolynomial, cf. [12, Proposition 3.9] for a proof.

Corollary 4.6 has been stated for the first time in [12] but the proof provided therecontained a gap filled in [5] using arguments different from ours.

Remark 4.7. For n D 3 the two examples described in Corollary 4.5, modulo linearequivalence, are the unique homaloidal polynomials without any assumption on deg.P / and/oron P 0 by a result of Dolgachev, see [9, Theorem 4]. For n D 4 there exist irreducible homa-loidal polynomials of degree 3 whose associated Cremona transformation is of type .2; 3/. Onesuch example is given by the equation of a special projection of the cubic scroll in P4 froma point lying in a plane generated by the directrix line and one of the lines of the ruling, see [7]for details and generalizations of this construction. For n � 4 there exist irreducible homaloidalpolynomials of any degree d � 2n � 5, see [7].

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[23] L. Pirio and J.-M. Trépreau, Sur l’algébrisation des tissus de rang maximal, Int. Math. Res. Not. IMRN (2014),DOI 10.1093/imrn/rnu066.

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Luc Pirio, IRMAR, UMR 6625 du CNRS, Université Rennes 1,Campus de Beaulieu, 35000 Rennes, France

e-mail: [email protected]

Francesco Russo, Dipartimento di Matematica e Informatica, Università degli Studi di Catania,Viale A. Doria 6, 95125 Catania, Italy

e-mail: [email protected]

Eingegangen 15. Oktober 2013, in revidierter Fassung 4. März 2014



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