2 The Cremona action of W p,q,r

We define the configuration space of ordered sets ofq+rpoints in the product of projective spacesPp,q:= (P^q−1)^q+rto be the GIT-quotient

Xp,q,r= (Pp,q)^q+r//SL(q)^p−1,

where the group SL(q)^p−1 acts naturally on the product Pp,q and diagonally on the product (Pp,q)^q+r. We choose a democratic linearization on the product (Pp,q)^q+r∼= (P^q−1)^(p−1)(q+r)defined by the invertible sheafO(1)^(p−1)(q+r). To ex-clude the trivial cases, we assume thatp,q,r>1. The varietyX_p,q,ris an irreducible rational variety of dimensionD= (p−1)(q−1)(r−1).

The groupSp−1×S_q+racts naturally onX_p,q,rby permuting thep−1 factors of Pp,qandq+rfactors of(Pp,q)^q+r. It is realized as the subgroup ofWp,q,rgenerated by the Coxeter generatorss1, . . . ,sp−2andsp, . . . ,sp+q+r−1. Following Coble, Mukai extends this action to a homomorphism

cr_p,q,r:Wp,q,r→Bir(X_p,q,r)∼=Aut_C(C(z₁, . . . ,zD)) (1) by defining the action of the remaining generatorsp−1and checking that the rela-tions between the generators are preserved. Recall that the standard Cremona trans-formationT in projective space P^q−1 is a birational transformation given by the formula

A general points set can be represented inXp,q,r by a unique ordered points set (p₁, . . . ,p_q+r)with the firstq+1 points equal to thereference points

[1,0, . . . ,0], . . . , [0, . . . ,0,1], [1, . . . ,1].

We use projective transformations in each copy ofP^q−1to assume that the projec-tions of the firstq+1 pointsp₁, . . . ,pq+1are the reference points inP^q−1. Then we applyT to the rest of the points. Note thatT is not defined at the intersections of the pull-backs of coordinate hyperplanes in the first factorP^q−1. One checks that this action of generators preserves the defining relations of the Coxeter group and defines the homomorphism (1). We call this homomorphism theCremona actionof Wp,q,r.

LetPbe an ordered set ofq+rdistinct points inPp,qand let π_P:V_P→X^q+r

be its blow-up. We considerV_P up to a birational isomorphism which is an iso-morphism outside a closed subset of codimension≥2 (apseudo-isomorphism). The action of such a birational isomorphism on the groupH²(V_P,Z)∼=Pic(V_P)is well-defined. Denote byE_i∼=P^pq−p−qthe exceptional divisor over the pointp_iand let e_i= [E_i]be its class inH²(V_P,Z). Let pr_i:Pp,q→P^q−1be the projection to thei-th factor andh_i= [π^∗(pr^∗_i(H))],whereHis a hyperplane inP^q−1. Then

h₁, . . . ,hp−1,e₁, . . . ,e_q+r form a basis ofH²(V_P,Z). We call it ageometric basis. Let

h¹, . . . ,h^p−1,e¹, . . . ,e^q+r

be its dual basis inH₂(V_P,Z). We can realize it by taking−eⁱto be the class of a line inEiandhⁱto be the homology class of a line inP^q−1embedded inPp,qunder the inclusion map in thei-th factorιi:P^q−1→(P^q−1)^p−1. Let

α₁=hp−2−hp−1, . . . ,αp−2=h₁−h₂, (4)

αp−1=h₁−e₁−. . .−e_q,

αp=e₁−e₂, . . . ,αp+q+r−2=eq+r−1−e_q+r, and

α¹=−h^p−2+h^p−1, . . . ,α^p−2=−h¹+h²,

α^p−1= (q−2)h¹+ (q−1)h²+. . .+ (q−1)h^p−1+e¹+. . .+e^q, α^p=e²−e¹, . . . ,α^p+q+r−2=−eq+r−1+e_q+r.

We immediately check that the matrix(α_i,α^j) +2Ip+q+r−2is the incidence matrix of the graphTp,q,r. The two bases

α= (α₁, . . . ,αp+q+r−2), α^∨= (α¹, . . . ,α^p+q+r−2)

form a root basis. The Weyl group of this root basis acts onH²(V_P,Z)(resp. on H₂(V_P,Z))as the group generated by the simple reflections

s_i:x→x+ (x,αⁱ)α_i(resp.sⁱ:y→y+ (y,α_i)αⁱ).

Lete=e₁+. . .+e_q+r.It is immediately checked that 1≤i≤p−2 :s_i(h_i) =h_i+1, s(h_j) =h_j,j6=i,i+1,

s_i(e_j) =e_j, j=1, . . . ,q+r, i=p−1 :s_i(h₁) = (q−1)h₁−(q−2)e,

s_i(h_j) =h_j+ (q−1)(h₁−e),j6=1,

s_i(e_j) =h₁−e+e_j, 1≤j≤q, s_i(e_j) =e_j, j>q.

i>p:s_i(h_j) =hj, j=1, . . . ,p−1,

s_i(e_p−1+i) =ep+i,s_i(e_j) =e_j,j6=p−1+i,p+i.

It follows that the Weyl group is isomorphic to the Coxeter groupW_p,q,rwith Coxeter generatorss_i.

Note that the following vectors are preserved under the action:

K_VP =−q(h₁+. . .+hp−1) + (pq−p−q)(e₁+. . .+e_q+r)∈H²(V_P,Z), k_VP =−q(h¹+. . .+h^p−1) +e¹+. . .+e^q+r∈H₂(V_P,Z).

One recognizes inK_VP the canonical class ofV_P. One can check thatαis a basis of the orthogonal complement ofk_VP inH²(V_P,Z)andα^∨is a basis of the orthogonal complement ofK_VP inH₂(V_P,Z).

Another way to look at this is to define a linear map H²(V_P,Z)→H₂(V_P,Z), γ→γ^∨, on the basis(h_i,e_j)by

h_i7→(q−1)(h¹+. . .+h^p−1)−hⁱ,i=1, . . . ,p−1, e_i7→ −eⁱ,i=1, . . . ,q+r.

Thenx·y:= (x,y^∨) defines a structure of a quadratic lattice onH²(V_P,Z) with the Gram matrix in the basis(h₁, . . . ,hp−1,e1, . . . ,eq+r)given by block-sum of the square matrix

A_p,q:=

This implies that the sublatticek_V^⊥

P with a basis α is an even lattice of signature (t₊,t₋)(or(t₊,t−,t₀)ift₀6=0):

It is convenient to introduce an abstract quadratic latticeI_p,q,r defined in a basis (h₁, . . . ,h_p−1,e₁, . . . ,e_q+r)by the matrixAp,q⊕ −I_q+r, a vector W(Ep,q,r)as the group of orthogonal transformations of the latticeEp,q,rgenerated by reflections with respect to the vectorsα_i:

si:x7→x+ (x·αi)α_i.

This is the Coxeter group corresponding to theT_p,q,r-graph. For simplicity of nota-tion, we setW(Ep,q,r) =Wp,q,r.

A choice of a geometric basis inH²(V_P,Z)defines ageometric markingofV_P, i.e. an isometry of latticesϕ:I_p,q,r→H²(V_P,Z)such thatϕ(K_p,q,r) =K_VP.Under this isometry the latticeEp,q,ris mapped ontoK_V^⊥

P.

It is easy to see thatW_p,q,r acts transitively on the root basis α ofEp,q,r. An element of theWp,q,r-orbit of any element from the canonical basis is called aroot.

A root is calledpositiveif it can be written as a linear combination of the canonical basis with non-negative coefficients. One can show that any rootα is either positive or−αis positive.

Lemma 2.1Let α = ∑_i=1^p−1dih_i − ∑^q+r_j=1mie_j be a positive root. Let d=d₁+. . .+dp−1. Suppose that one of the numbers d_iis positive. Then

(i) qd=∑^q+r_j=1m_j;

(ii)(q−1)d²−∑_i=1^p−1d²_i −∑^q+r_k=1m²_k=−2;

(iii)(q−1)d−d₁<m₁+. . .+m_q,if m₁≥m₂≥. . .≥m_q+rand d₁≤. . .≤dp−1; (iv) assume d>0, then d_i≥0,i=1, . . . ,p−1,and m_j≥0,j=1, . . . ,q+r.

Proof The first equality follows from the condition thatα·K_p,q,r=0. The second one follows from the condition thatα²=−2.

(iii) The ordering of the coefficients implies thatα◦αi≥0 fori6=p−1. One checks that

α·αp−1= (q−1)d−d₁−

q

∑

j=1

m_j. (5)

Assume the inequality does not hold. Thenα·α_i≥0 for allα_i. This means thatα belongs to the fundamental chamber of the root system. The proof that it is impos-sible is the same as in the proof of the corresponding statement for the casep=2 in [9], p. 75.

(iv) It is checked immediately thatd≥0 for a positive root. Let us use induction ond. Assumed=1. Then (i) and (ii) give

1−

p−1

∑

d_i²=

q

∑

m_j(m_i−1).

The left-hand side is non-positive, the right-hand-side is non-negative. This gives that one of thed_iis equal to 1, all others are zeros. Also,m_j=1 or 0. This checks the assertion.

It is known thats_itransforms the set of positive roots withαideleted to a subset

Note that a vector inI_p,q,rsatisfying (i)-(iv) from the lemma is not necessarily a root. An example in the casep=2,q=3 is given in [9]. some points coincide, andαp−1 is effective only if the firstq+1 points are in a hyperplane. A points set for which all such roots are not effective will be called a regular point set. It follows from Lemma 2.1 that, for rootsα withd=∑d_i>0, the condition that its image is effective reads as the condition of the existence of a hypersurface of multi-degree(d₁, . . . ,dp−1)passing through the pointsp_i∈Pwith multiplicity≥m_i. Obviously it is a closed condition. Assume∆(α) =Pp,qfor some α with∑d_i>0. Without loss of generality, we may assume thatm₁≥. . .≥m_q+r

andd1≤. . .≤dp−1. Also we may assume thatPis a regular set. Now we takeQ

from the projective equivalence class of cr_p,q,r(sp−1)([P]). Thenϕ_Q(sp−1(α))is an effective root (the transform underTof the hypersurface definingϕ_P(α)). Property (iii) of a positive root together with (6) implies that the hypersurface defined by the rootϕ_Q(sp−1(α))has multi-degree(d₁⁰, . . . ,d⁰_p−1)withd⁰=∑d_i⁰<d. Now we can use induction ond. We know that∆(s_p−1(α))is a proper subset, hence we find a regularQsuch that the rootsp−1(α)is not effective.

Definition 1A points set P is called unnodal if under the geometric marking ϕ_P:I_p,q,r→Pic(V_Q)defined byP, no root is mapped to an effective divisor class.

It follows from Lemma 2.2 that the set of unnodal points sets is the complement of the union of proper closed subsets∆(α), whereαis a positive root inEp,q,r. This set is infinitely countable if the latticeEp,q,ris not negatively definite.

Note that we do not know whether the closed subsets∆(α)are hypersurfaces in Pp,qwhenαis a root different fromα_i,i6=p−1. It is true in the cases whenPp,qis a surface.

Proposition 1LetPbe an unnodal point set. Then, for any w∈W_p,q,r, the marking

wϕ:=ϕ◦w⁻¹is a geometric marking on V_P defined by the points setQsuch that cr_p,q,r([P]) = [Q].

Proof Let P be a regular set of points. We use that, for Coxeter generator s_i∈W_p,q,r, cr_p,q,r(s_i)is defined at [P]. If s_i6=sp−1, then ^siϕ_P =ϕ_Q, where Q is obtained fromP by either permuting the points, or as the image of an automor-phism ofPp,qpermuting the(p−1)factors. Ifi=p−1, then[Q]is defined by the points set (3), whereT is the standard Cremona transformation ofPp,q. Note that the divisor classesE_i⁰representingsp−1(e_i),i=1, . . . ,q,are not contractible onV_P if dimPp,q>2. However, they become contractible when we apply toV_Pa pseudo-isomorphism. The geometric marking corresponding to a points setQrepresenting crp,q,r(s_i)([P])is equal to^siϕ. So, if we show thatQis again a regular set, we are done.

Letl(w)be the minimal length ofwas the product of Coxeter generators. This is well-defined in any Coxeter group. Let us prove by induction onl(w)that the image of a geometric basis defined byP underwis a geometric basis. Writewas the product of generatorss_ik· · ·s_i1, wherek=l(w). Assumei₁6=p−1. Then, as we have already observed,s_i1 transforms a geometric basis to a geometric basis. It is defined by a regular points setQsuch that cr_p,q,r(s_i1)([P]) = [Q].

Assume i1 = p−1. Consider the birational transformation crp,q,r(s_p−1) of X_p,q,r. It transforms the point [P] corresponding to a normalized points set P= (p₁, . . . ,p_q+1, . . . ,p_q+r)to the point[Q], where

Q= (p₁, . . . ,pq+1,T(p_q+2), . . . ,T(p_q+r)) = (p⁰₁, . . . ,p⁰_q+r).

Assume thatQ is not regular. If it does not satisfy the first condition of regular-ity, then some points in this set coincide. But this could happen only if one of the points p_i,i>q+1,inP lies in an exceptional divisor of T (i.e. the closure of

points which are mapped to the set of indeterminacy ofT⁻¹=T). It is easy to see that this set consists of the union of the pre-images of hyperplanes in the first copy ofP^q−1which are spanned by all points pr₁(p₁), . . . ,pr₁(p_q+1)except one, sayp_j. Sinceh₁−e₁−. . .−e_q+e_j−e_iis not an effective divisor, the images of the points {p₁, . . . ,pq,pi} \ {p_j}under the first projection are not contained in a hyperplane.

So the new points setQconsists of distinct points. Next assume that the second con-dition of regularity is not satisfied. This means that the projections of some points p⁰_j

1, . . . ,p⁰_j

q+1to somet-factor lie in a hyperplane. Applying cr_p,q,r(s_i),1≤i≤p−2, we may assume thatt=1. It follows from the definition of the transformationT that T^∗(h₁) =2h₁−e₁−. . .−e_q. It agrees with the action ofsp−1on the geometric ba-sis. If the pointsp⁰_j

1, . . . ,p⁰_j

q+1 are projected to a hyperplane in the first factor, then the pre-image of this hyperplane underT is an effective divisor in the class

2h₁−e1−. . .−eq−ej₁−. . .−e_jq+1=sp−1(h₁−ej₁−. . .−e_jq+1), where we replacee_jk withe_iif j_k=ifor somei≤q. Sinceh₁−e_j1−. . .−e_jq+1 is a root, andsp−1(h₁−e_j1−. . .−e_jq+1)is again a root, it cannot be effective. This proves thats_i1is well defined on[P]and transforms it to[Q], whereQis a regular point set. It remains to apply induction onl(w).

Proposition 2Let P be an unnodal points set and w∈W_p,q,r be such that crp,q,r(w)([P]) = [P].Then there exists a pseudo-automorphismτ:V_P− →V_P such that w=ϕ_P⁻¹◦τ^∗◦ϕ_P.

Proof Let σ : V_P → Pp,q be a birational morphism contracting the divisors E₁, . . . ,Eq+r to pointsp₁, . . . ,pq+r∈P. Then there exists a pseudo-automorphism Φ :V_P− →V_P⁰ and a contractionσ⁰:V_P⁰ →Pp,qof divisors E_i⁰ to the same set of pointsP such that Φ^∗(E_i⁰) =ϕ_P◦w◦ϕ_P⁻¹. Since two blow-ups of the same closed subscheme are isomorphic, there exists an isomorphismΨ :V_P− →V_P⁰ which sendsE_i toE_i⁰. The compositionτ=Φ◦Ψ⁻¹:V_P− →V_P is a pseudo-automorphism ofV_Pwhose existence is asserted in the Proposition.

Corollary 1LetPbe an unnodal set inPp,q. Then the stabilizer subgroup (W_p,q,r)_P:={w∈W_p,q,r: cr_p,q,r(w)([P]) = [P]}

is isomorphic to a subgroup of the group Aut_ps(V_P) of the group of pseudo-automorphisms of V_P.

LetΦ ∈Aut_ps(V_P)and letΦ^∗ be its action onH²(V_P,Z). We say thatΦ is Cremona-likeif there existsw∈W_p,q,r such that

w=ϕ_P⁻¹◦Φ^∗◦ϕ_P.

Let Aut_cr(V_P)be the subgroup of Aut_ps(V_P)of Cremona-like transformations. We have a natural homomorphism

Aut_cr(V_P)→Wp,q,r,Φ7→ϕ_P⁻¹◦Φ^∗◦ϕ_P.

It is clear that its kernel is isomorphic to the group of automorphisms ofV_P lifted from automorphisms ofPp,q.

Remark 1In the case p=2,q=3, one can prove, using Noether’s Theorem on generators of the planar Cremona group, that Autcr(V_P) =Aut(V_P).

It seems that A. Coble [2] and S. Kantor [19] claimed that Aut_cr(V_P) = Aut_ps(V_P) in the cases p = 2 and q is arbitrary. Their claim was based on their theory ofpunctualorregularCremona transformation ofP^q−1. A punctual Cremona transformation is a product of projective transformations and the standard Cremona transformations. By Noether’s Theorem, any planar Cremona transformation is punctual. Kantor says that a Cremona transformation Φ ofP³ has no fundamental curves of the 1st kind if the graph ofΦ transforms any fundamental curve to a fundamental curve of the inverse transformation.

He claimed that any such transformation is punctual (citing from [16], p. 318:

“the so-called proof is admittedly “gewagt”, and merits a stronger adjective”).

Another “equivalent definition” ([13], p. 192) of a punctual transformation requires that all base conditions follow from conditions at points. No rigorous proof of equivalence of these definitions is available. What we need is to prove (or disprove) the following assertion:

Let σ_i : V_Pi− → Pp,q be two blow-up varieties and Φ : V_P1− → V_P2 be a pseudo-isomorphism. Then the Cremona transformation σ2◦Φ ◦σ₁⁻¹ :Pp,q− → Pp,q is a composition of the standard transformation (3) and automorphisms ofPp,q.