GIT quotients of products of projective planes

GIT Quotients of Products of Projective Planes

FRANCESCAINCENSI(*)

ABSTRACT- We study the quotients for the diagonal action ofSL3(C) on t hen-fold productofP²(C): we are interested in describing how the quotient changes when we vary the polarization (i.e. the choice of an ample linearized line bundle). We illustrate the different techniques for the construction of a quo- tient, in particular the numerical criterion for semi-stability and the ``ele- mentary transformations'' which are resolutions of precisely described sin- gularities (casenˆ6).

Introduction.

Consider a projective algebraic varietyXactedonbya reductivealgebraic groupG. Geometric Invariant Theory (GIT) gives a construction of aG-in- variantopen subsetU of X for which the quotientU==Gexists andU is maximalwith thisproperty (roughlyspeaking,Uisobtained fromXthrowing away ``bad'' orbits). However the openG-invariantsubsetUdepends on the choice of aG-linearized ample line bundle. Given an ampleG-linearized line bundleL2Pic^G(X) overX, one defines the set of semi-stable points as

X^SS(L):ˆ fx2Xj 9n>0 ands2G(X;Lⁿ)^Gs:t:s(x)6ˆ0g;

and the set of stable points as

X^S(L):ˆfx2X^SS(L)jGxis closed inX^SS(L) and the stabilizerGxis finiteg:

Then itis possible to introduce a categorical quotientX^SS(L)==Gin which two points areidentifiedif the closure oftheir orbitsintersect. Moreover asshown in [10],X^SS(L)==Gexists as a projective variety and contains theorbit space X^S(L)=Gas a Zariski open subset.

(*) Indirizzo dell'A.: Dipartimento di Matematica, UniversitaÁ di Bologna, Piazza di Porta S. Donato, 5 - 40126 Bologna, Italy

E-mail: incensi@dm.unibo.it

QUESTION. - If one fixes X;G and the action ofG onX, but lets the linearized ample line bundle L vary in Pic^G(X), how do the open set X^SS(L)Xand the quotientX^SS(L)==Gchange?

Dolgachev-Hu [4] and Thaddeus [11] proved that only a finite number of GIT quotients can be obtained whenLvaries and gave a general de- scription of the maps relating the various quotients.

In this paper we study the geometry of the GIT quotients for

XˆP²(C). . .P²(C)ˆP²(C)ⁿ. We give examples fornˆ5 andnˆ6.

The contents of the paper are more precisely as follows.

Section 1 treats the general caseXˆP²(C)ⁿ: firstof all the numerical criterion of semi-stability is proved (Proposition 1.1). By means of this it is possible to show that only a finite number of quotients X^SS(m)==G exists (Subsection 1.2). At the end of the section we introduce the ele- mentary transformations which relate the different quotients.

Section 2 is concerned with the casenˆ5. Theorem 2.8 contains the main result of Section 2: we show that there are precisely six different quotients.

Section 3 discusses the casenˆ6: the main results of this Section are concerned with the number of different geometric quotients that may be obtained (it is less than or equal to 38: Table 3.1) and with the singula- rities that may appear in the quotients. In particular there are only two different types of singularities: in Subsection 3.2 they are described, using the EÂtale Slice theorem. Theorem 3.2 collects these results. At the end of the Section two examples show how these singularities are re- solved by ``crossing the wall''.

1. The general caseXˆP²(C)ⁿ.

LetGbe the groupSL₃(C) acting on the varietyXˆP²(C)ⁿand lets be the diagonal action

s: G P²(C)ⁿ ! P²(C)ⁿ g ; (x₁;. . .;x_n) 7! (gx₁;. . .;gx_n)

A line bundle L over X is determined by LˆL(m):ˆL(m1;. . .;mn)ˆ

ˆNⁿ

iˆ1p_i(O_P²_(C)(m_i)); m_i2Z8i, where p_i:X!P²(C) is t he i-th projec- tion. In particularLis ample iffm_i>0; 8i.

Moreover since eachp_iis an G-equivariantmorphism, Ladmits a ca- nonicalG-linearization:

Pic^G(X)Zⁿ:

Thus apolarizationis completely determined by the line bundleL.

Recall that a pointx2Xis said to besemi-stablewith respect to the polarizationmiff there exists aG-invariant section of some positive tensor power ofL,g2G(X;L^k)^G, such thatg(x)6ˆ0. A semi-stable point isstable if its orbit is closed and has maximal dimension. Thecategorical quotientof the open set of semi-stable points exists and is denoted byX^SS(m)==G:

X^SS(m)==G  Proj M¹

kˆ0

G(X;L^k)^G

! : The open setX^S(m)=GofX^SS(m)==Gis ageometric quotient.

We set X^US(m)ˆXnX^SS(m), the closed set of unstable points and X^SSS(m)ˆX^SS(m)nX^S(m), the set of strictly semi-stable points.

1.1 ±Numerical Criterion of semi-stability.

After fixing a polarizationL(m), we want to describe the set of semi- stable pointsX^SS(m): using the Hilbert-Mumford numerical criterion, we prove the following

PROPOSITION1.1. Let x2X andjmj:ˆPⁿ

iˆ1m_i. Then we have

x2X^SS(m), X

k;xkˆy

mkjmj 3 X

j;xj2r

m_j2jmj 3 8>

>>

><

>>

>>

: …1†

for every point y2P²(C)and for every line rP²(C).

PROOF. Fixing projective coordinates on the i-th copy of P²(C), [xi0:xi1:xi2], a pointx2X P(G(X;L(m)))ˆP^N(C)

, is described by homogeneous coordinates of this kind:

Yⁿ

iˆ1

x_i0^jix^k_i1ⁱx^m_i2^{i (ji‡ki)} where 0j_i;k_im_i,j_i‡k_im_i.

Letl_a0;a1;a2be the one-parameter subgroup ofGdefined byl_a0;a1;a2(t)ˆ

ˆdiag(t^a0;t^a1;t^a2) wherea0‡a1‡a2ˆ0; we can assumea0a1a2. The subgroupla0;a1;a2acts on every component ofC^N‡1, multiplying by

t^a0P

iji‡a1P

iki‡a2P

i(mi (ji‡ki)):

By the definition of the numerical function of Hilbert-Mumfordm_L(x;l), we are interested in determining the minimum value of

a₀Xⁿ

iˆ1

j_i‡a₁Xⁿ

iˆ1

k_i‡a₂Xⁿ

iˆ1

m_i (j_i‡k_i)

… †:

This should be obtained whenj_iˆk_iˆ0, 8i; butif there are somex_i2ˆ0, then the minimum value becomes:

a2

X

i;xi26ˆ0

mi‡a1

X

j;xj2ˆ0;xj16ˆ0

mj‡a0

X

k;xk2ˆxk1ˆ0

mk: …2†

Thusx2Xis semi-stable for the action ofla0;a1;a2if and only if expression (2) is less than or equal to zero.

Let a0ˆb₀‡b₁; a1ˆ b₀; a2ˆ b₁; itfollows thatb₁ 2b₀,b₁b₀eb₁0.

The expression (2) can be rewritten and the condition for semistability is

b₀ X

k;xk2ˆxk1ˆ0

m_k X

j;xj2ˆ0;xj16ˆ0

m_j 0

@

1

A‡b₁ X

k;xk2ˆxk1ˆ0

m_k X

i;xi26ˆ0

m_i

! 0 …3†

Fig. 1. - Planeb₀;b₁:

The figure 1 shows that every couple (b₀;b₁) that satisfies (3) is a po- sitive linear combination ofv1ˆ(1;1) ev2ˆ( 1;2). Thus the relation (3) must be verified in the two casesb₀ˆb₁ˆ1 eb₀ˆ 1;b₁ˆ2. After a few calculations we obtain thatxis semi-stable for the action of alll_a0;a1;a2if and only if

X

h;xhˆy

m_h jmj=3; y2P²(C) X

l;xl2r

m_l 2jmj=3; rP²(C) 8>

>>

<

>>

>:

foryˆ[1:0:0] andrthe linex₂ˆ0. Since every one-parameter subgroup is conjugate to one of the formla0;a1;a2 the proposition follows. p REMARK 1.2. The case with all m_iˆ1 is a special case of[10] Pro- position4.3.

REMARK1.3. x2X^S(m)iff the numerical criterion(1)is verified with strict inequalities.

The numerical criterion can be restated as follows: if K and J are subsets off1;. . .;ng, then we can associate them with the numbers:

g_K^C(m) ˆ jmj 3X

k2K

mk; g_J^L(m) ˆ2jmj 3X

j2J

mj:

In particular we have:g_J^C(m) ˆ g_J^L0(m); whereJ⁰ˆ f1;. . .;ng nJ.

Now for every collection of disjoint subsetsK1;. . .;Kroff1;. . .;ngwith jK_lj 2, we consider the set of configurations (x₁;. . .;x_n) where the points indexed by each set K_l are coincidentand there are no further coin- cidences:

U^C_K1;...;Kr ˆx2Xjif i6ˆj; then xiˆxj,i;j2Kl for somel

: We write also

U^C_; ˆx2Xjx_i6ˆx_j if i6ˆj :

In the same way, for every collection of subsetsJ₁;. . .;Jsoff1;. . .;ngwith jJ_lj 3 andJ_l6J_pifl6ˆp, define

U_J^L_1;...;Js ˆ x2Xj if i; j; kare distinct; then

x_i;xj;xk are collinear ,i; j; k2Jl for somel

:

Here by ``collinear'' we mean that there exists a linercontainingxi;xj;xk; we do not require that these points be distinct.

These definitions have the effect that the subsets U_K^C_1;...;Kr \U_J^L_1;...;Js correspond to points having precisely specified sets of coincident and collinear points. Note that the points of the subsets U_K^C_1;...;Kr have ne- cessarily some ``implied collinearities'' (for example, if x1ˆx2 then x1;x2;x3are collinear). It will be convenient to writeV_K^C_1;...;Kr for the subset

of U_K^C_1;...;Kr consisting of points for which there are no non-implied colli-

nearities. We write also

V_J^L_1;...;Js ˆU_J^L_1;...;Js\U^C_;

for the set of points with collinearities given by J1;. . .;Js and no coin- cidences.

REMARK1.4. We have U^C_K1;...;Kr\U^L_J1;...;JsX^SS(m)if and only if m_ijmj

3 for all i; g^C_Kl(m)0 for 1lr; g_J^L_l(m)0 for 1ls: Moreover, if any of these inequalities fails, then

U^C_K1;...;Kr\U_J^L_1;...;Js\X^SS(m)ˆ ;:

The same holds for X^S(m) if we replace all inequalities by strict in- equalities. In view of this, when studying X^SS(m)and X^S(m), it is suffi- cient to consider the subsets U_K^C\U_J^Lor even V_K^C and V_J^L. In fact

V_K^CX^SS(m) , m_ijmj

3 for all i and g_K^C(m)0; and

V_J^LX^SS(m) , mijmj

3 for all i and g_J^L(m)0; with similar statements for X^S(m).

1.2 ±Quotients.

PROPOSITION1.5. Let

U^GEN:ˆ fx2Xjx1;. . .;xn in general positiong X;

(i.e. every four points among fx1;. . .;xng are a projective system of

P²(C)). Then

1. X^SS(m)6ˆ ; , U^GENX^SS(m) , m_ijmj

3 for all i;

2. X^S(m)6ˆ ; , U^GEN X^S(m), m_i5jmj

3 for all i;

Moreover, if n5,

X^S(m)6ˆ ; , dim (X^SS(m)==G)ˆ2(n 4):

PROOF. Except for the final statement, this follows from Remark 1.4.

Since X^S(m)=G is a geometric quotient, it is obvious that X^S(m)6ˆ ; ) )dim (X^SS(m)==G)ˆ2(n 4). On the other hand, if X^S(m)ˆ ; but X^SS(m)6ˆ ;, we musthave m_iˆ jmj=3 for some i. We can suppose without loss of generality thatiˆ1. Every orbitinU^GENcontains a point of the form

1 0 0 1 . . . a 0 1 0 1 . . . b 0 0 1 1 . . . c 0

@

1 A;

witha;b;c6ˆ0. Acting by the one parameter subgroupl_{2; 1; 1}and letting t!0, we obtain

1 0 0 0 . . . 0 0 1 0 1 . . . b 0 0 1 1 . . . c 0

@

1 A:

This point belongs to the closure of the original orbit and remains semi- stable. It follows that

X^SS(m)==G (P¹(C))^{n 1}_SS

(m₂;. . .;m_n);

which has dimension less than or equal ton 4. p We know that the quotient X^SS(m)==G depends on the choice of the polarization L(m): moreover Dolgachev-Hu [4] and Thaddeus [11] have proved that when L(m) varies, then there exists only afinite number of different quotients.

Now we give a proof of the same result in our case.

COROLLARY 1.6. There are finitely many different quotients X^SS(m)==G.

PROOF. It follows from Proposition 1.5 and Remark 1.4 that X^SS(m)ˆU^GEN [ U^SS(m);

whereU^SS(m):ˆS

U^C_K1;...;Kr\U_J^L_1;...;JsjU_K^C_1;...;Kr \U^L_J1;...;Js X^SS(m) . In particular we can construct only a finite number of different setsU^SS(m) and as a consequence there exists a finite number of different open setsX^SS(m);

in conclusion only a finite number of quotientsX^SS(m)==Gexists. p REMARK 1.7. Ifn3, thenX^S(m)ˆ ;; moreoverX^SS(m)ˆ ;except when nˆ3 and m₁ˆm₂ˆm₃, in which case X^SS(m)ˆU^GEN and X^SS(m)==G is a point. If nˆ4 and mi5jmj=3 for all i, then X^S(m)ˆ

ˆX^SS(m)ˆU^GENandX^S(m)=Gis a point. OtherwiseX^S(m)ˆ ;and either X^SS(m)ˆ ;orX^SS(m)==Gis a point.

1.3 ±Elementary transformations.

Let m be a polarization such that 3 divides jmj and X^S(m)6ˆ ;;X^S(m)₄X^SS(m); letus consider ``variations'' ofmas follows:

mb ˆm(0;. . .;0;|{z}1

i

;0;. . .;0):

We can have two different kind of variations, depending on the valuejmjb : 1.mb ^‡1!ⁱ m (i.e.jmj b 2 mod 3);

2.mb !¹ⁱ m (i.e.jmj b 1 mod 3) .

In both cases we haveX^S(m)b ˆX^SS(m); studying the relations betweenb valuesg_J^C(m);b g_K^L(m) and valuesb g_J^C(m);g_K^L(m), we observe that

1. mb ^‡1!ⁱ m

X^S(m)b X^SS(m); X^S(m)b ˆX^SS(m)n [

i=2J;g_J^C(m)ˆ0_g_J^L(m)ˆ0

V_J;

X^S(m)X^S(m)b ; X^S(m)ˆX^S(m)b n [

i2H;g_H^C(bm)ˆ2_g_H^L(bm)ˆ1

V_H;

whereV_JisV_J^Cifg_J^C(m)ˆ0 orV_J^Lifg_J^L(m)ˆ0 and in the same way V_H isV_H^Cifg_H^C(m)b ˆ2 orV_H^Lifg_H^L(m)b ˆ1.

2. mb !¹ⁱ m

X^S(m)b X^SS(m); X^S(m)b ˆX^SS(m)n [

i2J;g_J^C(m)ˆ0_g_J^L(m)ˆ0

V_J;

X^S(m)X^S(m)b ; X^S(m)ˆX^S(m)b n [

i=2H;g_H^C(bm)ˆ1_g_H^L(bm)ˆ2

V_H;

whereV_JisV_J^Cifg^C_J(m)ˆ0 orV_J^Lifg_J^L(m)ˆ0 and in the same way V_H isV_H^Cifg_H^C(m)b ˆ1 orV_H^Lifg_H^L(m)b ˆ2.

At the end, we can illustrate the inclusions of the open sets of stable and semi-stable points, with the following diagrams:

The inclusionsX^S(m)X^S(m)b X^SS(m) induce a morphism u:X^S(m)=Gb !X^SS(m)==G;

…4†

which is an isomorphism over X^S(m)=G, while over X^SS(m)==G n n X^S(m)=G

it is a contraction of subvarieties. In fact, consider a point j2 X^SS(m)==G

n X^S(m)=G

: this is the image inX^SS(m)==Gof different, strictly semi-stable orbits, that all have in their closure a closed, minimal orbit Gx, for a certain configurationxˆ(x1;. . .;xn)2X^SSS(m). In parti- cular this configurationxhasjJjcoincident points, and the othersn jJj collinear; by the numerical criterion, we get g_J^C(m)ˆ0 and g_J^L0(m)ˆ0, whereJindicates the coincident points, whileJ⁰ˆ f1;. . .;ng nJindicates the collinear ones.

If there are no further coincidences, we can assumexhas the form

1 . . . 1 0 0 0 0 . . . 0

0 . . . 0 1 0 1 1 . . . 1

0 . . . 0 0 1 b₁ b₂ . . . b_{n jJj 2}

0

@

1

A;b_k 2C; 8k: The orbits O that contain Gx in their closure, are characterized by g_J^C(m)ˆ0 org_J^L0(m)ˆ0; there are two different cases:

1.g_J^C(m)ˆ0: orbits look like O1ˆG

1 . . . 1 0 0 a₁ a₂ . . . a_{n jJj 2}

0 . . . 0 1 0 1 1 . . . 1

0 . . . 0 0 1 rb₁ rb₂ . . . rb_{n jJj 2}

0

@

1

A;r2C;ak 2C:

2.g_J^L0(m)ˆ0: orbits look like

O₂ˆG 1 1 . . . 1 0 0 0 0 . . . 0

0 d₁ . . . d_{jJj 1} 1 0 1 1 . . . 1

0 e₁ . . . e_{jJj 1} 0 1 b₁ b₂ . . . b_{n jJj 2}

0

@

1

A;d_k;e_k2C:

Now, calculatingu ¹(j), it follows that:

u ¹(j)ˆu ¹(f(x));

by the numerical criterion, only one between V_J^C and V_J^L0 is included in X^S(m).b

Dealing with an elementary transformation of the first type (mb ^‡1!ⁱ m), then

- ifi2J )u ¹(j)ˆu ¹f(V_J^C[V_J^L0)

ˆbf(orbits of typeO₁): Whenn5, this has dimension:

dˆn jJj 3: …5†

In fact, let us consider the minimal closed orbitGx: all the orbits that containGxin their closure and are stable inX^S(m), are characterizedb by the coincidence ofjJjpoints (O₁orbits).

- ifi2J⁰ )u ¹(j)ˆu ¹

f(V_J^C[V_J^L0)

ˆbf(orbits of typeO2): Now the dimensiondofu ¹(j) is

dˆ2…n jJ⁰j 1† 1: …6†

Dealing with an elementary transformation of the second type (mb !¹ⁱ m), then

i2J)dˆ2…n jJ⁰j 1† 1; i2J⁰)dˆn jJj 3: …7†

2. XˆP²(C)⁵.

2.1 ±Number of quotients.

Let us study the case nˆ5: XˆP²(C)⁵. Let mˆ(m₁;. . .;m₅) be a polarization such that

05m_i51

3; m_im_i‡1; jmj ˆ1: …8†

After normalization ofjmjand possible permutation of the factors, this is equivalent by Proposition 1.5 to assuming thatX^S(m)6ˆ ;. Itis easy to see thatx2Xis unstable (that is, not semi-stable) if any of the following holds

- three of thexiare coincident;

- four of thexiare collinear;

- there are two coincident pairs ofx_i;

- any of the pairsx_i;x_jwithijˆ12;13;14;23;24 are coincident.

In fact, if any of these possibilities satisfies the semi-stability condition, there existskwithm_k1

3, contradicting (8). It follows that the following sets are always included inX^S(m):

V₁₃₅^L ; V₁₄₅^L ; V₂₃₅^L ; V₂₄₅^L ; V₃₄₅^L ; …9†

while the following sets may or may not be included inX^S(m):

V₁₅^C; V₂₅^C; V₃₄^C; V₃₅^C; V₄₅^C; V₂₃₄^L ; V₁₃₄^L ; V₁₂₅^L ; V₁₂₄^L ; V₁₂₃^L : …10†

In view of the excluded sets listed above and Remark 1.4, these are the only sets we need to consider in order to determineX^S(m) andX^SS(m). More- over, the sets in (10) pair off in an obvious way and, for each pair, either one member of the pair is contained in X^S(m) and the other member is con- tained inX^US(m) or both members are contained inX^SSS(m).

We consider firstthe case in which X^S(m)ˆX^SS(m), so that X^SSS(m)ˆ ;: then there are precisely six different possibilities and we will show that there are exactly six different Geometric Quotients. In fact

0. inU^S(m) there may be only setsV_J^L: an example is the polarization mˆ…1=5;1=5;1=5;1=5;1=5†;

1. if inU^S(m) there is one setV_K^C, it isV₄₅^C: in fact, fori6ˆj, we have m_i‡m_j m₄‡m₅, sog_ij^C(m)>0)g^C₄₅(m)>0.

Example:mˆ(1=4;1=4;1=4;1=8;1=8) ;

2. if inU^S(m) there are two setsV_K^C, they areV₄₅^C andV₃₅^C: the argument is similar to the previous one.

Example:mˆ(3=11;3=11;2=11;2=11;1=11) ;

3. if inU^S(m) there are three setsV_K^C, we can have two cases:

(a) V₄₅^C;V₃₅^C andV₂₅^C , examplemˆ(3=10;1=5;1=5;1=5;1=10) ; (b) V₄₅^C;V₃₅^C andV₃₄^C , examplemˆ(3=10;3=10;1=5;1=10;1=10) . 4. if in U^S(m) there are four sets V_K^C, they are V₄₅^C;V₃₅^C;V₂₅^C and V₁₅^C.

Example:mˆ(1=4;1=4;1=4;2=9;1=36) ;

5. the case of allV_K^Csets inU^S(m) is impossible, becauseV₄₅^C;V₃₅^C;V₃₄^C; V₂₅^C are incompatible.

We have found six cases:

0: U^S(m)S

fV₂₃₄^L ;V₁₃₄^L ;V₁₂₄^L ;V₁₂₃^L ;V₁₂₅^L ;V₁₃₅^L ;V₁₄₅^L ;V₂₃₅^L ;V₂₄₅^L ;V₃₄₅^L g 1: U^S(m)S

fV₂₃₄^L ;V₁₃₄^L ;V₁₂₄^L ;V₁₂₅^L ;V₄₅^C;V₁₃₅^L ;V₁₄₅^L ;V₂₃₅^L ;V₂₄₅^L ;V₃₄₅^L g; 2: U^S(m)S

fV₂₃₄^L ;V₁₃₄^L ;V₁₂₅^L ;V₃₅^C;V₄₅^C;V₁₃₅^L ;V₁₄₅^L ;V₂₃₅^L ;V₂₄₅^L ;V₃₄₅^L g; 3a: U^S(m)S

fV₂₃₄^L ;V₁₂₅^L ;V₂₅^C;V₃₅^C;V₄₅^C;V₁₃₅^L ;V₁₄₅^L ;V₂₃₅^L ;V₂₄₅^L ;V₃₄₅^L g; 3b: U^S(m)S

fV₂₃₄^L ;V₁₃₄^L ;V₃₄^C;V₃₅^C;V₄₅^C;V₁₃₅^L ;V₁₄₅^L ;V₂₃₅^L ;V₂₄₅^L ;V₃₄₅^L g; 4: U^S(m)S

fV₁₂₅^L ;V₁₅^C;V₂₅^C;V₃₅^C;V₄₅^C;V₁₃₅^L ;V₁₄₅^L ;V₂₃₅^L ;V₂₄₅^L ;V₃₄₅^L g: …11†

Then there are only six different open sets of stable points and thus six geometric quotients.

Now supposeX^S(m)6ˆX^SS(m). Then one or more of the pairs in (10) is contained in X^SSS(m). For such a pair, there are two distinct types of strictly semi-stable orbit:

- an orbitO₁withx_k1ˆx_k2;Kˆ fk₁;k₂g:O₁ˆV_K^C; - orbitsO2 withxi1;xi2;xi3collinear,i1;i2;i32K⁰.

OrbitO1 and all orbitsO2 contain in their closure a closed, minimal, strictly semi-stable orbit O12, that is characterized by xk1ˆxk2 and x_i1;x_i2;x_i3 collinear:

In the categorical quotient X^SS(m)==G, orbits O1 andO2 determine the samepoint; in factO12 (O1\O2).

Let us examine the stable case more accurately: we know that only one betweenO₁andO₂is included inX^S(m); whenO₁is included, itdetermines a point of the geometric quotient. In fact if for exampleV₄₅^C X^S(m), then f(V₄₅^C) P²(C)⁴_S

(m1;m2;m3;m4‡m5)=SL3(C) which is a point(see Remark 1.7). When orbits O2 are included in X^S(m), they determine a P¹(C) inX^S(m)=G. In factif for exampleV₁₂₃^L X^S(m), then we can assume

O₂ˆG 1 0 1 0 a

0 1 1 0 b

0 0 0 1 1 0

@

1

A;(a;b)2C²n f(0;0)g:

Applying toO₂ a projectivityG_l ofP²(C) that fixes the line that contains x1;x2;x3(Gldiag(l;l;l ²);withl2C), itfollows that

Glx3

1 0 1 0 l³a 0 1 1 0 l³b

0 0 0 1 1

0 B@

1 CA:

Ifa6ˆ0, then we can assumel³ˆa ¹; thus we obtainx5ˆ[1:a ¹b:1]; in the same way ifb6ˆ0, thenx5ˆ[ab ¹:1:1].

Then itis clear thatf(O₂)P¹(C).

In the semi-stable case when V_K^C;V_K^L0X^SS(m), we know from the above thatV_K^C\V_K^L06ˆ ;is a single non-singular pointofX^SS(m)==G, just as in the stable case whenV_K^Cis included in the open set of stable points.

In this way it follows that every categorical quotientX^SS(m)==G, where X^SS(m)U^GEN[

V_J^C;V_I^L;. . .;

|‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚} V_K^C;V_K^L0;. . .;V_H^C;V_H^L0

|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚} ; stable sets semi-stable sets is isomorphic to a geometric oneX^S(m⁰)=G, where

X^S(m⁰)U^GEN[ fV_J^C;V_I^L;. . .;V_K^C; ;V_H^Cg:

The polarizationm⁰is obtained frommusing elementary transformations such that for eachV_K^C\V_K^L0 6ˆ ;inX^SS(m), thenV_K^C X^S(m⁰). This is al- ways possible because the number of different quotients is finite.

THEOREM 2.8. Let XˆP²(C)⁵: then there are six non trivial quo- tients.

Moreover a quotient X^SS(m)==G is isomorphic to one of the following:

P²(C)with four points blown up (P²(C)₄) P²(C)with three points blown up (P²(C)₃) P²(C)with two points blown up (P²(C)₂) P²(C)with a point blown up (P²(C)₁) P¹(C)P¹(C) (P¹(C)²);

P²(C)

PROOF. The six different open sets of stable points (11) correspond to six different quotients:

0: X^S(m)=GP²(C) with four points blown up 1: X^S(m)=GP²(C) with three points blown up 2: X^S(m)=GP²(C) with two points blown up 3a: X^S(m)=GP²(C) with a point blown-up 3b: X^S(m)=GP¹(C)P¹(C)

4: X^S(m)=GP²(C)

The proof examines the different cases pointed out in the description of the open set of stable pointsX^S(m):

X^S(m)ˆU^GEN [ U^S(m): Case4.

X^S(m)U^GEN[ fV₁₂₅^L ;V₁₅^C;V₂₅^C;V₃₅^C;V₄₅^C;V₁₃₅^L ;V₁₄₅^L ;V₂₃₅^L ;V₂₄₅^L ;V₃₄₅^L g Stable configurations havex1x2x3x4in general position, whilex5is free in P²(C) (in particular it may be coincident with the other points).

Then X^S(m)=GP²(C):

Case3a.

X^S(m)U^GEN[ fV₁₂₅^L ;V₂₃₄^L ;V₂₅^C;V₃₅^C;V₄₅^C;V₁₃₅^L ;V₁₄₅^L ;V₂₃₅^L ;V₂₄₅^L ;V₃₄₅^L g There are two different kinds of stable configurations:

- x₁x₂x₃x₄in general position,x₅ cannotbe coincidentwithx₁ (i.e.

applying the projectivity of P²(C) that sends x₁;x₂;x₃;x₄ to [1:0:0];[0:1:0];[0:0:1];[1:1:1], thenx52P²(C)n f[1:0:0]g);

- x2x3x4collinear (``complementary'' condition ofx5ˆx1); using a projectivity ofP²(C) we obtain:

1 0 0 0 1

0 1 0 1 a

0 0 1 1 b

0

@

1

A; (a;b)2C²n f(0;0)g:

Then with another projectivity Gl of P²(C), that fixes the line containingx2x3x4, we get: (G_lˆdiag(l ²;l;l);l2C)

1 0 0 0 1

0 1 0 1 l³a 0 0 1 1 l³b 0

@

1 A

Ifa6ˆ0, then assumingl³ˆa ¹, we obtainx₅ˆ[1:1:a ¹b]; in the same way ifb6ˆ0, takingl³ˆb ¹, we getx5ˆ[1:ab ¹:1].

Passing to the quotient we get a cover ofP¹(C).

Comparing this case to the previous one, in X^S(m) the set U₁₅^C (that determines a point of the quotient) is substituted by U₂₃₄^L that gives P¹(C) in the quotient. Then

X^S(m)=GP²(C) with a blow-up:

Case3b. In this case, ifx1;x2;x3;x4are in general position,x5 cannotbe collinear withx₁;x₂. As in the previous case, the equalityx₁ˆx₅ is replaced by the collinearity ofx₂;x₃;x₄, giving rise to a blowing- up of the corresponding point ofP²(C). The same applies to the equalityx2ˆx5. The proper transform of the line joiningx1;x2

corresponds to the equalityx3ˆx4, which is allowed, so we must blow down this line to obtainP¹(C)P¹(C).

The other cases are analogous to the first two. p

2.2 ±QuotientsP²(C)⁵==G.

The following diagram shows some birational maps between quotients (the polarization is given in brackets with the corresponding quotient as a subscript); for example if mˆ(22211), then X^S(m)ˆP²(C)₃ (i.e. P²(C) with three points blown-up) and there is a morphism

u:X^S(22211)=GˆX^S(44422)=GˆP²(C)3!X^SS(44322)==GˆP¹(C)P¹(C) In factmb ˆ(44422) is an elementary transformation ofmˆ(44322) in the sense of Section 1.3 anduis the map given by (4).

The diagram has been obtained by direct calculation (some of the techniques will be illustrated in Section 3); in particular it includes some cases whereX^S(m)ˆ ;which are notincluded in the earlier discussion and where the quotient has dimension smaller than two. For example:

X(21111)(P¹(C))⁴(1111) that determines a one-dimensional categorical quotient, while X(33111)P⁰(C) that determines a zero-dimensional ca- tegorical quotient (compare the proof of Proposition 1.5)

3. XˆP²(C)⁶.

3.1 ±Number of quotients.

Now we study the casenˆ6:XˆP²(C)⁶; as in the previous case we first determine how many different quotients we can get when X^S(m)ˆX^SS(m) and the polarization varies.

For a polarization mˆ(m1;. . .;m6) such that X^S(m)6ˆ ; and X^S(m)ˆX^SS(m), then

X^S(m)ˆU^GEN [ U^S(m):

We want to describe the structure of the sets U^S(m); assume that 05mi51

3,mimi‡1 ,jmj ˆ1:

We are interested in those sets V_K^C_1;...;Kr that are included in X^S(m):

some arealwaysincluded inX^S(m):

V₃₆^C; V₄₆^C; V₅₆^C;

and othersmaybe included inX^S(m): the general setsV_K^C

V₁₅^C; V₁₆^C; V₂₃^C; V₂₄^C; V₂₅^C; V₂₆^C; V₃₄^C; V₃₅^C; V₄₅^C; V₁₅₆^C ; V₂₅₆^C ; V₃₄₅^C ; V₃₄₆^C ; V₃₅₆^C ; V₄₅₆^C ;

and also their ``combinations''V_K^C_1;K2, withjK1j ˆ jK2j ˆ2, disjointsubsets

off1;. . .;6g. As we shall see the number of different setsU^S(m) is 38: so the

number of chambers in which theG-ample cone of Dolgachev and Hu [4] is divided is less than or equal to 38.

First of all the minimum number of general sets V_K^C with jKj ˆ2, in- cluded in X^S(m) is five: in fact for example consider only the sets V₃₆^C;V₄₆^C;V₅₆^C that are always included inX^S(m), then obviously

m1‡m6>1

3 ;m2‡m5>1

3 ;m3‡m4>1

3 ) X⁶

iˆ1

m_i>1 : impossible:

In a similar way it is impossible to have only four sets V_K^C(jKj ˆ2) in X^S(m).

Then for five setsV_K^C, we haveV₁₆^C;V₂₆^C;V₃₆^C;V₄₆^C;V₅₆^C: in fact with another 5-tuple (for exampleV₄₅^C;V₂₆^C;V₃₆^C;V₄₆^C;V₅₆^C), we havejmj>1;which is im- possible. Moreover with these combinations, it is impossible to obtain a set asV_K^C withjKj ˆ3.

Going on with the calculations, we are able to construct the following table,

TABLE3.1.

V_K^C;

jKj ˆ2 NoV_K^C;

jKj ˆ3 1 setV_K^C;

jKj ˆ3 2 setsV_K^C;

jKj ˆ3 3 setsV_K^C;

jKj ˆ3 4 setsV_K^C; jKj ˆ3 V₁₆^C;V₂₆^C;V₃₆^C;

V₄₆^C;V₅₆^C 1 3

11(222221) No⁽⁾ No No No

V₁₆^C;V₂₆^C;V₃₆^C; V₄₅^C;V₄₆^C;V₅₆^C

3 1

14(333221)

V₄₅₆^C 1

17(444221) No⁽⁾ No No

V₃₄^C;V₃₅^C;V₃₆^C; V₄₅^C;V₄₆^C;V₅₆^C

3 1 8(221111)

V₄₅₆^C 1

11(332111)

V₄₅₆^C ;V₃₅₆^C 1

14(442211)

V₄₅₆^C ;V₃₅₆^C ; V₃₄₆^C 1

17(552221)

V₄₅₆^C ;V₃₅₆^C ; V₃₄₆^C ;V₃₄₅^C 1

10(331111) V₂₅^C;V₂₆^C;V₃₅^C;

V₃₆^C;V₄₅^C;V₄₆^C; V₅₆^C

3 1

11(322211)

V₄₅₆^C 1

14(433211)

V₄₅₆^C ;V₃₅₆^C 1

17(543311)

V₄₅₆^C ;V₃₅₆^C ; V₂₅₆^C 1

19(644311)

No⁽⁾

V₂₆^C;V₃₄^C;V₃₅^C; V₃₆^C;V₄₅^C;V₄₆^C;

V₅₆^C

1 3

14(432221)

V₄₅₆^C 1

17(543221)

V₄₅₆^C ;V₃₅₆^C 1

26(875321)

V₄₅₆^C ;V₃₅₆^C ; 1

16(542221) No⁽⁾ V₁₆^C;V₂₆^C;V₃₅^C;

V₃₆^C;V₄₅^C;V₄₆^C; V₅₆^C

3 1

17(443321)

V₄₅₆^C 1

20(554321)

V₄₅₆^C ;V₃₅₆^C 1

26(775421) No⁽⁾ No

V₁₆^C;V₂₆^C;V₃₄^C; V₃₅^C;V₃₆^C;V₄₅^C;

V₄₆^C;V₅₆^C

3 1

13(332221)

V₄₅₆^C 1

16(443221)

V₄₅₆^C ;V₃₅₆^C 1

19(553321)

V₄₅₆^C ;V₃₅₆^C ; V₃₄₆^C 1

25(774331)

No⁽⁾

V₁₆^C;V₂₅^C;V₂₆^C; V₃₅^C;V₃₆^C;V₄₅^C;

V₄₆^C;V₅₆^C

3 1

16(433321)

V₄₅₆^C 1

26(766421)

V₄₅₆^C ;V₃₅₆^C 1

26(765521)

V₄₅₆^C ;V₃₅₆^C ; V₂₅₆^C 1

25(755521)

No⁽⁾

V₂₅^C;V₂₆^C;V₃₄^C; V₃₅^C;V₃₆^C;V₄₅^C;

V₄₆^C;V₅₆^C

3 1

31(965542)

V₄₅₆^C 1

26(865322)

V₄₅₆^C ;V₃₅₆^C 1

13(432211) No^(y) No

Segue

that shows all the possible cases (in the ``admissible'' cells we exhibit an ex- ample of a polarization that realizes the geometric quotient). In particular it is not possible to have more than ten setsV_K^C(jKj ˆ2) inX^S(m): we would obtainjmj51.

3.2 ±Singularities.

In this section we study the singularities which appear in the catego- rical quotients whenX^SSS(m)6ˆ ;.

We suppose always that X^S(m)6ˆ ;, so thatm_i5jmj=3 for alli. Suppose that jmj is divisible by 3 and that there exist strictly semi-stable orbits (included inX^SSS(m)); then we can have different cases depending on some

``partitions'' of the polarizationm2Z⁶_>0: TABLE3.1.Segue.

V_K^C;

jKj ˆ2 NoV_K^C;

jKj ˆ3 1 setV_K^C;

jKj ˆ3 2 setsV_K^C;

jKj ˆ3 3 setsV_K^C;

jKj ˆ3 4 setsV_K^C; jKj ˆ3 V₁₅^C;V₁₆^C;V₂₅^C;

V₂₆^C;V₃₅^C;V₃₆^C; V₄₅^C;V₄₆^C;V₅₆^C

3 1

10(222211)

V₄₅₆^C 1

13(333211)

V₄₅₆^C ;V₃₅₆^C 1

16(443311)

V₄₅₆^C ;V₃₅₆^C ; V₂₅₆^C 1

25(766411)

V₄₅₆^C ;V₃₅₆^C ; V₂₅₆^C ;V₁₅₆^C 1

22(555511) V₂₄^C;V₂₅^C;V₂₆^C;

V₃₄^C;V₃₅^C;V₃₆^C; V₄₅^C;V₄₆^C;V₅₆^C

3 1

17(533222)

V₄₅₆^C 1

10(322111) No^(yy) No No

V₂₃^C;V₂₄^C;V₂₅^C; V₂₆^C;V₃₄^C;V₃₅^C; V₃₆^C;V₄₅^C;V₄₆^C;

V₅₆^C

1 3

7(211111) No^(yyy) No No No

() This case is notpossible, because there is notany available term;

()V₃₄₅^C is notincluded in X^S(m), because otherwise m3‡m4‡m551 3; m2‡m651

3)m1>1

3, which is impossible;

(^y)V₂₅₆^C ;V₃₄₅^C ;V₃₄₆^C 6X^S(m);

(^yy)V₂₄₆^C ;V₂₅₆^C ;V₃₄₅^C ;V₃₄₆^C ;V₃₅₆^C 6X^S(m);

(^yyy)V₂₃₆^C ;V₂₄₆^C ;V₂₅₆^C ;V₃₄₅^C ;V₃₄₆^C ;V₃₅₆^C ;V₄₅₆^C 6X^S(m).

1. there are two distinct indices i;j such that mi‡mj ˆ jmj=3 ; as a consequence, for the other indices we havemh‡mk‡ml‡mnˆ

ˆ2jmj=3.

InX^SS(m)==Gthese orbits determine a curveCijP¹(C):

If there do not exist indicesh;l, distinct from each other and fromi andjwithm_h‡m_lˆ jmj=3, then all the orbits are closed.

1.1 particular case: m_i‡m_jˆm_h‡m_lˆm_k‡m_nˆ jmj=3 for distinct indexes (i.e. there is a ``special'' minimal, closed orbit other than the orbits previously seen, characterized by xiˆxj;xhˆxl;xk ˆxn ).

2. there are three distinct indicesh;i;jsuch thatmh‡mi‡mj ˆ jmj=3;

as a consequence for the other indices it holdsmk‡ml ‡mnˆ2jmj=3 (i.e. there is a minimal, closed orbit such thatxhˆxiˆxj, andxk;xl;xn

collinear and distinct for the numerical criterion).

Let us study minimal, closed orbits and what they determine in X^SS(m)==G.

3.2.1 - x_iˆx_j and x_h; x_k; x_l; x_n collinear.

Consider a polarizationmˆ(m₁;. . .;m₆) as previously indicated and an orbitGxsuch thatxiˆxj(mi‡mjˆ jmj=3), and the other four points x_h;x_k;x_l;xn collinear (m_h‡m_k‡m_l‡mnˆ2jmj=3).