A variation on a conjecture of Faber and Fulton

0.39 MB·

by  Michele Bolognesi

A VARIATION ON A CONJECTURE OF FABER AND FULTON
MICHELE BOLOGNESI AND ALEX MASSARENTI
Abstract. In this paper we study the geometry of GIT configurations of n ordered
points on P1 both from the the birational and the biregular viewpoint. In particular,
we prove the analogue of the F-conjecture for GIT configurations of points on P1, that
7
is we show that every extremal ray of the Mori cone of effective curves on the quotient
1 (P1)n//PGL(2),takenwiththesymmetricpolarization, isgeneratedbyaonedimensional
0
boundarystratumofthemodulispace. Onthewaytothisresultwedevelopsometechnical
2
machinery that we use to compute the canonical divisor and the Hilbert polynomial of

n (P1)n//PGL(2) in its natural embedding.
a
J

1
3

] Contents
G
A Introduction 1
. 1. GIT quotients of (P1)n 4
h
t 2. Birational geometry of GIT quotients of (P1)n and the F-conjecture 8
a
3. Degrees of projections of Veronese varieties 19
m
4. Symmetries of GIT quotients 26
[
References 32

1
v
8
6
0
0 Introduction
0
. In oneof themost celebrated papers[DM69]in the history of algebraic geometry Deligne
2
and Mumford proved that there exists an irreducible scheme M coarsely representing
0 g,n
7 the moduli functor of n-pointed genus g smooth curves. Furthermore, they provided a
1 compactification M of M adding the so-called Deligne-Mumford stable curves as
g,n g,n
:
v boundary points. Afterwards other compactifications of Mg,n have been introduced, see
i for instance [Has03].
X
Inthispaperweareinterested inthecompactification ofM given bytheGITquotient
r 0,n
a Σ := (P1)m+3//PGL(2) of configurations of n = m+3 ordered points on P1 with respect
m
to the symmetric polarization on (P1)n. The aim of our notation is to stress the dimension
of the space. In particular, for m = 3 we obtain the celebrated Segre cubic 3-fold Σ ⊂ P4
3
[Do15].
Date: February 2, 2017.
2010 Mathematics Subject Classification. Primary 14D22, 14H10, 14H37; Secondary 14N05, 14N10,
14N20.
Key words and phrases. GIT quotients,Moduli of curves, Mori Dream Spaces, F-conjecture.
1
2 MICHELEBOLOGNESIANDALEXMASSARENTI
The moduli spaces M are among the most studied objects in algebraic geometry.
g,n
Despite this, many natural questions about their biregular and birational geometry remain
unanswered.
The F-conjecture, where as far as we know F stays for Fulton and Faber, for M is one
g,n
of the long-standing conjectures in the field. Its statement is the following:
Conjecture. [GKM02, Conjecture 0.2] A divisor on M is ample if and only if it has
g,n
positive intersection with all 1-dimensional strata. In other words, any effective curve in
M is numerically equivalent to an effective combination of 1- strata.
g,n
Inotherwords,aswestatedintheabstract,anyextremalrayoftheMoriconeofeffective
curves NE(M ) is generated by a one dimensional boundary stratum. In [GKM02] A.
g,n
Gibney, S. Keel and I. Morrison managed to reduce the conjecture for any genus g to the
genus zero case, that is M . Anyway, so far the conjecture is known up to n= 7 [KMc96,
0,n
Thm. 1.2(3)] (see [FGL16] for a more recent general account), and the very intricate
combinatorics of the moduli space M for higher n seem an obstacle pretty difficult to
0,n
avoid if one wants to try his chance.
In this paper we go in a somewhat orthogonal direction. Instead of trying to show the
conjecture for n > 7, we modify it slighlty by allowing a little coarser moduli space into
the picture. In fact, we consider the GIT quotient Σ . This quotient offers an alternate
m
compactification of M , which is a little coarser than M on the boundary, and in fact
0,n 0,n
it is the target a birational morphism M → Σ , that we will recall in the body of the
0,n m
paper, see also [Bo11].
Nevertheless, also Σ has a stratification of its boundary locus, similar to that of M ,
m 0,n
and one can ask exactly the same question of Conjecture . We tackle this problem taking
advantage of the fact that Σ is a Mori Dream Space, while we know that M is not
m 0,n
a Mori Dream Space for n ≥ 10 [CT15, Corollary 1.4], [GK16, Theorem 1.1], [HKL16,
Addendum 1.4]. Mori Dream Spaces, introduced by Y. Hu and S. Keel [HK00], form a
class of algebraic varieties that behave very well from the point of view of the minimal
model program. In particular, their cones of curves and divisors are polyhedral and finitely
generated.
By Proposition 2.7 if n = m+3 is even then Pic(Σ ) ∼= Z and its cones of curves and
m
divisors are 1-dimensional. On the other hand, if n = m+3 is odd then Pic(Σ ) ∼= Zm+3
m
andthebirationalgeometry ofΣ getsmoreinteresting. Inthiscasewemanagetodescribe
m
the cones of nef and effective divisors of Σ and their dual cones of effective and moving
m
curves. The first main result of this paper says that the analogue of the F-conjecture for
GIT configurations of points holds.
Theorem 1. If n = m+3 is odd then the Mori cone NE(Σ ) is generated by classes of
m
1-dimensional boundary strata.
In Theorem 2.33 we will also describeprecisely what are these 1-dimensional strata. The
mainingredients of theproofof Theorem1 is aconstruction of our GITquotients as images
Σ ⊂ PN of rational maps induced by certain linear systems on the projective space Pm,
m
duetoC.Kumar[Ku00,Ku03],acarefulanalysisoftheMorichamberdecompositionofthe
movable cone of certain blow-ups of the projective space and some quite refined projective
geometry of the GIT quotients. More precisely C. Kumar realized Σ as the closure of the
m
image of the rational map induced by the linear system L of degree g hyersurfaces of
2g−1
A VARIATION ON A CONJECTURE OF FABER AND FULTON 3
P2g−1 having multiplicity g −1 at 2g +1 general points if m = 2g −1 is odd, and as the
closure of the image of the rational map induced by the linear system L of degree 2g+1
2g
hyersurfaces of P2g having multiplicity 2g−1 at 2g+2 general points if m = 2g is even. In
particular, N = h0(P2g−1,L ) if m = 2g −1 is odd, and N = h0(P2g,L ) if m = 2g is
2g−1 2g
even.
The tools developed turn out to be useful in resolving a couple of other problems related
to GIT quotients of configuration of points on P1. Furthermore, thanks to recent results
on the dimension of linear system on the projective space due to M. C. Brambilla, O.
Dumitrescu and E. Postinghel [BDP15, BDP16], we obtain some explicit formulas for the
Hilbert polynomial of Σ ⊂ PN. These results are resumed in Corollaries 3.4, 3.5. We
m
would like to mention that an inductive formula for the degree of these GIT quotients
had already been given in [HMSV09], while a closed formula for the Hilbert function of
GIT quotients of evenly weighted points on the line had been given in [HH14]. The main
results on the geometry of Σ ⊂ PN in Sections 2.1, 2.9, 4, and Corollaries 3.4, 3.5 can be
m
summarized as follows.
Theorem 2. Let us consider the GIT quotient Σ ⊂ PN. If m = 2g−1 is odd we have
m
Pic(Σ ) ∼= Z, K ∼= O (−2),
2g−1 Σ2g−1 Σ2g−1
and the the Hilbert polynomial of Σ ⊆ PN is given by
2g−1
g−2
gt+2g−1 2g+1 t(g−r−1)+2g−1−r−1
h (t) = + (−1)r+1 .
Σ2g−1 2g−1 r+1 2g−1
(cid:18) (cid:19) r=0 (cid:18) (cid:19)(cid:18) (cid:19)
X
In particular
g−2
2g+1
deg(Σ ) = g2g−1+ (−1)r+1 (g−r−1)2g−1.
2g−1
r+1
r=0 (cid:18) (cid:19)
X
If m = 2g is even we have
Pic(Σ ) ∼=Z2g+3, K ∼= O (−1),
2g Σ2g Σ2g
and the the Hilbert polynomial of Σ ⊆PN is given by
2g
⌊2g−1⌋
(2g+1)t+2g 2 2g+2 t(2g−2r−1)+2g−r−1
h (t) = + (−1)r+1 .
Σ2g 2g r+1 2g
(cid:18) (cid:19) r=0 (cid:18) (cid:19)(cid:18) (cid:19)
X
In particular
⌊2g−1⌋
2 2g+2
deg(Σ ) = (2g+1)2g + (−1)r+1 (2g−2r−1)2g.
2g
r+1
r=0 (cid:18) (cid:19)
X
Furthermore, the automorphism group of Σ is isomorphic to the symmetric group on
m
n = m+3 elements S for any m ≥ 2.
n
4 MICHELEBOLOGNESIANDALEXMASSARENTI
Plan of the paper. The paper is organized as follows. In Section 1 we recall some well-
knownfactsandprovesomepreliminaryresultsonGITquotient, modulispacesofweighted
pointed rational curves and we clarify the relations between them. In Section 2 we prove
the analogue of the F-conjecture for Σ with m = 2g even. In Section 3 we work out
m
explicit formulas for the Hilbert polynomial a the degree of Σ ⊂ PN. Finally, in Section
m
4 we compute the automorphism groups of Σ .
m
Acknowledgments. We thank Carolina Araujo, Cinzia Casagrande, Han-Bom Moon and
Elisa Postinghel for useful conversations.
The authors are members of the Gruppo Nazionale per le Strutture Algebriche, Geo-
metriche e le loro Applicazioni of the Istituto Nazionale di Alta Matematica ”F. Severi”
(GNSAGA-INDAM).
1. GIT quotients of (P1)n
The main characters of the paper are the GIT quotients (P1)n//PGL(2), that we review
now very quickly. For details there are plenty of very good references on this subject
[MFK94, Do03, Do12, HMSV09, Bo11, DO88].
Let us consider the diagonal PGL(2)-action on (P1)n. An ample line bundle L endowed
with a linearization for the PGL(2)-action is called a polarization. Such a polarization on
(P1)n is completely determined by an n-tuple b = (b ,...,b ) of positive integers:
1 n
L = ⊠n O (b ).
i=1 P1 i
Now let us set |b| = b + ... + b . A point x ∈ (P1)n is said to be b-semistable if for
1 n
some k > 0, there exists a PGL(2)-invariant section s ∈ H0((P1)n,L⊗k)PGL(2) such that
X := {y ∈ (P1)n :s(y) 6=0} is affineand contains x. Asemistable pointx ∈ (P1)n is stable
s
if its stabilizer under the PGL(2) action is finite and all the orbits of PGL(2) in X are
s
closed. A categorical quotient of the open set ((P1)n)ss(b) of semistable points exists, and
this is what we normally denote by X(b)//PGL(2). We will omit to specify the polarization
when it is (1,...,1). If b is odd, then H0((P1)n,L⊗k)PGL(2) = 0 for odd k, and
(1.0) (P1)n(b)//PGL(2) = (P1)n(2b)//PGL(2).
Therefore, by replacing b by 2b if necessary, we assume that b is even.
1.1. Linear systems on Pn. Throughoutthepaperwe willdenote byL (m ,...,m )the
n,d 1 s
linear system of hypersurfacesof degree d in Pn passingthrough s general points p ,...,p ∈
1 s
Pn with multiplicities respectively m ,...,m . It was pointed out by Kumar [Ku03, Section
1 s
3.3] that the GIT quotients (P1)n(b)//PGL(2) can be obtained as the images of certain
rational polynomial maps defined on Pn−3.
Theorem 1.2. [Ku03, Theorem 3.4] Let us assume that b < b for any i = 1,...,n,
i j6=i j
and let p ,...,p ∈ Pn−3 be general points.
1 n−1 P
- If |b| = 2b is even let
L = L (b−b −b ,...,b−b −b ,...,b−b −b )
n−3,b−bn n 1 i n n n
be the linear system of degree b−b hypersurfaces in Pn−3 with multiplicity b−b −b
n i n
at p .
i
A VARIATION ON A CONJECTURE OF FABER AND FULTON 5
- If |b| is odd let
L = L (b−2b −2b ,...,b−2b −2b ,...,b−2b −2b )
n−3,b−2bn n 1 i n n n
be the linear system of degree b−2b hypersurfaces in Pn−3 with multiplicity b −
n
2b −2b at p .
i n i
Finally, let φ : Pn−3 99K P(H0(Pn−3,L)∗) be the rational map induced by L. Then φ
L L
maps birationally Pn−3 onto (P1)n(b)//PGL(2). In other words (P1)n(b)//PGL(2) may be
realized as the closure of the image φ in P(H0(Pn−3,L)∗).
L
Example1.3. Forinstance,ifn = 6andb = ... = b = 1thenb = 3,andL = L (1,...,1).
1 6 3,2
In this case the rational map φ is given by the quadrics in P3 passing through five general
L
points, and (P1)n(b)//PGL(2) ⊂ P4 is the Segre cubic 3-fold. This is a very well-known
classical object, see [Do15, AB15, BB12] for some historic perspective and applications.
If n = 5 and b = ... = b = 1 then L = L (1,...,1) that is the linear system of plane
1 5 2,3
cubics through four general points. In this case the quotient is a del Pezzo surfaceof degree
five.
1.4. Moduli of weighted pointed curves. In[Has03], Hassett introducedmodulispaces
of weighted pointed curves. Given g ≥ 0 and rational weight data A[n] = (a ,...,a ),
1 n
0 < a ≤ 1, satisfying 2g −2+ n a > 0, the moduli space M parametrizes genus
i i=1 i g,A[n]
g nodal n-pointed curves {C,(x ,...,x )} subject to the following stability conditions:
1 n
P
- each x is a smooth point of C, and the points x ,...,x are allowed to coincide
i i1 ik
k
only if a ≤ 1,
j=1 ij
- the twisted dualizing sheaf ω (a x +···+a x ) is ample.
C 1 1 n n
P
In particular, M is one of the compactifications of the moduli space M of genus g
g,A[n] g,n
smooth n-pointed curves.
1.5. For fixed g,n, consider two collections of weight data A[n],B[n] such that a ≥ b for
i i
any i = 1,...,n. Then there exists a birational reduction morphism
ρ : M → M
B[n],A[n] g,A[n] g,B[n]
associating to a curve [C,s ,...,s ] ∈ M the curve ρ ([C,s ,...,s ]) obtained by
1 n g,A[n] B[n],A[n] 1 n
collapsing components of C along which ω (b s +...+b s ) fails to be ample, where ω
C 1 1 n n C
denotes the dualizing sheaf of C.
1.6. Furthermore, for any g, consider a collection of weight data A[n] = (a ,...,a ) and a
1 n
subset A[r] := (a ,...,a ) ⊂ A[n] such that 2g−2+a +...+a > 0. Then there exists
i1 ir i1 ir
a forgetful morphism
π :M → M
A[n],A[r] g,A[n] g,A[r]
associating to a curve [C,s ,...,s ]∈ M the curve π ([C,s ,...,s ]) obtained by
1 n g,A[n] A[n],A[r] 1 n
collapsing components of C along which ω (a s +...+a s ) fails to be ample.
C i1 i1 ir ir
One of the most elegant aspects of the theory of rational pointed curves is the relation
with rational normal curves and their projective geometry. This has been outlined by
Kapranov in [Ka93]. Here below we briefly recall this, and his construction of M as an
0,n
iterated blow-up of Pn−3.
6 MICHELEBOLOGNESIANDALEXMASSARENTI
Kapranov’s blow-up construction. We follow [Ka93]. Let (C,x ,...,x ) be a genus zero
1 n
n-pointed stable curve. The dualizing sheaf ω of C is invertible, see [Kn83]. By [Kn83,
C
Corollaries 1.10and1.11]thesheafω (x +...+x )isveryampleandhasn−1independent
C 1 n
sections. Then it defines an embedding φ : C → Pn−2. In particular, if C ∼= P1 then
deg(ωC(x1 +...+xn)) = n−2, ωC(x1 +...+xn) ∼= φ∗OPn−2(1) ∼= OP1(n−2), and φ(C)
is a degree n−2 rational normal curve in Pn−2. By [Ka93, Lemma 1.4] if (C,x ,...,x ) is
1 n
stable the points p = φ(x ) are in linear general position in Pn−2.
i i
This fact combined with a careful analysis of limits in M of 1-parameter families
0,n
contained in M are the key for the proof of the following theorem [Ka93, Theorem 0.1].
0,n
Theorem1.8. Letp ,...,p ∈ Pn−2 bepointsinlineargeneralposition, andletV (p ,...,p )
1 n 0 1 n
betheschemeparametrizing rationalnormalcurvesthrough p ,...,p . ConsiderV (p ,...,p )
1 n 0 1 n
as a subscheme of the Hilbert scheme H parametrizing subschemes of Pn−2. Then
- V (p ,...,p )∼= M .
0 1 n 0,n
- Let V(p ,...,p ) be the closure of V (p ,...,p ) in H. Then V(p ,...,p ) ∼= M .
1 n 0 1 n 1 n 0,n
Kapranov’s construction allows to translate many questions about M into statements
0,n
on linear systems on Pn−3. Consider a general line L ⊂ Pn−2 through p . There exists a
i i
unique rational normal curve C through p ,...,p , and with tangent direction L in p .
Li 1 n i i
Let [C,x ,...,x ]∈ M be a stable curve, and let Γ ∈ V (p ,...,p ) be the corresponding
1 n 0,n 0 1 n
rational normal curve. Since p ∈ Γ is a smooth point, by considering the tangent line T Γ
i pi
we get a morphism
f : M −→ Pn−3
(1.9) i 0,n
[C,x ,...,x ] 7−→ T Γ
1 n pi
Furthermore, f is birational and defines an isomorphism on M . The birational maps
i 0,n
f ◦f−1
j i
M
0,n
fi fj
Pn−3 fj◦fi−1 Pn−3
arestandardCremonatransformationsofPn−3 [Ka93,Proposition2.12]. Foranyi = 1,...,n
the class Ψ is the line bundle on M whose fiber on [C,x ,...,x ] is the tangent line
i 0,n 1 n
T C. From the previous description we see that the line bundle Ψ induces the birational
pi i
morphism fi :M0,n → Pn−3, that is Ψi = fi∗OPn−3(1). In [Ka93] Kapranov proved that Ψi
is big and globally generated, and that the birational morphism f is an iterated blow-up of
i
the projections from p of the points p ,...,pˆ,...p and of all strict transforms of the linear
i 1 i n
spaces that they generate, in order of increasing dimension.
Construction 1.10. Fix (n−1)-points p ,...,p ∈Pn−3 in linear general position:
1 n−1
(1) Blow-up the points p ,...,p ,
1 n−1
(2) Blow-up the strict transforms of the lines hp ,p i, i ,i = 1,...,n−1,
i1 i2 1 2
.
.
.
(k) Blow-upthestricttransformsofthe(k−1)-planeshp ,...,p i,i ,...,i = 1,...,n−1,
i1 ik 1 k
.
.
.
A VARIATION ON A CONJECTURE OF FABER AND FULTON 7
(n−4) Blow-up the strict transforms of the (n − 5)-planes hp ,...,p i, i ,...,i =
i1 in−4 1 n−4
1,...,n−1.
Now, consider the moduli spaces of weighted pointed curves X [n] := M for k =
k 0,A[n]
1,...,n−4, such that
- a +a > 1 for i =1,...,n−1,
i n
- a +...+a ≤ 1 for each {i ,...,i } ⊂ {1,...,n−1} with r ≤ n−k−2,
i1 ir 1 r
- a +...+a > 1 for each {i ,...,i } ⊂ {1,...,n−1} with r > n−k−2.
i1 ir 1 r
While apologizing for the new notation, we try to justify it by remarking that X [n]
k
is isomorphic to the variety obtained at the kth step of the blow-up construction. The
composition of these blow-up morphism here above is the morphism f : M → Pn−3
n 0,n
induced by the psi-class Ψ . Identifying M with V(p ,...,p ), and fixing a general
n 0,n 1 n
(n−3)-plane H ⊂ Pn−2, the morphism f associates to a curve C ∈ V(p ,...,p ) the point
n 1 n
T C ∩H.
pn
In [Has03, Section 2.1.2] Hassett considers a natural variation of the moduli problem of
weightedpointedrationalstablecurvesbyconsideringweightsofthetypeA[n] =(a ,...,a )
1 n
such that a ∈ Q, 0 < a ≤1 for any i = 1,...,n, and n a = 2.
i i i=1 i
By [Has03, Section 2.1.2] we may construct an explicit family of sucheweighted curves
P
C(A) → M over M as an explicit blow-down of the universal curve over M .
0,n 0,n 0,n
Furthermore, if a < 1 for any i = 1,...,n we may interpret the geometric invariant theory
i
quoetient (P1)n//PGL(2) with respect to the linearization O(a1,...,an) as the moduli space
M e associated to the family C(A).
0,A[n]
Remark 1.11. Note that we may interpret the GIT quotient (P1)n(b)//PGL(2) as a mod-
e
uli space M0,Ae[n] by taking the weights ai = |2b|bi. Conversely, given the space M0,Ae[n]
with (a ,...,a ) = (α1,..., αn) such that n a = 2 we may consider the GIT quotient
1 n β1 βn i=1 i
(P1)n(b)//PGL(2) with b = a M, where M = LCM(β ).
i i P i
Remark 1.12. Let M0,Ae[n] be a moduli space with weights ai summing up to two, and let
M beamodulispacewithweightsa ≥ a foranyi = 1,...,n. By[Has03,Theorem8.3]
0,A[n] i i
there exists a reduction morphism ρAe[n],A[n] :M0,A[n] → M0,eAe[n] operating as the standard
reduction morphisms in 1.5. e
Proposition 1.13. Let φ : Pn−3 99K (P1)n(b)//PGL(2) ⊂ P(H0(Pn−3,L)∗) be the rational
L
map in Theorem 1.2, and let f : M → Pn−3 in (1.9). Then there exits a reduction
i 0,n
morphism ρ:M → (P1)n(b)//PGL(2) making the following diagram
0,n
M
0,n
ρ
fn
Pn−3 φL (P1)n(b)//PGL(2)
commutative.
Proof. Asobservedin[Ku03]viathetheoryofassociated points, each pointx ∈ Pn−3 which
is linearly general with respect to the n−1 fixed points in Pn−3 defines a configuration of
8 MICHELEBOLOGNESIANDALEXMASSARENTI
pointsontheuniquerationalnormalcurveofdegreen−3passingthroughthe(n−1)+1 = n
points. Moreover, this configuration is the image of x in (P1)n(b)//PGL(2) via φ . Via the
L
identification V (p ,...,p ) ∼= M of Theorem 1.8, one easily obtains the claim. (cid:3)
0 1 n 0,n
From the next section on, we will always omit the vector b of the polarization since it
will always be either (1,...,1) or (2,...,2) when we consider an odd number of points,
according to (1.0).
2. Birational geometry of GIT quotients of (P1)n and the F-conjecture
Inthissection wewillstudysomebirationalaspectsofthegeometry oftheGITquotients
we introduced. In particular, we will describe their Mori cone and show that its extremal
rays are generated by 1-dimensional strata of the boundary.
Let us now fix a suitable notation for the GIT quotients. We will denote by Σ the GIT
m
quotient (P1)m+3//PGL(2).
2.1. The odd dimensional case. We start by studyingthe case where the GIT quotients
parametrize an even number of points, that is Σ , for g ≥ 2. This reveals to be less
2g−1
complicated than the even dimensional one, that we will consider eventually. In any case it
is worth looking at it. In fact it will allow us to prove another interesting result in Theorem
2.7 that, as far as we know, does not seem to have appeared in the literature. First of all,
we need a preliminary result.
Let us define L := L (g−1,...,g−1) as the linear system of degree g forms on
2g−1 2g−1,g
P2g−1 vanishing with multiplicity g at 2g+1 general points p ,...,p ∈ P2g−1. In [Ku00,
1 2g+1
Theorem 4.1] Kumar proved that L induces a birational map
2g−1
(2.2) σ : P2g−1 99K P(H0(P2g−1,L )∗)
g 2g−1
and that the GIT quotient Σ , that is the Segre g-variety, in Kumar’s paper [Ku00], is
2g−1
obtained as the closure of the image of σ in P(H0(P2g−1,L )∗).
g 2g−1
Proposition 2.3. Let p ,...,p ∈ P2g−1 be points in general position, and let X2g−1 be
1 2g+1 g−2
the variety obtained the step g of Construction 1.10. Then we have the following commu-
tative diagram
2g−1
X
g−2
σeg
f
P2g−1 σg Σ2g−1 ⊂ PN.
That is, the blow-up morphism f :X2g−1 → P2g−1 resolves the rational map σ .
g−2 g
2g−1
Proof. By Construction 1.10 the blow-up X may be interpreted as the moduli space
g−2
M with A[2g+2] = 1,..., 1,1 .
0,A[2g+2] g g
Furthermore, byRemark1(cid:16).11 Σ2g−1 i(cid:17)sthesingularmodulispaceM0,Ae[2g+2] withweights
A[2g+2] = 1 ,..., 1 , 1 , and by Proposition 1.13 the morphism σ :X2g−1 → Σ is
g+1 g+1 g+1 g g−2 g
exactly the r(cid:16)eduction morphis(cid:17)m ρAe[2g+2],A[2g+2] : M0,A[2g+2] → M0,Ae[2g+2] defined just by
lowering the weights. e (cid:3)
A VARIATION ON A CONJECTURE OF FABER AND FULTON 9
This fairly simple result has some interesting consequences, that we will illustrate in the
rest of this section. Anyway, first we need a technical, and probably well-known, lemma.
Lemma 2.4. Let X be a normal projective variety and f : X 99K Y a birational map of
projective varieties not contracting any divisor. Then K ∼ f−1K .
X ∗ Y
Proof. Since X is normal f is defined in codimension one. Let U ⊆ X be a dense open
subset whose complementary set has codimension at least two where f is defined. Since f
does not contract any divisor then its exceptional set has at least codimension two, hence
we may assume that f is an isomorphism onto its image. Therefore K ∼ (f−1K ) ,
|U X|U ∗ Y |U
and since U is at least of codimension two we get the statement. (cid:3)
The first consequence of Proposition 2.3 is the following.
Lemma 2.5. The canonical sheaf of Σ ⊂ PN is K ∼= O (−2).
2g−1 Σ2g−1 Σ2g−1
Proof. Let X2g−1 be the blow-up of P2g−1 at 2g + 1 general points p ,...,p , and let
1 1 2g+1
f :X2g−1 99K Σ be the birational map induced by the morphism σ :X2g−1 → Σ
g 1 2g−1 g g−2 2g−1
2g−1
in Proposition 2.3. We will denote by H the pull-back in X of the hyperplane section
1
of P2g−1, and by E ,...,E the exceptional divisors. e
1 2g+1
The interpretation of σ as a reduction morphism in the proof of Proposition 2.3 yields
g
that f does not contract any divisor. Indeed f contracts just the strict transforms of the
g g
(g − 1)-planes generated by g of the blown-up points. Since σ is induced by the linear
e g
system L , the pull-back via f of a hyperplane section of Σ is the strict transform
2g−1 g 2g−1
of a hypersurface of degree g in P2g−1 having multiplicity g−1 at p ,...,p . Since
1 2g+1
2g+1 2g+1
−KX2g−1 ∼ 2gH −(2g−2) Ei = 2 gH −(g−1) Ei
1 !
i=1 i=1
X X
we have that −KX2g−1 ∼ fg∗OΣ2g−1(2). Now, in order to conclude it is enough to apply
1
Lemma 2.4 to the birational map f :X2g−1 99K Σ . (cid:3)
g 1 2g−1
Proposition 2.6. The divisor class group of Σ is Cl(Σ ) ∼= Z2g+2. Furthermore
2g−1 2g−1
Pic(Σ ) is torsion free.
2g−1
Proof. The classical case of the Segre cubic Σ ⊂ P4 has been treated in [Hu96, Section
3
3.2.2]. Hence we may assume that g ≥2.
LetY = H∩Σ beageneralhyperplanesection ofΣ . Sincedim(Sing(Σ )) = 0
2g−1 2g−1 2g−1
by Bertini’s theorem, see [Har77, Corollary 10.9] and [Har77, Remark 10.9.2], Y is smooth.
Note that X = (σ )−1(Y)⊂ X2g−1 is the strict transform via f of a general element of the
g g−2
linear system L inducing σ . Therefore, X is smooth and σ : X → Y is a divisorial
2g−1 g g|X
contraction between smooth varieties.
e
Sincedim(X) > 2, theGrothendieck-Lefschetz theorem(see forinstance[Ba78,Theorem
e
2g−1
A]) yields that the natural restriction morphism Pic(X ) → Pic(X) is an isomorphism.
g−2
Therefore Pic(X) ∼= Pic(X2g−1) ∼= Zh, with h = 1+(2g +1)+ 2g+1 +...+ 2g+1 . By
g−2 2 g
the interpretation of σ as a reduction morphism in Proposition 2.3 we know that the
g (cid:0) (cid:1) (cid:0) (cid:1)
codimension one part of the exceptional locus of σ consists of the exceptional divisors
g
the blown-up positiveedimensional linear subspaces of P2g−1. Therefore, Pic(Y) ∼= Z2g+2.
IndeedPic(Y)isgeneratedbytheimages viaσ ofthepull-backofthehyperplanesection
g|X e
e
10 MICHELEBOLOGNESIANDALEXMASSARENTI
of P2g−1 and of the exceptional divisors over the 2g +1 blown-up points. However, since
Σ is not smooth we can not conclude by the Grothendieck-Lefschetz theorem that its
2g−1
Picard group is isomorphic to Z2g+2 as well.
Ontheotherhand,thankstoaversionfornormalvarietiesoftheGrothendieck-Lefschetz
theorem [RS06, Theorem1], weget thatCl(Σ )∼= Cl(Y), andsinceY is smoothwehave
2g−1
Cl(Y) ∼= Pic(Y).
Finally, by [Kl66, Corollary 2] we have that, even when the ambient variety is singular,
the restriction morphism in the Grothendieck-Lefschetz theorem is injective. Therefore, we
have an injective morphismPic(Σ ) ֒→ Pic(Y) ∼= Z2g+2, and hence Pic(Σ ) is torsion
2g−1 2g−1
free. (cid:3)
This allows us to compute the Picard group of Σ . By different methods, this com-
2g−1
putation was also carried out in [MS16].
Proposition 2.7. The Picard group of the GIT quotient Σ ⊂ PN is Pic(Σ ) ∼=
2g−1 2g−1
ZhHi, where H is the hyperplane class. In particular, we have that Nef(Σ ) ∼= R≥0.
2g−1
Proof. Let σ : X2g−1 → Σ ⊂ PN betheresolution of the birational map σ : P2g+1 99K
g g−2 2g−1 g
Σ2g−1 in Proposition 2.3. Note that Pic(Xg2−g−21) ∼= Zρ(Xg2−g−21), where ρ(Xg2−g−21) = 1+(2g +
1)+ 2g+1 e+...+ 2g+1 , and that σ contracts all the exceptional divisors of the blow-up
2 g−1 g
f : X2g−1 → P2g−1 over positive dimensional linear subspaces. Now, let E ⊂ X2g−1 be
(cid:0)g−2 (cid:1) (cid:0) (cid:1) i 0
the exceptional divisor over the poinet p ∈ P2g−1, and let E its strict transform in X2g−1.
i i g−2
2g−1
Furthermore, denote by e the class of a general line in E ⊂ X .
i i 0
ByProposition2.3weknowthat, besidesthedivisorsoveer thepositivedimensionallinear
subspaces, σ also contracts the strict transforms S ,...,S , with r = 2g+1 , in X2g−1 of
g 1 r g g−2
the (g−1)-planes p ,...,p . Note that the contraction of S is given by the contraction
i1 ig i (cid:0) (cid:1)
of the strictetransforms of the degree g −1 rational normal curves in p ,...,p passing
(cid:10) (cid:11) i1 ig
through p ,...,p . In fact, any degree g −1 rational normal curve through p ,...,p is
i1 ig (cid:10) i1(cid:11) ig
contained in p ,...,p and a morphism contracts S to a point if and only if it contracts
i1 ig i
all these curv(cid:10)es to a poi(cid:11)nt. We may write the class in the cone N1(Xg2−g−21)R ∼= Rρ(Xg2−g−21) of
the strict transform of such a rational normal curve as
(2.8) (g−1)l−e −...−e
i1 ig
where l is the pull-back of a general line in P2g−1. Note that the classes in (2.8) generate
the hyperplane
2g+1
gl+ (g−1)e = 0
i
 
 Xj=1 
inN1(X12g−1)R ∼= R2g+2,whereX12g−1 istheblow-upofP2g−1 atp1,...,p2g+1. Therefore,the
2g−1
birational morphism σ : X → Σ contracts the locus spanned by classes of curves
g g−2 2g−1
generating asubspaceof dimension 2g2+1 +...+ 2gg−+11 +2g+1of N1(Xg2−g−21)R ∼= Rρ(Xg2−g−21),
and then e
(cid:0) (cid:1) (cid:0) (cid:1)
2g+1 2g+1
2g−1
ρ(Σ )= ρ(X )− +...+ +2g+1 = 1.
2g−1 g−2 2 g−1
(cid:18)(cid:18) (cid:19) (cid:18) (cid:19) (cid:19)