Table Of ContentON TWO MODULI SPACES OF SHEAVES SUPPORTED ON
QUADRIC SURFACES
5
1
0 MARIOMAICAN
2
b Abstract. Weshowthatthemodulispaceofsemi-stablesheavesonasmooth
e quadricsurface,havingdimension1,multiplicity4,Eulercharacteristic2,and
F firstChernclass(2,2), isthe blow-upattwo points ofacertainhypersurface
3 inaweightedprojectivespace.
2
] LetM be the moduli spaceofGiesekersemi-stable sheavesF onP1×P1 having
G
Hilbert polynomial PF(m)=4m+2, relative to the polarization O(1,1), and first
A
Chernclass c1(F)=(2,2). Let MP3(m2+3m+2)be the moduli space of Gieseker
h. semi-stable sheaves F on P3 having Hilbert polynomial PF(m) = m2 +3m+2.
t Such sheaves are supported on quadric surfaces. The purpose of this note is to
a
m show that MP3(m2+3m+2) is isomorphic to a certainhypersurface in a weighted
projective space (see Proposition 6) and to give an elementary proof of a result of
[
Chung and Moon [3] stating that M is the blow-up of MP3(m2+3m+2) at two
2 regular points.
v Let l, m, n be positive integers. Let V be a vector space over C of dimension l.
6 The reductive group G = GL(n,C)×GL(m,C) /C∗ acts by conjugation on the
5
5 vector space Hom(Cn,Cm⊗(cid:0) V) of m×n-matrices(cid:1)with entries in V. The resulting
2 good quotient
0
. N(V;m,n)=N(l;m,n)=Hom(Cn,Cm⊗V)ss//G
1
0 is called a Kronecker moduli space. Kronecker moduli spaces arise from the study
5 of moduli spaces of torsion-free sheaves, as in [4]. According to [10, Corollary 3.7]
1
and [3, Lemma 5.2], the map
:
v
i Hom(2OP3(−1),2OP3)ss −→MP3(m2+3m+2), hϕi7−→hCoker(ϕ)i,
X
is a good quotient modulo GL(2,C)×GL(2,C) /C∗. Thus, the above moduli
r
a spaceis isomorphicto N(4;2(cid:0),2). Accordingto[10,(cid:1)Remark3.9],MP3(m2+3m+2)
is rational; this result was reproved in [3] using the wall-crossing method.
Lemma 1. Assume that N(l;m,n) contains stable points. Then the same is true
of N(k;m,n) for all integers k > l, and, moreover, N(k;m,n) is birational to
A(k−l)mn×N(l;m,n).
Proof. LetU, V be vectorspacesoverCofdimensionk−l,respectively,l,andput
W = U ⊕V. The projection of W onto the second factor induces a G-equivariant
projection
π: Hom(Cn,Cm⊗W)−→Hom(Cn,Cm⊗V).
2010 Mathematics Subject Classification. 14D20,14D22.
Key words and phrases. Moduliofsheaves,Semi-stablesheaves,GeometricInvariantTheory.
1
2 MARIOMAICAN
From King’s criterion of semi-stability [8] we see that
π−1 Hom(Cn,Cm⊗V)s ⊂Hom(Cn,Cm⊗W)s.
(cid:0) (cid:1)
The left-hand-side, denoted by E, is a trivial G-linearized vector bundle over
Hom(Cn,Cm⊗V)s with fiber Hom(Cn,Cm⊗U). The geometric quotient map
Hom(Cn,Cm⊗V)s −→N(V;m,n)s
is a principal G-bundle, so we can apply [7, Theorem 4.2.14] to deduce that E
descendstoavectorbundleF overN(V;m,n)s. Clearly,F isthegeometricquotient
of E by G, hence F is isomorphic to an open subset of N(W;m,n)s. We conclude
that N(W;m,n) is birational to A(k−l)mn×N(V;m,n). (cid:3)
Proposition 2.
(i) For l ≥3, N(l;2,2) is rational.
(ii) For l ≥3 and n≥1, N(l;n,n+1) is rational.
Proof. According to [4, Lemma 25], N(3;2,2) is isomorphic to P5. Identifying P5
with the space of conic curves in P2, the stable points correspond to irreducible
conics. Applying Lemma 1, yields (i).
According to [5, Propositions 4.5 and 4.6], the subset of N(3;n,n+1) of matri-
ces whose maximal minors have no common factor is isomorphic to the subset of
HilbP2(n(n+1)/2) of schemes that are not contained in any curve of degree n−1.
Thus, N(3;n,n+1) is birationalto HilbP2(n(n+1)/2), so it is rational. Moreover,
N(3;n,n+1) consists only of stable points. Applying Lemma 1, yields (ii). (cid:3)
Proposition 3. For l≥3 and n≥1, N(l;n,n) is a rational variety.
Proof. The argument is inspired by [10, Remark 3.9]. In view of [4, Section 3],
N(3;n,n)containsstablepoints. Thisisduetothefactthatwehavetheinequality
x<n/n<1/x,wherexisthesmallersolutiontotheequationx2−3x+1=0. Thus,
we arein the contextof Lemma 1, which assertsthat N(l;n,n)is rationalfor l≥3
ifN(3;n,n)isrational. Wemay,therefore,restricttothecasewhenl=3. LetV be
avectorspaceoverCwithbasis{x,y,z}. Anelementϕ∈Hom(Cn,Cn⊗V)canbe
writtenuniquelyintheformϕ=ϕ x+ϕ y+ϕ z,whereϕ ,ϕ ,ϕ ∈Hom(Cn,Cn).
1 2 3 1 2 3
Let
Hom(Cn,Cn⊗V) ⊂Hom(Cn,Cn⊗V)s
0
be the open invariant subset given by the condition that ϕ be invertible. Let
1
X ⊂ Hom(Cn,Cn⊗V) be the closed subset given by the condition ϕ = I. The
0 1
group PGL(n,C) acts on X by conjugation. The composite map
X ֒→Hom(Cn,Cn⊗V) −→Hom(Cn,Cn⊗V) /G
0 0
is surjective and its fibers are precisely the PGL(n,C)-orbits. Thus, it factors
through a bijective morphism
X/PGL(n,C)−→Hom(Cn,Cn⊗V) /G.
0
In characteristic zero, bijective morphisms of irreducible varieties are birational.
We have reduced to the following problem. Let U be a complex vector space of
dimension2 and letPGL(n,C) actonHom(Cn,Cn⊗U) by conjugation. Then the
resulting good quotient is rational.
ON TWO MODULI SPACES OF SHEAVES SUPPORTED ON QUADRIC SURFACES 3
Choose a basis {y,z} of U. An element ψ ∈Hom(Cn,Cn⊗U) can be uniquely
written in the form ψ =yψ +zψ , where ψ ,ψ ∈Hom(Cn,Cn). Let
1 2 1 2
Hom(Cn,Cn⊗U) ⊂Hom(Cn,Cn⊗U)
0
be the open invariant subset given by the conditions that ψ have trivial stabilizer
andthatψ beinvertibleandhavedistincteigenvalues. LetY ⊂Hom(Cn,Cn⊗U)
1 0
betheclosedsubsetgivenbytheconditionthatψ beadiagonalmatrix. LetS,T ⊂
1
PGL(n,C) be the image of the canonical embedding of the group of permutations
of n elements, respectively, the subgroup of diagonal matrices. Then H =ST is a
closed subgroup of PGL(n,C) leaving Y invariant. The composite map
Y ֒→Hom(Cn,Cn⊗U) −→Hom(Cn,Cn⊗U) /PGL(n,C)
0 0
is surjective and its fibers are precisely the H-orbits. Thus, it factors through a
bijective morphism
Y/H −→Hom(Cn,Cn⊗U) /PGL(n,C)
0
that must be birational. We have reduced the problem to showing that Y/H is
rational.
Let Y ⊂ Y be the open H-invariant subset given by the condition that all
0
entries of ψ be non-zero. Concretely, Y = D×E, where D,E ⊂ Hom(Cn,Cn)
2 0
are the subset of invertible diagonalmatrices with distinct entries on the diagonal,
respectively, the subset of matrices without zero entries. The normal subgroup
T ≤ H acts trivially on D, hence (D×E)/T is a trivial bundle over D with fiber
E/T. The induced action of S = H/T is compatible with the bundle structure.
The stabilizer in S of any ψ ∈ D acts trivially on the fiber over ψ , because it is
1 1
trivial. It follows that (D×E)/T descents to a fiber bundle F over D/S. Clearly,
F is isomorphic to (D ×E)/H, hence (D ×E)/H is birational to D/S ×E/T.
Both D/S and E/T are rational, namely D/S is isomorphic to an open subset of
Sn(A1)≃An, while E/T ≃(A1\{0})n2−n+1. In conclusion, Y/H is rational. (cid:3)
Let r >0 and χ be integers. Let MP2(r,χ) denote the moduli space of Gieseker
semi-stable sheaves on P2 having Hilbert polynomial P(m) = rm+χ. It is well
known that MP2(r,0) is birational to N(3;r,r) and, if r is even, MP2(r,r/2) is
birational to N(6;r/2,r/2). We obtain the following.
Corollary 4. The moduli spaces MP2(r,0) and, if r is even, MP2(r,r/2), are ra-
tional.
The rationality of MP2(3,0) and MP2(4,2) is already known from [9].
The maps
det: Hom(C2,C2⊗V)−→S2V, det(ϕ)=ϕ ϕ −ϕ ϕ ,
11 22 12 21
and
e: Hom(C2,C2⊗V)−→Λ4V, e(ϕ)=ϕ ∧ϕ ∧ϕ ∧ϕ
11 22 12 21
are semi-invariant in the sense that for any (g,h)∈G and ϕ∈Hom(C2,C2⊗V),
det((g,h)ϕ)=det(g)−1det(h)det(ϕ), e((g,h)ϕ)=det(g)−2det(h)2e(ϕ).
Using King’s criterion of semi-stability [8], it is easy to see that ϕ is semi-stable if
∗
and only if det(ϕ) 6=0 and is stable if and only if det(ϕ) is irreducible in S V. In
the casewhendim(V)=3,the isomorphismN(V;2,2)→P(S2V)of[4]is givenby
hϕi7→hdet(ϕ)i.
4 MARIOMAICAN
In the sequel we will assume that dim(V) = 4 and that m = 2, n = 2. Choose
bases {x,y,z,w} of V and {v ,v ,v ,v } of V∗. Consider the semi-invariant func-
1 2 3 4
tions
ǫ,ρ: Hom(C2,C2⊗V)−→C, ǫ(ϕ)=iv1∧v2∧v3∧v4e(ϕ),
ρ(ϕ)=iv1∧v2∧v3∧v4(iv1det(ϕ)∧iv2det(ϕ)∧iv3det(ϕ)∧iv4det(ϕ)).
Here i denotes the internal product with a vector v ∈V∗.
v
Proposition 5. We have the relation ǫ2 =ρ.
Proof. Let {v′,v′,v′,v′} be another basis of V∗ and let υ ∈ GL(4,C) be the
1 2 3 4
change-of-basismatrix. With respectto this basis we define the functions ρ′ andǫ′
as above. Then ǫ′(ϕ)=det(υ)ǫ(ϕ) and ρ′(ϕ)=det(υ)2ρ(ϕ), hence ǫ(ϕ)2 =ρ(ϕ) if
and only if ǫ′(ϕ)2 =ρ′(ϕ). Put U =span{x,y,z} and let
π: Hom(C2,C2⊗V)−→Hom(C2,C2⊗U)
be the morphisminduced by the projectionofV =U⊕Cw ontothe firstfactor. It
is enough to verify the relation on the Zariski open subset given by the condition
that det(π(ϕ)) be irreducible. Changing, possibly, the basis of U, we may assume
that det(π(ϕ)) = x2−yz. Since π(ϕ) is stable, and since N(U;2,2) is isomorphic
to P(S2U), we have
x y x+aw y+bw
π(ϕ)∼ , so we may write ϕ= .
(cid:20) z x (cid:21) (cid:20) z+cw x+dw (cid:21)
We have
det(ϕ)=x2−yz+(a+d)xw−cyw−bzw+(ad−bc)w2,
e(ϕ)=(d−a)x∧y∧z∧w.
Since wearefree to choosethe basisofV∗, we choose{v ,v ,v ,v } to be the dual
1 2 3 4
of {x,y,z,w}. We have
∂
i det(ϕ)= det(ϕ)=2x+(a+d)w,
v1 ∂x
∂
i det(ϕ)= det(ϕ)=−z−cw,
v2 ∂y
∂
i det(ϕ)= det(ϕ)=−y−bw,
v3 ∂z
∂
i det(ϕ)= det(ϕ)=(a+d)x−cy−bz+2(ad−bc)w,
v4 ∂w
2 0 0 a+d
(cid:12) 0 0 −1 −c (cid:12)
ǫ(ϕ)=d−a, ρ(ϕ)=(cid:12) (cid:12)=(a−d)2.
(cid:12) 0 −1 0 −b (cid:12)
(cid:12) (cid:12)
(cid:12) a+d −c −b 2(ad−bc) (cid:12)
(cid:12) (cid:12)
In conclusion, ǫ(ϕ)2 =(d−a)2 =(cid:12) ρ(ϕ). (cid:12) (cid:3)
(cid:12) (cid:12)
ConsidertheactionofC∗ onS2V⊕Λ4V givenbyt(q,p)=(tq,t2p)andletPdenote
the weighted projective space (S2V ⊕Λ4V)\{0} /C∗. Consider the map
η: N(V;2,2)(cid:0)−→P, η(hϕi)=(cid:1)hdet(ϕ),e(ϕ)i.
Choose coordinates on P given by the choice of basis {x,y,z,w} of V. In view of
Proposition 5, the image of η is contained in the hypersurface H ⊂P given by the
equation res(q)=p2, where res(q) denotes the resultant of the quadratic form q.
ON TWO MODULI SPACES OF SHEAVES SUPPORTED ON QUADRIC SURFACES 5
Proposition 6. Assume that dim(V)=4. Then the map η: N(V;2,2)→H is an
isomorphism.
Proof. ThesingularpointsoftheconeHˆ ⊂S2V⊕Λ4V overH areoftheform(q,0),
whereq ∈S2V is a singularpointofthe vanishinglocus ofthe resultant. Itfollows
that Hˆ is regular in codimension 1. From Serre’s criterion of normality we deduce
thatH isnormal(conditionS2issatisfiedbecauseHˆ isahypersurfaceinasmooth
variety). Normality is inherited by a good quotient, hence H = (Hˆ \{0})/C∗ is
normal,too. Inviewofthe MainTheoremofZariski,itisenoughtoshowthatη is
bijective. Since N(V;2,2) is complete, and since N(V;2,2) and H are irreducible
of the same dimension, it is enough to show that η is injective.
Assume that η(hϕ i) = η(hϕ i). Varying ϕ and ϕ in their respective orbits,
1 2 1 2
we may assume that det(ϕ )=det(ϕ ) and e(ϕ )=e(ϕ ). If det(ϕ ) is reducible,
1 2 1 2 1
say det(ϕ )=uu′ for some u,u′ ∈V, then it is easy to see that
1
u u u u
ϕ ∼ 1 , ϕ ∼ 2
1 (cid:20) 0 u′ (cid:21) 2 (cid:20) 0 u′ (cid:21)
for some u ,u ∈ V. But then hϕ i = hϕ i = hdiag(u,u′)i. Assume now that
1 2 1 2
det(ϕ ) is irreducible. There exists a vector w ∈ V and a subspace U ⊂ V such
1
that V = U ⊕Cw and det(π(ϕ )) is irreducible (notations as at Proposition 5).
1
As mentioned at Proposition 5, we may choose a basis {x,y,z} of U such that
det(π(ϕ ))=x2−yz, forcing
1
x y
π(ϕ )∼π(ϕ )∼ .
1 2 (cid:20) z x (cid:21)
Thus, we may write
x+a w y+b w x+a w y+b w
ϕ = 1 1 , ϕ = 2 2 .
1 (cid:20) z+c1w x+d1w (cid:21) 2 (cid:20) z+c2w x+d2w (cid:21)
The relation det(ϕ ) = det(ϕ ) yields the relations b = b , c = c , a +d =
1 2 1 2 1 2 1 1
a +d . The relation e(ϕ ) = e(ϕ ) yields the relation a −d = a −d . We
2 2 1 2 1 1 2 2
conclude that ϕ =ϕ , hence hϕ i=hϕ i. (cid:3)
1 2 1 2
Remark 7. It was already known to Le Potier [10, Remark 3.8] that the map
det: N(V;2,2)−→P(S2V)
is a double cover branched over the locus of singular quadratic surfaces.
Inthesequel,wewillusetheabbreviationsO(r,s)=OP1×P1(r,s),ω =ωP1×P1,and
FD = Ext1 (F,ω) for a sheaf F on P1 ×P1 of dimension 1. We quote below [3,
O
Proposition 3.8].
Proposition 8. The sheaves F giving points in M are precisely the sheaves having
one of the following three types of resolution:
ϕ
(1) 0−→2O(−1,−1)−→2O−→F −→0,
(2) 0−→O(−2,−1)−→O(0,1)−→F −→0,
(3) 0−→O(−1,−2)−→O(1,0)−→F −→0.
6 MARIOMAICAN
Thispropositionwasprovedin[3]bythewall-crossingmethod,however,itwasalso
nearly obtained in [1]. At [1, Lemma 20] it is mistakenly claimed that all sheaves
in M have resolution (1). At a closer inspection, the argument of [1, Lemma
20] shows that the sheaves in M satisfying the conditions H0(FD(1,0)) = 0 and
H0(FD(0,1)) = 0 are precisely the sheaves given by resolution (1). Indeed, the
exact sequence (50) in [1] reads
(4) 0−→H−→2O−→F −→0,
whereHisalocallyfreesheafofrank2anddeterminantω. Dualizingthissequence,
we get the exact sequence
(5) 0−→2O(−2,−2)−→HD ≃H∗⊗ω ≃H⊗det(H)∗⊗ω ≃H−→FD −→0.
From this we get the relations
h1(H(1,0))=h0(FD(1,0)) and h1(H(0,1))=h0(FD(0,1)).
The vanishing of H1(H(1,0)) and H1(H(0,1)) implies that H ≃ 2O(−1,−1), in
which case (4) yields resolution (1).
According to [11, Theorem 13], if F gives a point in M, then FD(0,1) and
FD(1,0) give points in the moduli space M′ of semi-stable sheaves on P1 × P1
having Hilbert polynomial P(m)=4m and first Chern class c =(2,2). We claim
1
that the sheaves E giving points in M′ and satisfying the condition H0(E) 6= 0
are precisely the structure sheaves of curves E ⊂ P1 ×P1 of type (2,2). By the
argument of [1, Lemma 9], O gives a stable point in M′. Conversely, if E gives a
E
point in M′ and H0(E) 6= 0, then, by the argument of [6, Proposition 2.1.3], there
is an injective morphism O → E for a curve C ⊂ P1 ×P1. If C did not have
C
type (2,2), then the semi-stability of E would get contradicted. Thus, C has type
(2,2) and, comparing Hilbert polynomials, we see that O ≃ E. In conclusion, if
C
H0(FD(0,1))6=0, then F ≃O (0,−1)D ≃O (0,1), hence F has resolution(2). If
E E
H0(FD(1,0))6=0, then F ≃O (−1,0)D ≃O (1,0), hence F has resolution (3).
E E
We denote by M ,M ,M ⊂M the subsets of sheaves given by resolution (1),
0 1 2
(2), respectively, (3). Clearly, M is open and M , M are divisors isomorphic to
0 1 2
P8. Let Hom(2O(−1,−1),2O) denote the subset of injective morphisms.
0
Corollary 9. The canonical map from below is a good quotient modulo G:
γ: Hom(2O(−1,−1),2O) −→M , γ(ϕ)=hCoker(ϕ)i.
0 0
Proof. According to [2, Lemma 1], for any coherent sheaf F on P1×P1 there is a
spectralsequence convergingto F in degree zero andto 0 in degrees different from
zero, similar to the Beilinson spectral sequence. Its first level Eij is given by
1
Eij =0 if i>0 or i<−2,
1
E0j =Hj(F)⊗O, E−2,j =Hj(F(−1,−1))⊗O(−1,−1),
1 1
and by the exact sequences
Hj(F(0,−1))⊗O(0,−1)−→E−1,j −→Hj(F(−1,0))⊗O(−1,0).
1
If F gives a point in M , then
0
H0(F)≃C2, H1(F)=0, H0(F(−1,−1))=0, H1(F(−1,−1))≃C2,
H0(F(0,−1))=0, H1(F(0,−1))=0, H0(F(−1,0))=0, H1(F(−1,0))=0.
ON TWO MODULI SPACES OF SHEAVES SUPPORTED ON QUADRIC SURFACES 7
Thus, E has only two non-zero terms: E−2,1 = 2O(−1,−1) and E0,0 = 2O. The
1 1 1
relevant part of E is represented in the following table:
2
2O(−1,−1) 0 0
❘❘❘❘❘❘❘❘ϕ❘❘❘❘❘❘❘))
0 0 2O
The sequence degenerates at E , hence ϕ is injective and Coker(ϕ) ≃ F. This
3
shows that resolution (1) can be obtained from the Beilinson spectral sequence of
F. Arguingasat[6,Theorem3.1.6],wecanseethatresolution(1)canbeobtained
for local flat families of sheaves in M , hence γ is a categorical quotient. By the
0
uniquenessofthecategoricalquotient,wededucethatγisagoodquotientmap. (cid:3)
WefixvectorspacesV andV overCofdimension2andwemaketheidentifications
1 2
P1×P1 =P(V )×P(V ), H0(O(r,s))=SrV∗⊗SsV∗, V =V∗⊗V∗.
1 2 1 2 1 2
Let
W⊂Hom 2O(−1,−1)⊕O(−1,0)⊕O(0,−1), O(−1,0)⊕O(0,−1)⊕2O
(cid:0) (cid:1)
be the open subset of injective morphisms ψ for which Coker(ψ) is Gieseker semi-
stable. We represent ψ by a matrix
1⊗u 1⊗v a 0
12 12 1
ψ = ψ11 ψ12 = u11⊗1 v11⊗1 0 a2 ,
(cid:20) ψ21 ψ22 (cid:21) f11 f12 u21⊗1 1⊗u22
f f v ⊗1 1⊗v
21 22 21 22
where a ,a ∈C, u ,v ∈V∗, f ∈V. The algebraic group
1 2 ij ij j ij
G= Aut(2O(−1,−1)⊕O(−1,0)⊕O(0,−1))×Aut(O(−1,0)⊕O(0,−1)⊕2O) /C∗
acts o(cid:0)n W by conjugation. We represent elements of G by pairs (g,h), where(cid:1)
g 0 h 0
g = 11 , h= 11 ,
(cid:20) g21 g22 (cid:21) (cid:20) h21 h22 (cid:21)
g ∈Aut(2O(−1,−1)), h ∈Aut(2O), etc.
11 22
Proposition 10. The canonical map θ: W → M, θ(ψ) = hCoker(ψ)i is a good
quotient modulo G.
Proof. Let W ⊂ W be the open subset given by the condition that ψ be in-
0 12
vertible. Concretely, W is the set of morphisms ψ such that ψ is invertible and
0 12
α(ψ)=ψ −ψ ψ−1ψ is injective. In view of Proposition8, M =θ(W ). The
21 22 12 11 0 0
restricted map θ : W →M is the composition
0 0 0
α γ
W −→Hom(2O(−1,−1),2O) −→M ,
0 0 0
where γ is the good quotient map from Corollary 9. Let G EG be the closed
0
normal subgroup given by the conditions g =cI, h =cI, c∈C∗. We have the
11 22
relation α(hψg−1) = h α(ψ)g−1, hence α is constant on the orbits of G . Since
22 11 0
any ψ ∈W is equivalent to
0
0 I
,
(cid:20) α(ψ) 0 (cid:21)
it follows that the fibers of α are precisely the G -orbits,and that α has a section.
0
We deduce that α is a geometric quotient modulo G . Since γ is a good quotient
0
8 MARIOMAICAN
moduloG/G ,weconcludethatθ isagoodquotientmoduloG. LetMs ⊂M be
0 0 0 0
the subset of stable points. Since γ−1(Ms)→Ms is a geometric quotient modulo
0 0
G/G , we deduce that θ−1(Ms)→Ms is a geometric quotient modulo G.
0 0 0
Assumenowthatψ ∈W\W . DenoteF =Coker(ψ). Thenψ 6=0,otherwise
0 12
Coker(ψ ) would be a destabilizing subsheaf of F. Thus, W\W is the disjoint
22 0
union of two subsets W and W . The former is given by the relations a 6= 0,
1 2 1
a = 0; the latter is given by the relations a = 0, a 6= 0. Assume that ψ ∈ W .
2 1 2 1
Then u , v are linearly independent, otherwise F would have a destabilizing
11 11
quotient sheaf of slope zero. Likewise, u , v are linearly independent, otherwise
22 22
F would have a destabilizing subsheaf of slope 1. Consider the morphism
ξ ∈Hom(2O(−1,−1)⊕O(0,−1), O(0,−1)⊕2O),
u ⊗1 v ⊗1 0
11 11
ξ = f11−a−11u21⊗u12 f12−a−11u21⊗v12 1⊗u22 .
f −a−1v ⊗u f −a−1v ⊗v 1⊗v
21 1 21 12 22 1 21 12 22
Clearly, F ≃Coker(ξ). Applying the snake lemma to the exact diagram
0 0
0 //O(0,(cid:15)(cid:15)−1) (cid:20)11⊗⊗uv2222(cid:21) //2O(cid:15)(cid:15) [−1⊗v22 1⊗u22] ////O(0,1)
(cid:15)(cid:15) (cid:15)(cid:15)
0 //2O(−1,−1)⊕O(0,−1) ξ //O(0,−1)⊕2O ////F
O(−2,−1)(cid:31)(cid:127) (cid:20)−uv1111⊗⊗11(cid:21) //2O(−1(cid:15)(cid:15) ,−1)[u11⊗1 v11⊗1]//O(0,(cid:15)(cid:15)−1) //0
(cid:15)(cid:15) (cid:15)(cid:15)
0 0
we obtain resolution (2). This shows that θ(W ) ⊂ M . It is now easy to see
1 1
thatthe restrictedmapW →M is surjectiveandthatits fibersarepreciselythe
1 1
G-orbits. By symmetry, the same is true of the restricted map W →M .
2 2
Let Ms ⊂M be the open subset of stable points and Ws =θ−1(Ms). We have
proved above that the fibers of the restricted map θs: Ws →Ms are precisely the
G-orbits. Since Ms is normal (being smooth), we can apply [12, Theorem 4.2] to
deduce that θs is a geometric quotient modulo G. Finally, since M = M ∪Ms,
0
we deduce that θ is a good quotient map. (cid:3)
Choose bases {u ,v } of V∗ and {u ,v } of V∗. Then x = u ⊗u , y = v ⊗u ,
1 1 1 2 2 2 1 2 1 2
z =u ⊗v , w =v ⊗v form a basis of V. An easy calculationshows that the set
1 2 1 2
of injective morphisms
Hom(2O(−1,−1),2O) ⊂Hom(C2,C2⊗V)
0
is the subset of matrices whose determinant is not a multiple of xw−yz. Thus,
Hom(2O(−1,−1),2O) //G≃N(V;2,2)\det−1{hxw−yzi}.
0
ON TWO MODULI SPACES OF SHEAVES SUPPORTED ON QUADRIC SURFACES 9
According to Remark 7, det−1{hxw−yzi} consists of two points ν and ν , where
1 2
ǫ(ν )=1, ǫ(ν )=−1. We saw at Corollary 9 that γ induces an isomorphism
1 2
Hom(2O(−1,−1),2O) //G−→M .
0 0
The inverse of this isomorphism is denoted by
β : M −→N(V;2,2)\{ν ,ν }.
0 0 1 2
It is natural to ask whether M is the blow-up of N(V;2,2) at ν and ν . This is,
1 2
indeed,oneofthemainresultsin[3],whereablowing-downmapβ: M→N(V;2,2)
isconstructedviaFourier-Mukaitransformsofsheaves,inviewoftheidentification
of N(V;2,2) with MP3(m2+3m+2). We give below an alternate construction.
Proposition 11. The map β extends to a blowing-down map β: M→N(V;2,2)
0
with exceptional divisor M ∪M and blowing-up locus {ν ,ν }.
1 2 1 2
Proof. Recall that on M =W //G, β is induced by the map sending ψ to
0 0 0
ψ −a−1 u21⊗1 1⊗u 1⊗v −a−1 1⊗u22 u ⊗1 v ⊗1 .
21 1 (cid:20) v21⊗1 (cid:21) 12 12 2 (cid:20) 1⊗v22 (cid:21) 11 11
(cid:2) (cid:3) (cid:2) (cid:3)
Equivalently, β is induced by the map sending ψ to
0
a ψ −a−1a u21⊗1 1⊗u 1⊗v − 1⊗u22 u ⊗1 v ⊗1
2 21 1 2(cid:20) v21⊗1 (cid:21) 12 12 (cid:20) 1⊗v22 (cid:21) 11 11
(cid:2) (cid:3) (cid:2) (cid:3)
which is defined on W ∪ W . Clearly, this map factors through a morphism
0 1
M ∪M →N(V;2,2), which maps M to the class of the matrix
0 1 1
1⊗u x y
2 u ⊗1 v ⊗1 = ,
(cid:20) 1⊗v2 (cid:21) 1 1 (cid:20) z w (cid:21)
(cid:2) (cid:3)
that is, to ν . Analogously, β extends to a morphism defined on M ∪M , which
1 0 0 2
maps M to the class of the matrix
2
u ⊗1 x z
1 1⊗u 1⊗v = ,
(cid:20) v1⊗1 (cid:21) 2 2 (cid:20) y w (cid:21)
(cid:2) (cid:3)
thatis,toν . Thetwomorphismsconstructedthusfargluetoamorphismβ: M→
2
N(V;2,2). Since ν and ν are smooth points, β is a blow-down. (cid:3)
1 2
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CenterforGeometryanditsApplications,PohangUniversityofScienceandTech-
nology,Pohang 790-784,RepublicofKorea
E-mail address: [email protected]
