Finiteness of isomorphic classes in the set of moduli schemes of sheaves on a surface

0.09 MB·English

by  Kimiko Yamada

FINITENESS OF ISOMORPHIC CLASSES IN THE SET OF
MODULI SCHEMES OF SHEAVES ON A SURFACE
KIMIKO YAMADA
0
1
0
2 Abstract. When a non-singular complex projective surface X satisfies that
n KX ∼ 0, we shall show that there are only finitely many isomorphic classes as
a abstractschemes in the set of moduli schemes of H-semistable sheaves with fixed
J Chern classesα on X, where H runs overthe set of all α-generic polarizations on

5 X.
1

1. Introduction
]
G
A Let X be a non-singular projective surface over C, α an element of (r,c1,c2) ∈
. Z>0 × NS(X) × Z with r > 0, and H an ample line bundle (polarization) on X.
h
Then there is a coarse moduli scheme M(H,α) of H-semistable sheaves of type α on
t
a X. It is projective over C. Here we say that M(H,α) and M(H′,α) are isomorphic
m
by definition if (i) any sheaf of type α on X is H-stable (resp. H-semistable) if and
[ ′ ′
only if it is H -stable (resp. H -semistable), and (ii) any H-semistable two sheaves

1
of type α are S-equivalent with respect to H-semistability if and only if they are
v
′
S-equivalent with respect to H -semistability. The set
9
2
6 (cid:8)M(H,α) (cid:12) H: α-generic polarization on X(cid:9)/(isomorphism by definition)
(cid:12)
2
is countable and can be infinite. However we shall prove
.
1
0 Theorem 1.1. When r > 0 and K ∼ 0, the set
X
0
1 (cid:8)M(H,α) (cid:12) H: α-generic polarization on X(cid:9)/(isomorphism as abstract schemes)
: (cid:12)
v
is finite.
i
X
r From [5], M(H,α) and M(H′,α) are birationally equivalent if 2rc2 − (r − 1)c21
a is sufficiently large with respect to H and H′. When X is minimal and κ(X) > 0
there is a moduli-theoretic analogy of minimal model program of M(H,α) by [7].
If K ∼ 0, then we can use Theorem and M(H,α) and M(H′,α) are connected
X
by Mukai flops. Thus when X is minimal and κ(X) ≥ 0, one can use there results
′
about birational relation between M(H,α) and M(H ,α).
The author expresses hearty thanks to Prof. S. Mukai and Prof. Y. Namikawa
for their invaluable suggestions and comments.
2. Proof of Theorem
We begin with preliminary. The notion of α-walls (or walls with respect to α)
appeared in [1], [2] and [4]. A connected component of the complement of the union
ofall α-walls inthe amplecone Amp(X)of X iscalled anα-chamber. Apolarization
on X is said to be α-generic if it is contained in no α-wall.
1991 Mathematics Subject Classification. Primary 14J60; Secondary 14D20.
1
2 KIMIKOYAMADA
′
Now let H and H be α-generic polarizations. As mentioned in Introduction, the
set
(1) (cid:8)M(H,α) (cid:12) H: α-generic polarization on X(cid:9)/(isomorphism by definition)
(cid:12)
is countable and can be infinite. Indeed, in case where r = 2, any α-wall W equals
Wη = (cid:8)L ∈ Amp(X) (cid:12) L·η = 0(cid:9) with η ∈ NS(X) such that 0 < −η2 ≤ 4c2 − c21.
(cid:12) ′
When α-generic polarizations H and H lie in adjacent α-chambers separated by W,
′
a sheaf E of type α is H-semistable and not H -semistable if and only if E is given
by a nonsplit extension
(2) 0 −→ OX(D)⊗IZl −→ E −→ OX(c1 −D)⊗IZr −→ 0,
where Z and Z are zero-dimensional subscheme in X such that l(Z ) + l(Z ) =
l r l r
c2 −c21/4+η2/4 and D is a divisor such that 2D−c1 ∼ η. When KX ∼ 0, it holds
that
2
(3) −χ(OX(c1 −D)⊗IZr,OX(D)⊗IZl) = −η /2+l(Zl)+l(Zr)−χ(OX)
2 2 2
= 2c2−c1/2−l(Zl)−l(Zr)−χ(OX) > 2c2−c1/2−c2+c1/4−χ(OX) > −χ(OX),
where the first inequality holds since −η2/4 = c2 −c21/4−l(Zl)−l(Zr) > 0 by the
Hodge index theorem, and the second inequality holds from Bogomolov’s inequality.
1
Thus if KX ∼ 0 and χ(OX) ≤ 0, then (3) implies Ext (OX(c1−D)⊗IZr,OX(D)⊗
I ) 6= 0 and a H-semistable sheaf of type α which is not H′-semistable does exist.
Zl
′
As a result M(H,α) and M(H ,α) are not isomorphic by definition if α-generic
′
polarizations H and H are separated by a α-wall. On the other hand, Matsuki-
Wentworth [4] gaveanexample of anAbeliansurfacewhere thereareinfinitely many
α-walls for some α = (2,c1,c2). Consequently the set (1) can be infinite.
Now we prove Theorem 1.1. We shall use the following three facts.
Fact 2.1. [[8], Lemma 1.1] Let L be a line bundle on X. When a polarization H is α-
generic, the mapE 7→ E⊗L inducesan isomorphism⊗L : M(H,α) → M(H,α(L)),
where α(L) is the Chern class of E ⊗L for a sheaf E of type α.
Fact 2.2. [[4], Lemma 1.5’] If Λ ⊂ Amp(X) is a finite rational cone, then only
finitely many α-walls intersect1 with Λ.
Fact 2.3. [[6], [3]] When K ∼ 0, there is a finite rational cone Λ ⊂ Amp(X) such
X
that Aut(X)·Λ ⊃ Amp(X).
In general c1(τ∗(E)) 6= c1(E) in NS(X) for τ ∈ Aut(X), so τ does NOT induce
an isomorphism between M(H,α) and M(τ∗(H),α). Thus Theorem 1.1 does not
follow Fact 2.2 and Fact 2.3 themselves. For α = (r,c1,c2) ∈ Z>0×NS(X)×Z, put
′ ′
Aut(X,c1) = (cid:8)τ ∈ Aut(X) (cid:12) c1 ≡ τ∗c1 mod rNS(X)(cid:9). Then Aut(X)/Aut(X,c1)
(cid:12) ′
is finite since the map ι : Aut(X)/Aut(X,c1) → NS(X)/rNS(X) defined by
′
ι(τ Aut(X,c1)) = [c1 − τ∗c1] is injective and r > 0. Let τ1,...,τN be represen-
′
tative elements of Aut(X)/Aut(X,c1). By Fact 2.3, there is a finite rational cone
Λ ⊂ Amp(X) such that Aut(X) · Λ ⊃ Amp(X). Let Λ′ ⊂ Amp(X) be the fi-
∗ ∗
nite rational cone spanned by τ (Λ),...,τ (Λ). By Fact 2.2, only finitely many
1 N
′
α-chambers, say D1,...,DM, intersect with Λ. Let Hj be a polarization contained
in D .
j
FINITENESS OF ISOMORPHIC CLASSES 3
Claim 2.4. For any α-generic polarization H, we have some 1 ≤ j ≤ M such that
M(H) ≃ M(H ) as abstract schemes.
j
Proof. Since Aut(X) · Λ ⊃ Amp(X), some τ ∈ Aut(X) satisfies τ∗H ∈ Λ. For
′
some 1 ≤ i ≤ N, [τ] = [τi] in Aut(X)/Aut(X,c1), so one has τ = τi · τ0 with
τ0 ∈ Aut′(X,c1). Then τ∗H ∈ Λ implies that τ0∗H ∈ τi∗Λ ⊂ Λ′. The map E 7→ τ0∗E
deduces an isomorphism
τ0∗ : M(H,α) ≃ M(τ0∗H,(r,τ0∗c1,c2)).
′
Because τ0 ∈ Aut(X,c1), some L ∈ Pic(X) satisfies τ0∗c1 = c1 −rL in NS(X). By
Fact 2.1, the map E 7→ E ⊗L deduces
⊗L : M(τ0∗H,(r,τ0∗c1,c2)) ≃ M((r,τ0∗c1 +rL = c1,c2),τ0∗H).
(One can check that c2(τ0∗E ⊗L) = c2.) Since τ0∗c1 = c1 −rL, one can verify that
τ0∗H is α-generic. At last, M(τ0∗H,α) = M(Hj,α) for some 1 ≤ j ≤ M because
τ0∗H ∈ Λ′. (cid:3)
This claim ends the proof of Theorem 1.1.
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E-mail address: [email protected]
Department of mathematics, Kyoto University, Japan