SINGULARITIES OF MODULI OF STABLE SHEAVESON SOME ELLIPTIC SURFACESKIMIKO YAMADA
Abstract. Let X be some type of elliptic surface X over C with κ(X) = 1,and M (c) the coarse moduli scheme of rank-two stable sheaves with Chern classes(c1 , c2 ) = (0, c) on X. Then M (c) allows only canonical singularities. By using it,
we hope eventually to calculate the Kodaira dimension of M (c).

1. Introduction
Let be O(1) an ample line bundle on a non-singular projective surface X over
C. A torsion-free sheaf E on X is O(1)-stable (resp. semistable) if for any proper
subsheaf F of E one has χ(F (n))/rk(F ) < χ(E(n))/rk(E) (resp. ≤) when n ≫ 0.
There exists the coarse moduli scheme M (c) of O(1)-stable rank-two sheaves withChern classes (c1 , c2 ) = (0, c) ∈ Pic(X) × Z by Gieseker-Maruyama. If c is odd, thenM (c) is projective over C. By Donaldson and Zuo, if c is suﬃciently large w.r.t. Xand O(1), then M (c) is normal, l.c.i., and of dimension ext1 (E, E)0 − ext2 (E, E)0with E ∈ M (c).
In this article, we shall consider the following question, and report the following
theorem.
Question 1.1. (1) How is the birational property of M (c), e.g. its Kodaira dimension κ(M (c))? (2) Does M (c) allow only canonical singularities?
(See Definition 2.1 for definition of terms)

Theorem 1.2. Let X be a minimal elliptic surface over P1 s.t. (i) χ(OX ) =1, (ii) its singular fibers are either rational integral curve with one node (I1 ) or
multiple fiber with smooth reduction (nI0 ), and (iii) X has just two multiple fibers
with multiplicities (2, m), where m is odd and m ≥ 3. In particular, κ(X) is 1. LetO(1) be c-suitable, that is, O(1) is so close to the fiber class f of the elliptic fibration
X → P1 , that O(1) and f is not divided by any c-wall ([1, Def. 2.1]).
Then M (c) admits only canonical singularities.
For some history of Question 1.1, see Section 3. As explained there, this questionis settled mainly in the one case where pg (X) ̸= 0 and one can use generically nondegenerate two-forms, or in the another case where moduli of sheaves is related toThis work was supported by the Grants-in-Aid for Young Scientists (B), JSPS, No. 23740037.
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some more-clariﬁed scheme, e.g. Hilbert scheme of points, but neither case ﬁts thesituation of Theorem 1.2. Expecting to calculate Kodaira dimension of M (c) byusing its original deﬁnition and Theorem 1.2, the author is now trying to estimate
the dimension of pluricanonical maps of M (c), and hopes to report it somewhere
else. We end with notifying that such a plan worked well in the following.
Fact 1.3. [6, Y] Let M be a moduli scheme of stable sheaves with fixed Chern classes
on an Enriques surface or a hyper-elliptic surface. If its expected dimension is notless than 7, then M admits only canonical singularities. Moreover, if M is compact,
then its Kodaira dimension is zero. Also the characteristic of singular points of M
is obtained at [6, Lem. 13(a)].
Notation. For a line bundle L, Exti (E, E ⊗ L)◦ denotes the kernel of trace map
Exti (E, E⊗L) → H i (L). Denote dim Exti (E, F ) and dim Exti (E, E⊗)◦ by exti (E, F )and exti (E, E ⊗ L)◦ respectively.
2. Ideas in the proof of Theorem 1.2
Let us begin with recalling the deﬁnition of some terms.
Definition 2.1. (1) Given any variety V0 , deﬁne its Kodaira dimension κ(V0 ) tobe max{dim ΦmKṼ m ∈ N}, where Ṽ is a desingularization of a completion of V0 .
Kodaira dimension is birational invariant.
(2) A normal variety V is said to admit only canonical singularities when (i) KV is
Q-Cartier, and (ii) if ϕ : Ṽ → V is a desingularization with except divisors Ei , then∑(ai ≥ 0).
KṼ = ϕ∗ KV + i ai Ei
When V does so and V is complete, κ(V ) equals max{dim ΦmKṼ
need not consider its desingularization Ṽ in calculating κ(V ).

m ∈ N}, so weLemma 2.2. Under assumptions in Theorem 1.2, any sheaf E ∈ M satisfies thatext2 (E, E)◦ = hom(E, E(KX ))◦ ≤ 1 .
The next fact results from deformation theory of sheaves and singularities theory.
Fact 2.3 ([6] Lem. 2.5.). Let E be a stable sheaf on a projective surface.
(1) If hom(E, E(KX ))◦ = 0, then moduli M is non-singular at E.
(2) Suppose hom(E, E(KX ))◦ = 1 so Hom(E, E(KX ))◦ = C · f . Then f : E →E(KX ) define a map H 1 (f− ) : Ext1 (E, E) → Ext1 (E, E(KX )) by H 1 (f− )(α) =f ◦ α − α ◦ f . If rkH 1 (f− ) ≥ 3, then M admits only canonical singularity at E.
Thus it’s important to estimate rkH 1 (f− ). Let k(P1 ) denote the function ﬁeldof P1 and k(P1 ) its algebraic closure. We set η = η(P1 ) = Spec(k(P1 )), η̄ =Spec((k(P1 ))), Xη = X ×P1 η, and Xη̄ = X ×P1 η̄. Xη̄ is a nonsingular elliptic

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curve over η̄. Any sheaf F on X induces Fη on Xη , and Fη̄ on Xη̄ . For E ∈ M (c),Eη̄ is degree-zero semi-stable vector bundle on Xη̄ since O(1) is c-suitable, and soAtiyah’s classiﬁcation of vector bundles on an elliptic curve deduces the following.
Lemma 2.4. For E ∈ M (c), one of the following holds:(A) Eη̄ is decomposable, that is, Eη̄ ≃ OXη̄ (F ) ⊕ OXη̄ (−F ) on Xη̄ . Moreover,(A-1) OXη̄ (F ) is not rational over k(P1 ). Let C → P1 be the double cover consisting of nonsingular curves which corresponds to the stabilizer subgroup of OXη̄ (F )in Gal(k(P1 )/k(P1 )). Then OXη̄ (F ) is rational over η ′ = Spec(k(C)).
(A-2) OXη̄ (F ) is rational over k(P1 ).
(B) Eη̄ is indecomposable on Xη̄ .
Let E be a singular point of M (c), and then there exists a traceless homomorphism
f : E → E(KX ) by Fact 2.3 (1). We study E and f with Lemma 2.4 in mind, andget the following.
Proposition 2.5. Under assumptions in Theorem 1.2, any singular point E ∈M (c) satisfies the following: In Lemma 2.4, only Case (A-1) occurs; any tracelesshomomorphism f : E → E(KX ) satisfies det f ̸= 0; the determinant det f ∈ Γ(2KX )induces double covers C → P1 and γ : Y = X ×P1 C → P1 , and decompositions ofγ ∗ E on Y(1)0 −→ F± −→ γ ∗ E −→ G± −→ 0,that extend decompositions of Eη̄ on Xη̄
0 −→ Ker(f ± s) −→ Eη̄ −→ Im(f ± s) −→ 0,where ±s are eigenvalues of fη̄ : Eη̄ → E(KX )η̄ ≃ Eη̄ .
To estimate the rank of H 1 (f− ) : Ext1 (E, E) → Ext1 (E, E(KX )), we look intoRHom(F± , G± ), and so on. Since only Case (A-1) occurs by Proposition 2.5, subsheaves F± ⊂ γ ∗ E do not descend to subsheaves of E. Consequently several cohomology groups coming from RHom(F± , G± ) etc. vanish, and thus we can obtain
good estimation of rkH 1 (f− ) from below. In such a way, we can prove Theorem 1.2.
3. Appendix: History of Question 1.1
Here we note some historical background of Question 1.1; refer to [4, Section 11]
for more. When X is a minimal surface with KX = 0, i.e. a K3 surface or atorus, moduli scheme M of rank-two stable sheaves is of Kodaira dimension zeroby [4, Thm. 11.1.7.]. If X is a minimal surface of general type, the expecteddimension of moduli scheme ext1 (E, E)0 − ext2 (E, E)0 is even, and |KX | contains
a reduced curve, then M is of general type for c2 ≫ 0 by [5]. In these works, they

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utilize generically non-degenerate two-forms, a generalization of symplectic forms
introduced by Mukai. When X is an Enriques surface or hyper-elliptic, see Fact 2.3.
Let X be an elliptic surface. When c1 (E) · f is odd, M is non-singular. If inaddition X has just two multiple ﬁbers, then moduli M is birationally equivalent toSymt (J e+1 (X)) by [2, Thm. 3.14], where c1 (E) · f = 2e + 1. This work uses the factthat E ∈ M is obtained by a sequence of elementary transforms of a special sheaf
V0 s.t. V0 |f is stable for every ﬁber f .
When c1 · f is even and X is an elliptic surface over P1 with just two multiple
ﬁbers (plus some conditions), then M birationally ﬁbers over some projective space
whose ﬁbers are isomorphic to ﬁnite union of Jacobian of some hyperelliptic curves
by [1, Section 7]. They construct this ﬁbration using the spectral cover induced bya stable sheaf (cf. [3, p. 229]). Some upper bound of κ(M ) is obtained there, but
κ(M ) itself is still unknown.
Question 1.1 is unsolved yet in the following cases: (a) X is an Enriques surface,but moduli of stable sheaves is not compact. (b) X is of Kodaira dimension one,but c1 (E) · f is even. (c) X is of general type, but conditions in [5] do not hold;for example, the expected dimension of moduli is odd, or pg (X) = 0. (d) Most ofresults above holds when c2 ≫ 0. How is the case where c2 is not suﬃciently large?
References
1. R. Friedman, Rank two vector bundles over regular elliptic surfaces, Invent. Math. 96 (1989),
no. 2, 283–332.
2.
, Vector bundles and so(3)-invariants for elliptic surfaces, J. Amer. Math. Soc. 8 (1995),no. 1, 29–139.
3.
, Algebraic surfaces and holomorphic vector bundles, Springer-Verlag, New York, 1998.
4. D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Friedr. Vieweg & Sohn,
1997.
5. J. Li, Kodaira dimension of moduli space of vector bundles on surfaces, Invent. Math. 115(1994), no. 1, 1–40.
6. K. Yamada, Singularities and Kodaira dimension of moduli scheme of stable sheaves on Enriquessurfaces, Kyoto J. Math. 53 (2013), no. 1, 145–153.
E-mail address: yamada@xmath.ous.ac.jp
Department of applied mathematics, Faculty of Science, Okayama University ofScience, Okayama, Japan