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the same singular fibers, and their discriminant loci are the same. The Jacobian <strong>of</strong>
Y can be identified with J (Proposition 4.5.2), and therefore we get an isomorphism
D b coh(Y ) ∼ = D b coh( ¯ J, ¯ β),
where ¯ β is the unique element <strong>of</strong> Br( ¯ J) that extends β, the element <strong>of</strong> Br(J s ) that
corresponds to Y → S. But Theorem 4.5.2 and Remark 6.5.2 tell us that β = α k ,
and thus ¯ β = ¯α k . Therefore, we have
which is what we wanted.
D b coh( ¯ J, ¯α) ∼ = D b coh(X) ∼ = D b coh(Y ) ∼ = D b coh( ¯ J, ¯α k ),
Pro<strong>of</strong>. Choose a relatively ample sheaf OX(1) such that the degree <strong>of</strong> its restriction
to any fiber <strong>of</strong> X → S is n. (The fact that this can be done is a well known fact
in the theory <strong>of</strong> elliptic fibrations. See, for example, [13, Section 1], where n is
denoted by δS.) We’ll consider M under this particular polarization.
The important result (which we prove later) is that if C is an I2 curve, polarized
by a polarization <strong>of</strong> degree n, then there is a family on C × C, flat over the second
component, whose fibers are stable <strong>sheaves</strong> <strong>of</strong> rank 1, degree k on C, and thus C
is naturally embedded in the moduli space MC(1, k) <strong>of</strong> semistable <strong>sheaves</strong> <strong>of</strong> rank
1, degree k on C. Therefore, there is no contraction <strong>of</strong> one component, as was the
case when k = 0.
The whole analysis in Sections 6.3, 6.4, and 6.5 carries through without any
significant modification, except for the extra simplification caused by the fact that
there are no contractions on the I2 fibers. Thus X and Y are locally isomorphic
over S, and hence Y is smooth.
The standard technique <strong>of</strong> Mukai ([32, Appendix A]) can then be used to prove
that the moduli problem is fine in this case. Another way to see this fact is the
following: the obstruction to the existence <strong>of</strong> a universal sheaf is an element γ <strong>of</strong>
Br(Y ) (since there are no properly semistable <strong>sheaves</strong>). Let U = S − ∆, where ∆
is the discriminant locus <strong>of</strong> X → S (or Y → S). There is a natural restriction map
Br ′ (Y ) → Br ′ (YU), which is injective by the results in [18]. (Using the standard
purity statement [18, III, 6.2], we can remove the singular locus <strong>of</strong> ∆ from the
picture. Then we have the situation <strong>of</strong> removing a smooth divisor from a smooth
scheme, for which we get injectivity by a corrected version <strong>of</strong> [18, III, 6.2].) But
the results in Section 4.5 show that the restriction γ|YU is zero, and thus γ = 0.
Hence the moduli problem is fine.
A universal sheaf then exists, and it induces an integral transform for which
the usual criterion for being an equivalence applies, giving us
D b coh(X) ∼ = D b coh(Y ).