derived categories of twisted sheaves on calabi-yau manifolds

Remark 6.3.2. This theorem basically tells us that, as long as we do not have
reducible fibers, nothing very interesting happens: the relative moduli space will
only consist <strong>of</strong> some rearrangement <strong>of</strong> the fibers <strong>of</strong> the original space.
Note that the usual way to parametrize line bundles <strong>of</strong> degree zero on an elliptic
curve (or the other singular curves that the theorem deals with) is to associate to
a point Q ∈ C the line bundle O(Q − P ), where P is the origin. Our construction
associates O(P − Q) to Q ∈ C, an entirely isomorphic construction. We chose this
particular one for ease <strong>of</strong> notation later on. (When Q is singular, it is easy to refer
to O(−Q) as IQ, but there is no easy way to refer to its dual.) Note that this
convention is different from the one used in Chapter 4.
From here on we concentrate on understanding what is a corresponding result to
Theorem 6.3.1, for a curve <strong>of</strong> type I2, i.e. a curve which consists <strong>of</strong> two components,
l1 and l2, each isomorphic to P 1 , and meeting transversely at two points. Fix such
a curve C for the rest <strong>of</strong> the section.
On C fix the polarization whose restriction to each component P 1 is O P 1(1).
We’ll study <strong>sheaves</strong> F on C which are semistable (with respect to this polarization)
and have Hilbert polynomial P (F ; t) = 2t (note that this is the Hilbert polynomial
<strong>of</strong> OC by a simple calculation). This corresponds to studying semistable <strong>sheaves</strong>
<strong>of</strong> rank 1 and degree 0, under the correct definition <strong>of</strong> rank (see, for example, [25,
1.2.2]).
This may seem like an arbitrary choice <strong>of</strong> polarization, but in fact the next
lemma shows that the notion <strong>of</strong> (semi)stability does not depend on the choice <strong>of</strong>
polarization, and therefore this particular choice is just a convenience.
Recall the definition <strong>of</strong> the reduced Hilbert polynomial: the (usual) Hilbert
polynomial <strong>of</strong> a sheaf F <strong>of</strong> dimension n (the dimension <strong>of</strong> a sheaf is the dimension
<strong>of</strong> its support) is always <strong>of</strong> the form
P (F ; t) =
n
ait i
with an > 0, and the reduced Hilbert polynomial <strong>of</strong> F is defined to be
i=0
p(F ; t) = 1
P (F ; t).
an
Reduced Hilbert polynomials are compared lexicographically: p(F ; t) ≤ p(G ; t) if
and only if the leading coefficient <strong>of</strong> p(G ; t) − p(F ; t) is non-negative.
Lemma 6.3.3. Consider C polarized by two polarizations; for a coherent sheaf F
on C, write P (F ; t) and p(F ; t) for the usual and the reduced Hilbert polynomials,
respectively, with respect to the first polarization, and P ′ (F ; t) and p ′ (F ; t) with
respect to the second one. Then we have, for any coherent sheaf F on C,
p(F ; t) ≤ t ⇔ p ′ (F ; t) ≤ t,
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