14 TOM BRIDGELAND
6.1. Excepional collections. Suppose Z is a Fano variety and let X = ωZ
be the total space of the canonical bundle of Z with its projection π : X Z.
Suppose Z has a full exceptional collection (E0, ··· , En−1) such that for all p 0
and all i, j one has
HomZ
k
(Ei, Ej ωZ
p
) = 0 unless k = 0.
Such collections are called geometric and are known to exist in many interesting
examples. It was shown in [14] that there is then an equivalence
D Qcoh(X)

= D Mod(B),
as above, where B is the endomorphism algebra
B = EndX
(
n−1
i=0
π∗Ei
)
.
We give some examples.
Example 6.1. Take Z =
P2
and the geometric exceptional collection
O,O(1),O(2).
Then the algebra B is the path algebra of the quiver

3

3

3
with commuting relations. The numbers labelling the arrows denote the number
of arrows. Commuting relations means that if we label the arrows joining a pair
of vertices by symbols x, y, z then the relations are given by xy yx, yz zy and
xz zx.
Example 6.2. Take Z =
P1
×
P1
and the geometric exceptional collection
O,O(1, 0), O(0, 1), O(1, 1).
This leads to a quiver of the form

2
2

2

2

4
with some easily computed relations. Alternatively, one could take the collection
O,O(1, 0), O(1, 1), O(2, 1),
which gives the quiver

2

2

2

2
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