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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