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Exceptional sequences of line bundles and spherical twists: a toric example. (English) Zbl 1261.14008

Any smooth projective toric surface \(X\) can be obtained from a Hirzebruch surface by finitely many blow-ups, at least when \(X\neq \mathbb{P}^2\). Recall that any toric surface has a distinguished sequence of torus-invariant divisors which sum up to the anticanonical divisor. Abstracting their numerical properties gives the notion of a toric system. Also recall that an object \(E\) in the bounded derived category of coherent sheaves on a smooth projective variety \(Y\) is called exceptional if \(\text{Hom}(E,E)=\mathbb{C}\) and \(\text{Hom}(E,E[k])=0\) for all \(k\neq 0\). A finite collection of exceptional objects is an exceptional sequence if \(\text{Hom}(E_j,E_i[k])=0\) for all \(k\) and \(j>i\) and such a sequence is called full if it generates the whole category. Any exceptional sequence of line bundles on a toric surface gives rise to a toric system which is then called exceptional.
Given a toric system on a toric surface \(X\), there is a procedure called augmentation which produces a toric system on any blow-up of \(X\). In a previous paper the author and N. Ilten gave an example of a toric surface with an exceptional sequence which did not arise from an augmentation. The purpose of the paper under review is to show that this sequence is in fact full. The main idea is to use a spherical twist which maps this sequence to one which is obtained via augmentation and is therefore full.
The paper is organised as follows. In Section 2 the author recalls some basic facts on toric systems, the birational geometry of toric surfaces and exceptional sequences. In Section 3 spherical twists and the induced action on the Picard group are studied. The main result is proved in Section 4.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14J26 Rational and ruled surfaces
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References:

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