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Helix theory. (English) Zbl 1072.14020
The aim of this expository paper is to give a wide overview of helix theory, with a special focus on projective spaces and del Pezzo surfaces. The authors first introduce the concepts of exceptional collections, mutations, helices, orthogonal decompositions, braid group action and Serre operator in the context of a linear triangulated category \(\mathcal{D}\). Here, they prove periodicity of a full exceptional collection (i.e. a collection generating the whole \(\mathcal{D}\)).
The relevant case of the derived category \(\mathcal{D}^b(X)\) of coherent sheaves on a smooth projective variety \(X\) is considered. In particular, if \(X=\mathbb{P}^n\), applications such as Beilinson-type spectral sequences and Castelnuovo-Mumford regularity are given. Some open problems are also mentioned.
The paper then takes up the case of \(X\) being a del Pezzo surface, represented as the blow–up of \(\mathbb{P}^2\) at \(d\) points, \(e_i\) being the \(i-\)th exceptional divisor and \(h\) the pull-back of the hyperplane class on \(\mathbb{P}^2\). Here, any exceptional object of \(\mathcal{D}^b(X)\) is in fact a sheaf, up to a shift.
The main result presented is the constructibility of any exceptional collection. This means that, for all \(k\leq d+3\), any collection \((E_1,\ldots,E_k)\) can be included into a full exceptional collection \((E_1,\ldots,E_{d+3})\), which can be reduced by a finite number of mutations to \[ (\mathcal{O}_{e_1}(-1),\ldots,\mathcal{O}_{e_d}(-1), \mathcal{O},\mathcal{O}(h),\mathcal{O}(2\,h)). \] The key ingredients are the notions of stability and the analysis of rigid sheaves.
Finally, the authors underline the connections between helix theory, the solutions of Markoff Diophantine equations, and the theory of rigid local systems over the complement of a finite set of points in \(\mathbb{P}^1\).

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
18F30 Grothendieck groups (category-theoretic aspects)
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results