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On cyclic strong exceptional collections of line bundles on surfaces. (English) Zbl 1478.14032

This article studies exceptional collections of line bundles on (smooth projective) surfaces. For a variety \(X\) over a field \(k\), objects \(E_1,\ldots,E_m\) in \(\mathcal D^b(X)\) form an exceptinal collection if \(\mathrm{RHom}(E_i,E_i) = k\) for \(1\leq i\leq m\) and \(\mathrm{RHom}(E_j,E_i)=0\) for \(1\leq i<j\leq m\). One is interested in exceptional collections with further properties:
maximal length: if \(m\) equals the rank of \(K_0(X)_{\mathrm{num}}\);
full: if \(E_1,\ldots,E_m\) generate \(\mathcal D^b(X)\);
strong: if \(\mathrm{RHom}(E_i,E_j) = \mathrm{Hom}(E_i,E_j)\) for \(1 \leq i,j \leq m\);
cyclic strong: if \(E_l\otimes \omega_X,\ldots,E_m\otimes \omega_X, E_1,\ldots,E_{l-1}\) is a strong exceptional collection for all \(1 \leq l \leq m\).
It is conjectured that the existence of a full exceptional collection of line bundles is related to the rationality of \(X\), and moreover, that in this case any such collection of maximal length is automatically full. But such questions are delicate, and history showed that even in the (easiest) case of toric surfaces, many conjectures about such collections turned out to be wrong.
On the positive side, we have the existence of full exceptional collections of line bundles on rational surfaces and a criterion when strong ones exist, see [L. Hille and M. Perling, Compos. Math. 147, No. 4, 1230–1280 (2011; Zbl 1237.14043)]. Even more, the existence of a full strong exceptional collection of line bundles on a surface, implies that it is a rational surface, see M. Brown and I. Shipman, Mich. Math. J. 66, No. 4, 785–811 (2017; Zbl 1429.14013)].
The first main result of this article is, that the existence of a cyclic strong exceptional collection \(\mathcal L_1,\ldots,\mathcal L_m\) of line bundles on a surface \(X\) implies that \(X\) is weak del Pezzo. The authors give an explicit list which weak del Pezzo surfaces admit such a collection. In this case \(\bigoplus_i \mathcal L_i\) is a (particularly nice) generator of \(\mathcal D^b(X)\). The authors show that any object in \(\mathcal D^b(X)\) can be built out of its summands by taking cones at most two times. This implies that for these weak del Pezzo surfaces, the Rouqier dimension of \(\mathcal D^b(X)\) is two.
Another big part of the article is the question how (full) exceptional collections of line bundles can be constructed. A mutated version of Orlov’s blowup formula gives a procedure called standard augmentation, see [L. Hille and M. Perling, Compos. Math. 147, No. 4, 1230–1280 (2011; Zbl 1237.14043)], which starting with a full exceptional collection of line bundles on a surface \(X\) gives such collections on a blowup \(\tilde X\). The authors show that if an exceptional collection of line bundles on a rational surface is numerically cyclic strong, then it comes from a standard augmentation. If one drops the (numerical) cyclicity then this does not hold any more. In fact, the authors give an example of full strong exceptional collection of line bundles on a weak del Pezzo surface of degree two, which is not a standard augmentation.

MSC:

14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14J26 Rational and ruled surfaces
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