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Exceptional bundles associated to degenerations of surfaces. (English) Zbl 1282.14074

Given positive integers \(a,n\) with \(a<n\) and \((a,n)=1\), a Wahl singularity of type \(\frac{1}{n^2}(1,na-1)\) is the cyclic quotient singularity \(0\in \mathbb{A}^2/(\mathbb{Z}/n^2\mathbb{Z})\), where the generator \(1\) of \(\mathbb{Z}/n^2\mathbb{Z}\) acts by sending \((u,v)\) to \((\xi u,\xi^{na-1}v)\) and where \(\xi\) is a primitive \(n^2\)-th root of unity. Any such singularity admits a so-called \(\mathbb{Q}\)-Gorenstein smoothing, that is, a \(1\)-parameter deformation such that the general fibre is smooth and the canonical divisor of the total space is \(\mathbb{Q}\)-Cartier. The main result of the paper under review states, roughly speaking, that if \(X\) is a projective normal surface with a unique singularity \(P\in X\) of Wahl type \(\frac{1}{n^2}(1,na-1)\) and \(\mathcal{X}\) is a \(\mathbb{Q}\)-Gorenstein smoothing over a base \(T\), then under some technical conditions there exists a reflexive sheaf \(\mathcal{E}\) on the total space of the deformation restricting to an exceptional bundle \(F\) of rank \(n\) on a general fibre \(Y\). Furthermore, given a line bundle \(\mathcal{H}\) which is ample on the fibres of the deformation, \(F\) is slope stable with respect to \(\mathcal{H}_{|Y}\) and its Chern classes can be computed.
The rough idea of the proof of the main result is as follows. Consider the above \(1\)-parameter deformation \(\mathcal{X}\). After a base change there exists a proper birational morphism \(\pi: \widetilde{\mathcal{X}}\to \mathcal{X}\) whose exceptional locus is a normal surface \(W\) mapping to the singularity. This surface is an explicit weighted projective hypersurface determined by \(a\) and \(n\) and it is possible to construct an exceptional bundle \(G\) on \(W\) by degenerating it to a surface with a Wahl singularity of type \(\frac{1}{a^2}(1,ab-1)\) (where \(b=n\) mod \(a\)). On the other hand, the special fibre \(\widetilde{X}\) of \(\widetilde{\mathcal{X}}\) is a reducible surface which is a union of \(W\) and \(X'\), the strict transform of \(X\), intersecting along a smooth rational curve. Under the above technical assumptions there is a certain bundle on \(X'\) which can be glued to \(G\) to obtain a bundle \(E\) on \(\widetilde{X}\). The latter extends to a locally free sheaf on the total space.
The paper is organised as follows. In Section 2 the author collects some facts concerning Wahl singularities and their \(\mathbb{Q}\)-Gorenstein deformations. Section 3 presents a blowup construction showing the existence of \(\pi\) and describing the geometry, in particular the special fibre of \(\widetilde{\mathcal{X}}\), explicitly. The next section deals with the gluing procedure mentioned above while Section 5 establishes the existence of the bundle \(G\) on \(W\). In Section 6 the author considers the projective plane as an example and shows, roughly speaking, that any exceptional bundle on \(\mathbb{P}^2\) of rank greater than \(1\) comes from a deformation of a Wahl singularity and vice versa. Lastly, in the last section some facts on reflexive sheaves, toric geometry and other topics used in the paper are collected.

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14J10 Families, moduli, classification: algebraic theory
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References:

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