## 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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