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**Computing the walls associated to Bridgeland stability conditions on projective surfaces.**
*(English)*
Zbl 1307.14022

The paper under this review is on the wall-chamber structure of the space of Bridgeland stability conditions for general projective surfaces.

By the work of Bridgeland for \(K3\) and abelian surfaces, explicit construction of stability conditions is known, and various properties of the space of stability conditions have been studied. The wall and chamber structure is the one of those properties. A wall corresponds to a destabilizing object, and it is useful when one studies the birational nature of the moduli spaces of stable objects.

To make the explanation more precise, let us recall the so-called large volume limit. The stability conditions given by Bridgeland’s construction are parametrized by a Neron-Severi class and an ample divisor, When one choses the ample divisor very large, then the corresponding stability condition is equivalent to the Gieseker stability, and it often occurs that one already knows the property of the moduli space of stable objects. Then one can study the moduli space for another stability conditions by the wall-crossing analysis or by the application of Fourier-Mukai transforms.

A generalization of Bridgeland’s construction for arbitrary smooth projective surfaces are given by the work of D. Arcara and A. Bertram [J. Eur. Math. Soc. (JEMS) 15, No. 1, 1–38 (2013; Zbl 1259.14014)]. Using this generalization, the author obtains several general results on the structure of walls on the space of stability conditions.

One of the impressive features is the nested nature of the walls. Theorem 3.1, which the author calls Bertram’s Nested Wall Theorem, claims that for an arbitrary projective surface and a given stability condition, there is a plane in the space of stability condition such the radii of the walls on that plane are nested. Such phenomena had been already known under some specific conditions for abelian or \(K3\) surfaces, and the result in this paper gives a transparent view for general situations.

Another important result is on the bound of the radii of walls, given in Theorem 3.12. These theorems are shown using the several basic observations and calculations given in the section 2. Presentation of those technical parts often make it difficult to read the paper, but the author succeeded in giving brief and accurate accounts. I think this paper helps very much those studying the moduli spaces of Bridgeland stable objects for general projective surfaces using the wall-chamber structures.

By the work of Bridgeland for \(K3\) and abelian surfaces, explicit construction of stability conditions is known, and various properties of the space of stability conditions have been studied. The wall and chamber structure is the one of those properties. A wall corresponds to a destabilizing object, and it is useful when one studies the birational nature of the moduli spaces of stable objects.

To make the explanation more precise, let us recall the so-called large volume limit. The stability conditions given by Bridgeland’s construction are parametrized by a Neron-Severi class and an ample divisor, When one choses the ample divisor very large, then the corresponding stability condition is equivalent to the Gieseker stability, and it often occurs that one already knows the property of the moduli space of stable objects. Then one can study the moduli space for another stability conditions by the wall-crossing analysis or by the application of Fourier-Mukai transforms.

A generalization of Bridgeland’s construction for arbitrary smooth projective surfaces are given by the work of D. Arcara and A. Bertram [J. Eur. Math. Soc. (JEMS) 15, No. 1, 1–38 (2013; Zbl 1259.14014)]. Using this generalization, the author obtains several general results on the structure of walls on the space of stability conditions.

One of the impressive features is the nested nature of the walls. Theorem 3.1, which the author calls Bertram’s Nested Wall Theorem, claims that for an arbitrary projective surface and a given stability condition, there is a plane in the space of stability condition such the radii of the walls on that plane are nested. Such phenomena had been already known under some specific conditions for abelian or \(K3\) surfaces, and the result in this paper gives a transparent view for general situations.

Another important result is on the bound of the radii of walls, given in Theorem 3.12. These theorems are shown using the several basic observations and calculations given in the section 2. Presentation of those technical parts often make it difficult to read the paper, but the author succeeded in giving brief and accurate accounts. I think this paper helps very much those studying the moduli spaces of Bridgeland stable objects for general projective surfaces using the wall-chamber structures.

Reviewer: Shintaro Yanagida (Kyoto)

### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |

18E30 | Derived categories, triangulated categories (MSC2010) |

18E40 | Torsion theories, radicals |