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Determinantal Barlow surfaces and phantom categories. (English) Zbl 1323.14014

The study of structures related to the bounded derived category of coherent sheaves \(D^b(X)\) of a smooth projective variety \(X\) is an interesting point of investigation. The Grothendieck group of \(D^b(X)\) is generated by the objects of this category quotiented by the relation \(X_2 = X_1 \oplus X_3\) for all distinguished triangles \(X_1 \rightarrow X_2 \rightarrow X_3 \rightarrow X_1[1]\). The Hochschild homology is an homological construction that can be applied to subcategories of \(D^b(X)\); it can be useful to retrieve a lot of geometric information and can be exploited in order to get some invariants and to compare the derived categories of different schemes. A geometric phantom category is an admissible subcategory \(\mathcal{A}\) of \(D^b(X)\) with Hochschild homology \(HH_*(\mathcal{A})=0\) and Grothendieck group \(K_0(\mathcal{A})=0\).
This paper gives a deep and explicit descriptions of the derived category of a determinantal Barlow surface (generic in a small deformation neighbourhood of \(S_0\), a fixed determinantal Barlow surface). In particular, the authors describe a semiorthogonal decomposition of the bounded derived category by an exceptional sequence of \(11\) line bundle and a phantom category. The main point of interest is that there are very few examples of phantom categories in the literature and, as pointed out in the introduction of the paper, an interesting point for a further analysis would be to understand if the existence of phantom categories is exclusively a pathology or an interesting and potentially useful structure. It is worth to mention the very precise analysis of the determinantal Barlow surfaces. In particular the explicit computation of a basis of the Picard lattice and of the sections of the line bundles of the semiorthogonal decomposition.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J29 Surfaces of general type
18E30 Derived categories, triangulated categories (MSC2010)

Software:

Macaulay2
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Full Text: DOI arXiv

References:

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