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Representation type of surfaces in \(\mathbb{P}^3\). (English) Zbl 1456.14021

A possible way to measure the complexity of a given \(n\)-dimensional polarized variety \((X, \mathcal{O}_X (1))\) is to ask for the families of non-isomorphic indecomposable aCM (arithmetically Cohen-Macaulay) vector bundles that it supports (recall that a vector bundle \(\mathcal{E}\) on \(X\) is aCM if \(H^i(X,\mathcal{E}\otimes\mathcal{O}_X(t))= 0\) for all \(t\in\mathbb{Z}\) and \(i=1,\dots, n-1\)). The first result on this direction was Horrocks’ theorem which states that on the projective space the only indecomposable aCM bundle up to twist is the structure sheaf \(\mathcal{O}_{\mathbb{P}^n}\).
Inspired by analogous classifications in quiver theory and representation theory, a classification of polarized varieties as finite, tame and wild was proposed. ACM varieties of finite type (namely, supporting only a finite number of non-isomorphic indecomposable aCM vector bundles) were completely classified in [ D. Eisenbud and J. Herzog, Math. Ann. 280, No. 2, 347–352 (1988; Zbl 0616.13011]. If we look at the other extreme of complexity we would find the varieties of wild representation type, namely, varieties for which there exist \(r\)-dimensional families of non-isomorphic indecomposable aCM bundles for arbitrary large \(r\). Recently, the representation type of any reduce aCM polarized variety has been determined [D. Faenzi and J. Pons-Llopis, “The Cohen-Macaulay representation type of arithmetically Cohen-Macaulay varieties”, Preprint, arXiv:1504.03819].
In the article under review, the authors prove that every surface \(X\) with a regular point in the three-dimensional projective space of degree at least four is of wild representation type under the condition that either \(X\) is integral or Pic\((X)\) is \(\mathbb{Z}\)-generated by \(\mathcal{O}_X(1)\). Alongside, they also prove the interesting result that every non-integral aCM variety of dimension at least two is also very wild: namely there exist arbitrarily large dimensional families of pairwise non-isomorphic aCM non-locally free sheaves of rank one.

MSC:

14F06 Sheaves in algebraic geometry
13C14 Cohen-Macaulay modules
16G60 Representation type (finite, tame, wild, etc.) of associative algebras

Citations:

Zbl 0616.13011
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Full Text: DOI arXiv Euclid

References:

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