\(n\)-dimensional Fano varieties of wild representation type
A vector bundle \(F\) on a smooth projective variety \((X,\mathcal{O}_X)\) is called ACM if it has no intermediate cohomology, that is if \(\mathrm{H}^i(X,F\otimes\mathcal{O}_X(t))=0\) for all \(t\in\mathbb{Z}\) and \(0<i<\mathrm{dim}(X)\). Varieties supporting only a finite number of indecomposable ACM bundles, up to twist and isomorphism, are called of \textit{finite representation type} and are completely classified. On the opposite side, a variety is said of \textit{wild representation type} if there exist \(r\)-dimensional families of non-isomorphic indecomposable ACM sheaves, for arbitrarily large \(r\). In the paper under review the authors prove that all Fano blow-ups of \(\mathbb{P}^n\) at a finite number of points are of wild representation type. More precisely, for any \(r\geq n\), they construct families of simple ACM rank \(r\) vector bundles whose dimension is of order \(r^2\). In the case \(n=2\), i.e. when \(X\) is a Del Pezzo surface, these bundles are also proved to be \textit{B. Ulrich} [Math. Z. 188, 23--32 (1984; Zbl 0573.13013)], \(\mu\)-semistable and \(\mu\)-stable with respect to suitable polarizations.
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