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A remark on the Ueno-Campana’s threefold. (English) Zbl 1383.14010
Let \(A\) be an Abelian variety of dimension three, and \(G\) a finite group acting freely on \(A\) in codimension \(2\). In the paper under review, the authors prove that, if there exists a resolution \(X\) of the quotient variety \( A/G\) given by the blow-up of the singular points of \(A/G\) and such that the exceptional divisor at each singular point of \(A/G\) is irreducible, then \(X\) cannot be obtained as the blow-up of a smooth threefold along a smooth centre. In particular, they show that the Ueno-Campana’s threefold cannot be obtained as the blow-up of \(\mathbb{P}^3\), \(\mathbb{P}^2\times\mathbb{P}^1\), or \(\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\) along smooth centres, giving a negative answer to a question raised by Oguiso and Truong.

14J50 Automorphisms of surfaces and higher-dimensional varieties
14E07 Birational automorphisms, Cremona group and generalizations
14E30 Minimal model program (Mori theory, extremal rays)
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