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Non-rationality of the symmetric sextic Fano threefold. (English) Zbl 1317.14033
Faber, Carel (ed.) et al., Geometry and arithmetic. Based on the conference, Island of Schiermonnikoog, Netherlands, September 2010. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-119-4/hbk). EMS Series of Congress Reports, 57-60 (2012).
Summary: We prove that the symmetric sextic Fano threefold, defined by the equations \(\Sigma x_i = \Sigma x_i^2 = \Sigma x_i^3 = 0\) in \(\mathbb{P}^6\), is not rational. In view of the work of Y. Prokhorov [J. Algebr. Geom. 21, No. 3, 563–600 (2012; Zbl 1257.14011)], our result implies that the alternating group \(A_7\) admits only one embedding into the Cremona group \(\mathrm{Cr}_3\) up to conjugacy.
For the entire collection see [Zbl 1253.00019].

14E08 Rationality questions in algebraic geometry
14J45 Fano varieties
14E07 Birational automorphisms, Cremona group and generalizations
14M20 Rational and unirational varieties
13A50 Actions of groups on commutative rings; invariant theory
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