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Non-rationality of the \({\mathfrak S}_6\)-symmetric quartic threefolds. (English) Zbl 1327.14220
Summary: We prove that the quartic hypersurfaces defined by \(\sum x_i= t \sum x^4_i-(\sum x^2_i)^2= 0\) in \(\mathbb{P}^5\) are not rational for \(t\neq 0,2,4,6,{10\over 7}\).

MSC:
14M20 Rational and unirational varieties
14E08 Rationality questions in algebraic geometry
14K30 Picard schemes, higher Jacobians
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