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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