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Two rational nodal quartic 3-folds. (English) Zbl 1370.14013

Let \(\mathbb{W}\) be the 6-dimensional permutation representation of the permutation group \(S_6\). Choose coordinates \(x_0,\dots, x_5\) in \(\mathbb{W}\) so that they are permuted by \(S_6\). Then \({\mathbb{P}}^5={\mathbb{P}}({\mathbb{W}})\) is a projective space with homogeneous coordinates \(x_0,\dots, x_5\). It is assumed that the ground field is the field \(\mathbb{C}\) of complex numbers.
In \({\mathbb{P}}^5\), consider the quartic 3-fold \(X_t\) defined by \(x_0+x_1+x_2+x_3+x_4+x_5=0\) and \[ x_0^4+x_1^4+x_2^4+x_3^4+x_4^4+x_5^4=t(x_0^2+x_1^2+x_2^2+x_3^2+x_4^2+x_5^2)^2, \] where \(t \in \mathbb{C} \).
It is known that all \(X_t\) are singular. In December 2012, Alexei Bondal and Yuri Prokhorov posed the problem:
Determine all \(t \in \mathbb{C} \) such that \(X_t\) is rational.
In 2013, Beauville proved: If \(t\notin \{\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{7}{10}\} \), then \(X_t\) is not rational.
J. A. Todd [Q. J. Math., Oxf. Ser. 7, 168–174 (1936; Zbl 0015.04101)] proved that \(X_{ \frac{1}{2}}\) is rational. \(X_{ \frac{1}{4}}\) is also rational since its projectively dual variety is a Segre cubic.
In this paper, the authors prove that the quartic 3-folds \(X_{ \frac{1}{6}}\) and \(X_{ \frac{7}{10}}\) are rational. The proof is similar to the proof given by Todd in 1935. For the rationality of \(X_{ \frac{1}{6}}\), they construct an explicit \(S_5\)-birational map from \({\mathbb{P}}^3\) to \(X_{ \frac{1}{6}}\) that is a degeneration of J. A. Todd’s construction [Q. J. Math., Oxf. Ser. 6, 129–136 (1935; Zbl 0011.36803)]. For the rationality of \(X_{ \frac{7}{10}}\), they construct an explicit \({\mathcal{U}}_6\)-birational map from \({\mathbb{P}}^3\) to \(X_{ \frac{7}{10}}\) that is a special case of Todd’s construction.

MSC:

14E08 Rationality questions in algebraic geometry
14J30 \(3\)-folds
14E05 Rational and birational maps
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