Abstract
We study the unirationality property of an algebraic variety X (over \(\mathbb {C}\)) versus the so-called stable birational infinite transitivity of X. We show that in the case when X is a smooth quartic hypersurface, these two notions do not coincide.
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Notes
In fact, we show that S is acted by “ sufficiently many ” additive groups \(({{\textbf {k}}}_p,+)\), which is enough for our argument.
It is essential here that S is a \(\textrm{K3}\) surface (and not a CM elliptic curve for instance).
This projection depends a priori on D.
Here, one possibly needs to replace U with a smaller affine subset. Note however that D remains nilpotent due to \(D^2g = 0\).
Take for instance \(X_1\) to be a hyperplane section of (quasi-projective and R-flat) U.
This makes our reduction mod p considerations essential as compared to the case of \({{\textbf {k}}}= \mathbb {C}((t))\).
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Acknowledgements
I am grateful to F. Bogomolov, S. Galkin, and D. Huybrechts for their interest to my paper and for fruitful conversations. Some parts of the paper were prepared during my stay at the Oxford University (July 2014) and CIRM, Universit‘a di Trento (August 2014), whose hospitality I happily acknowledge. The work was supported by World Premier International Research Initiative (WPI), MEXT, Japan, Grant-in-Aid for Scientific Research (26887009) from Japan Mathematical Society (Kakenhi), and by the State assignment project FSMG-202-0013.
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