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Quadric surface bundles over surfaces and stable rationality. (English) Zbl 1397.14026
A variety $$X$$ is called stably rational if and only if $$X\times \mathbb{P}^r$$ is birational to $$\mathbb{P}^n$$. The property of being rational or stably rational is not deformation invariant in general. In fact, Hassett, Pirutka and Tschinkel have constructed smooth families $$X \rightarrow B$$ of complex projective 4-folds such that the general fiber is not stably rational but some special fibers are. The method used to construct these families uses the specialization method initially developed by C. Voisin [Invent. Math. 201, No. 1, 207–237 (2015; Zbl 1327.14223)] and later improved by J.-L. Colliot-Thélène and A. Pirutka [Ann. Sci. Éc. Norm. Supér. (4) 49, No. 2, 371–397 (2016; Zbl 1371.14028)].
In this paper, the author proves a simplified version of the specialization method in the case of quadric surface bundles. By using this method the author is able to construct stably rational and non stably rational quadric bundles over $$\mathbb{P}^2$$.

##### MSC:
 14E08 Rationality questions in algebraic geometry 14M20 Rational and unirational varieties 14J35 $$4$$-folds 14D06 Fibrations, degenerations in algebraic geometry
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