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Stable rationality of del Pezzo fibrations of low degree over projective spaces. (English) Zbl 1457.14082

Summary: The main aim of this article is to show that a very general three-dimensional del Pezzo fibration of degrees 1, 2, and 3 is not stably rational except for a del Pezzo fibration of degree 3 belonging to explicitly described two families. Higher-dimensional generalizations are also discussed and we prove that a very general del Pezzo fibration of degrees 1, 2, and 3 defined over the projective space is not stably rational provided that the anti-canonical divisor is not ample.

MSC:

14J26 Rational and ruled surfaces
14D06 Fibrations, degenerations in algebraic geometry
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