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1–16 The Lüroth Problem [B1] [B2] [B3] [B4] [B5] [B6] [B7] [BBK] [BCSS] [B-L] [Bl] [Bo] [B-S] [C] [C-G] [Ch] [Co] [C-O] [C-P1] [C-P2] [C-S] 25 A. Beauville, Variétés de Prym et jacobiennes intermédiaires. Ann. Sci. Éc. Norm. Supér. 10, 309–391 (1977) A. Beauville, Les singularités du diviseur de la jacobienne intermédiaire de l’hypersurface cubique dans P4 , in Algebraic Threefolds (Cime, Varenna, 1981). Lecture Notes in Mathematics, vol. 947 (Springer, Berlin/New York, 1982), pp. 190–208 A. Beauville, Variétés rationnelles et unirationnelles, in Algebraic Geometry – Open Problems (Proc.

Another example was found by Tregub [Tre93]: Suppose there is a quartic Veronese surface V ' P2 X meeting P transversally at three points. Then its proper transform VQ XQ is a section of q, giving rationality. To generalize this, we employ a basic property of quadric surfaces due to Springer (cf. 1] and [Swa89]): Let Q P3K be a quadric surface smooth over a field K. L/ ¤ ;. K/ ¤ ; and Q is rational over K via projection from a rational point. This applies when there exists a surface W X intersecting the generic fiber of q transversally in an odd number of points.

Thus the Stein factorization Q 2 / ! S ! X=P yields a degree two K3 surface—the double cover S ! P2 branched over B—and Q 2 / ! S. 11]): • the generic fiber of q is rational over K; • q admits a rational section; • r admits a rational section. The resulting birational map 1 W X Ü P2 P2 blows down the lines incident to the section of q, which are parametrized by a surface birational to S. Cubic fourfolds containing a plane have been re-examined recently from the perspective of twisted K3 surfaces and their derived categories [Kuz10, MS12, Kuz17].

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