Russo, Francesco; Staglianò, Giovanni Trisecant flops, their associated \(K3\) surfaces and the rationality of some cubic fourfolds. (English) Zbl 1521.14032 J. Eur. Math. Soc. (JEMS) 25, No. 6, 2435-2482 (2023). A very general cubic fourfold is conjectured to be irrational and the locus of rational ones, also conjecturally, should be the union of certain irreducible divisors \(\mathcal{C}_d\) (in their moduli \(\mathcal{C}\)), of special admissible cubic fourfolds of discriminant \(d\); their rationality relies on the existence of certain \(K3\) surface (further references on this Kuznetsov Conjecture can be found in the Introduction of the paper under review). In this paper, the authors take the point of view of Mori Theory to describe the cases \(d=14, 26, 38\) and \(42\), in fact the first four admissible values of \(d\) (cubics fourfolds in \(\mathcal{C}_d\), \(d=14,26,38\), were known to be rational); and furthermore to prove (see Theorem 5.12) the rationality of every cubic fourfold in \(\mathcal{C}_{42}\), the first not known case. The birational maps from \(X\) to a rational fourfold \(W\) are displayed in a, say, Mori Theory diagram (see (0.1)) in such a way that the role of the \(K3\) surface is very explicit: a non-minimal birational model in \(W\) can be constructed via some very peculiar linear systems of hyperplane sections. Reviewer: Roberto Muñoz (Madrid) Cited in 2 Documents MSC: 14E08 Rationality questions in algebraic geometry 14M20 Rational and unirational varieties 14M07 Low codimension problems in algebraic geometry 14N05 Projective techniques in algebraic geometry 14J28 \(K3\) surfaces and Enriques surfaces 14J70 Hypersurfaces and algebraic geometry Keywords:rationality of cubic fourfolds; flops; Mori theory Software:Macaulay2; GitHub; SpecialFanoFourfolds; Cremona; KoszulDivisorOnPic14M8 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Alzati, A., Russo, F.: Some extremal contractions between smooth varieties arising from pro-jective geometry. Proc. London Math. Soc. (3) 89, 25-53 (2004) Zbl 1063.14014 MR 2063658 · Zbl 1063.14014 [2] Ando, T.: On extremal rays of the higher-dimensional varieties. Invent. Math. 81, 347-357 (1985) Zbl 0554.14001 MR 799271 · Zbl 0554.14001 [3] Artin, M.: Algebraization of formal moduli. II. Existence of modifications. Ann. of Math. 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