Rationality of Fano Threefolds with Terminal Gorenstein Singularities. I

Prokhorov, Yuri

doi:10.1134/s0081543819060130

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2020

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Cited by 5 publications

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“…Many partial classification results of this sort can be found in the literature, see, e.g., [JR06], [JPR11], [CM13], [Pro16], [Pro17], [Pro19], [Pro22], [Tak22], [CKGS23]. However, none of these papers contains a full and detailed classification, with an accurate description of all geometric aspects.…”

Section: Introductionmentioning

confidence: 99%

Rationality Over Nonclosed Fields of Fano Threefolds With Higher Geometric Picard Rank

Kuznetsov¹,

Prokhorov

2022

J. Inst. Math. Jussieu

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We prove rationality criteria over nonclosed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$ . For the last type of such threefolds, we provide a unirationality criterion and construct examples of unirational but not stably rational varieties of this type.

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Section: Introductionmentioning

confidence: 99%

Rationality Over Nonclosed Fields of Fano Threefolds With Higher Geometric Picard Rank

Kuznetsov¹,

Prokhorov

2022

J. Inst. Math. Jussieu

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On the rationality of certain Fano threefolds

Ciliberto

2023

manuscripta math.

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In this paper we study the rationality problem for Fano threefolds $$X\subset {\mathbb P}^{p+1}$$ X ⊂ P p + 1 of genus p, that are Gorenstein, with at most canonical singularities. The main results are: (1) a trigonal Fano threefold of genus p is rational as soon as $$p\geqslant 8$$ p ⩾ 8 (this result has already been obtained in Przyjalkowski et al. (Izv Math 69(2):365–421, 2005), but we give here an independent proof); (2) a non-trigonal Fano threefold of genus $$p\geqslant 7$$ p ⩾ 7 containing a plane is rational; (3) any Fano threefold of genus $$p\geqslant 17$$ p ⩾ 17 is rational; (4) a Fano threefold of genus $$p\geqslant 12$$ p ⩾ 12 containing an ordinary line $$\ell $$ ℓ in its smooth locus is rational.

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