On the rationality of certain Fano threefolds

Pub Date : 2023-11-06 DOI:10.1007/s00229-023-01514-2

Ciro Ciliberto

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Abstract

Abstract 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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论法诺的某些合理性

摘要本文研究了具有至多正则奇点的格伦斯坦(Gorenstein)属P的Fano三倍$$X\subset {\mathbb P}^{p+1}$$ X∧P P + 1的合理性问题。主要结果是:(1)当$$p\geqslant 8$$ p小于8时，p属的三角形Fano三倍是有理的(这个结果已经在Przyjalkowski等人中获得(Izv Math 69(2): 365-421, 2005)，但我们在这里给出了一个独立的证明);(2)含有平面的$$p\geqslant 7$$ p小于7属的非三角形Fano三倍是有理的;(3)属$$p\geqslant 17$$ p大于或等于17的任何Fano三倍是合理的;(4)属$$p\geqslant 12$$ p小于12的Fano三倍在其平滑位点中包含一条普通线$$\ell $$ r是合理的。

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