Prym Varieties: Theory and Applications

Shokurov, V. V.

doi:10.1070/im1984v023n01abeh001459

Cited by 54 publications

(72 citation statements)

References 18 publications

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“…This question has been considered in depth for smooth 3-folds [Iskovskih and Manin 1971;Clemens and Griffiths 1972;Beauville 1977;Tyurin 1980;Sarkisov 1980;Shokurov 1983;Alekseev 1987;Corti 1995;Pukhlikov 1998;Iskovskikh and Prokhorov 1999;Corti 2000]. However, relatively mild singularities can force a 3-fold to be rational.…”

Section: )mentioning

confidence: 99%

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Nonrational nodal quartic threefolds

Cheltsov

2006

Pacific J. Math.

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It is known that the ‫-ޑ‬factoriality of a nodal quartic 3-fold in ‫ސ‬ 4 implies its nonrationality. We prove that a nodal quartic 3-fold with at most 8 nodes is ‫-ޑ‬factorial, while one with 9 nodes is not ‫-ޑ‬factorial if and only if it contains a plane. There are nonrational non-‫-ޑ‬factorial nodal quartic 3-folds. In particular, we prove the nonrationality of a general non-‫-ޑ‬factorial nodal quartic 3-fold that contains either a plane or a smooth del Pezzo surface of degree 4.

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Section: )mentioning

confidence: 99%

“…Then is a reduced divisor with at most simple normal crossings [Beauville 1977;Tyurin 1980;Sarkisov 1980Sarkisov , 1982Shokurov 1983;Corti 2000].…”

Section: The Proof Of Theoremmentioning

confidence: 99%

Nonrational nodal quartic threefolds

Cheltsov

2006

Pacific J. Math.

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“…When deg ∆ = 7, there are 4 deformation families of standard conic bundles and Case 30 corresponds to the generic even theta characteristic case. By [Sho83], the standard conic bundle X is non rational. 2.…”

Section: 22mentioning

confidence: 99%

“…There are some rationality criteria for strict Mori fibre spaces [Sho83,Ale87,Shr07], so that partial answers are known when a small Qfactorialisation of Y is a strict Mori fibre space. Terminal quartic hypersurfaces that contain a plane may in some cases be studied directly.…”

Section: Introductionmentioning

confidence: 99%

A classification of terminal quartic 3-folds and applications to rationality questions

Kaloghiros

2011

Math. Ann.

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Abstract. This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not Q-factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Qfactorialisation of Y . In this case, the generators of Cl Y / Pic Y are "topological traces " of K-negative extremal contractions on X. One can show, as an application of these methods, that a number of families of non-factorial terminal Gorenstein Fano 3-folds are rational. In particular, I give some examples of rational quartic hypersurfaces Y 4 ⊂ P 4 with rk Cl Y = 2 and show that when rk Cl Y ≥ 6, Y is always rational.

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“…Until now despite all the efforts it has remained unsolved. Analogs of quite a few geometrical characterizations of the Jacobians for the case of Prym varieties are either unproved or known to be invalid (see reviews [1,2] and references therein).…”

Section: Introductionmentioning

confidence: 99%