Factoriality Of Certain Hypersurfaces Of P4 With Ordinary Double Points

Ciliberto, Ciro; Gennaro, Vincenzo Di

doi:10.1007/978-3-662-05652-3_1

Cited by 21 publications

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“…We note that an analogue to Conjecture 13 for smooth surfaces on a nodal hypersurface in ‫ސ‬ 4 is proved in [Ciliberto and Di Gennaro 2004]. It is easy to see that Conjecture 13 holds for quadrics and cubics [Finkelnberg and Werner 1989].…”

Section: )mentioning

confidence: 84%

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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: 84%

“…The same method can be applied to any nodal hypersurface in ‫ސ‬ 4 . The following result is implied by Theorem 24 (see [Werner 1987;Dimca 1990;Ciliberto and Di Gennaro 2004]). …”

Section: Proposition 16 Letmentioning

confidence: 99%

Nonrational nodal quartic threefolds

Cheltsov

2006

Pacific J. Math.

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“…First of all as in [1], by the hypotheses, V is projectively normal. Hence if X ⊂ V is a smooth projective surface, to prove that X is a complete intersection, it is sufficient to prove that the surface residual to X in the complete intersection of V with a hypersurface of degree 0 is a complete intersection.…”

Section: Pietro Sabatinomentioning

confidence: 99%

Some remarks on factoriality of certain hypersurfaces in $\mathbb{P}^4 $

Sabatino

2005

Arch. Math.

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Let V be a reduced and irreducible hypersurface of degree k 3. In this paper we prove that if the singular locus of V consists of δ 2 ordinary double points, δ 3 ordinary triple points and if δ 2 + 4δ 3 < (k − 1) 2 , then any smooth surface contained in V is a complete intersection on V .Refining an argument of Severi (see [8]), Ciliberto and Di Gennaro prove in [1] that any surface contained in a hypersurface of P 4 with few ordinary double points is a complete intersection.In what follows we will discuss the possibility of extending the argument in [1] when the singularities involved are not only ordinary double points. More precisely we want to prove the following statement: Theorem 1. Let V ⊂ P 4 be a reduced and irreducible hypersurface of degree k 3. If the singular locus of V consists of δ 2 ordinary double points, δ 3 ordinary triple points and if δ 2 + 4δ 3 < (k − 1) 2 , then any smooth projective surface contained in V is a complete intersection on V .If the singularities involved are of order greater then three we obtain a weaker result: Proposition 2. Let V ⊂ P 4 be a reduced and irreducible hypersurface of degree k 3. Suppose that the singular locus of V consists of δ ordinary points of order at mostr and let δ r be the number of singular points of order r, 2 r r. If r r=2 (r − 1) 2 δ r < (k − 1) 2

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“…On the other hand, the following result is proved in [8]. Then V is nodal and contains the plane x = y = 0.…”

Section: Introductionmentioning

confidence: 95%

On factoriality of nodal threefolds

Cheltsov

2005

J. Algebraic Geom.

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We prove the Q-factoriality of a nodal hypersurface in P 4 of degree n with at most (n−1) 2 4 nodes and the Q-factoriality of a double cover of P 3 branched over a nodal surface of degree 2r with at most (2r−1)r 3 nodes. for fruitful conversations. Special thanks goes to V. Kulikov for Lemma 38. The author would also like to thank the referee for useful comments.All varieties are assumed to be projective, normal, and defined over C. 1 A 3-fold is called nodal if all its singular points are ordinary double points. 2 A variety is called Q-factorial if a multiple of every Weil divisor on the variety is a Cartier divisor.

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Factoriality Of Certain Hypersurfaces Of P4 With Ordinary Double Points

Cited by 21 publications

References 6 publications

Nonrational nodal quartic threefolds

Nonrational nodal quartic threefolds

Some remarks on factoriality of certain hypersurfaces in $\mathbb{P}^4 $

On factoriality of nodal threefolds

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