Nonrational nodal quartic threefolds

Cheltsov, Ivan

doi:10.2140/pjm.2006.226.65

Cited by 39 publications

(67 citation statements)

References 71 publications

(152 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Definition 3.20 A normal Gorenstein variety X has terminal (respectively canonical) singularities if for a given resolution of singularities f : Y → X all the coefficients a i ∈ Z in (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) are positive (respectively non-negative).…”

Section: Terminal and Canonical 3-fold Gorenstein Singularitiesmentioning

confidence: 99%

“…Thus Y ⊂ G is an ample divisor; it then follows easily from the Lefschetz theorems that the restriction H 2 (G) → H 2 (Y) is an isomorphism and H 3 (Y) is torsion-free. To see this consider the long exact cohomology sequence of the pair G, Y : Example 7.4 (Small resolution of a generic quartic containing a quadric surface) See also Cheltsov [13,Example 10]. Fix a quadric surface Q = Q 2 2 ⊂ P 4 and let Q ⊂ X ⊂ P 4 be a general quartic 3-fold containing Q.…”

Section: Example 72mentioning

confidence: 99%

“…Table 8.1 lists the possible number of nodes e, the defect σ , the number of projective small resolutions s, the number of planes P contained in the nodal cubic and b 3 (Y) denotes the third Betti number of any projective small resolution of the nodal cubic (if any exists); the latter is computed using (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) and the fact that b 3 = 10 for a non-singular cubic 3-fold.…”

Section: Weak Del Pezzo 3-folds From Nodal Cubicsmentioning

confidence: 99%

“…(iv) (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) and non-negativity of b 3 (Y) immediately implies e − σ ≤ 5; Table 8.1 shows that there are 5 different possible combinations of e and σ realising e − σ = 5 (which forces b 3 (Y) = 0).…”

mentioning

confidence: 99%

See 3 more Smart Citations

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds

et al. 2013

View full text Add to dashboard Cite

We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular attention to a subclass of weak Fano 3-folds that we call semi-Fano 3-folds. Semi-Fano 3-folds satisfy stronger cohomology vanishing theorems and enjoy certain topological properties not satisfied by general weak Fano 3-folds, but are far more numerous than genuine Fano 3-folds. Also, unlike Fanos they often contain P 1 s with normal bundle O(−1) ⊕ O(−1), giving rise to compact rigid holomorphic curves in the associated ACyl Calabi-Yau 3-folds.We introduce some general methods to compute the basic topological invariants of ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds, and study a small number of representative examples in detail. Similar methods allow the computation of the topology in many other examples.All the features of the ACyl Calabi-Yau 3-folds studied here find application in [17] where we construct many new compact G 2 -manifolds using Kovalev's twisted connected sum construction. ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds are particularly well-adapted for this purpose.

show abstract

Section: Terminal and Canonical 3-fold Gorenstein Singularitiesmentioning

confidence: 99%

Section: Example 72mentioning

confidence: 99%

Section: Weak Del Pezzo 3-folds From Nodal Cubicsmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds

et al. 2013

View full text Add to dashboard Cite

show abstract

“…The group SL 5 acts naturally on P 4 and this Grassmannian, linearized via the Plücker embedding. Consider the loci of stable and semistable points U s ⊂ U ss ⊂ Gr (2 Γ(O P 4 (2))) and the resulting GIT quotient M = Gr (2 Γ(O P 4 (2)))/ /SL 5 Principal results of [27] include:…”

Section: Basic Properties Of Quartic Del Pezzo Surfacesmentioning

confidence: 99%

Quartic del Pezzo surfaces over function fields of curves

Hassett

Tschinkel²

2013

Open Mathematics

View full text Add to dashboard Cite

We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.

show abstract

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

Kaloghiros

2011

Math. Ann.

View full text Add to dashboard Cite

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.

show abstract

Nonrational nodal quartic threefolds

Cited by 39 publications

References 71 publications

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds

Quartic del Pezzo surfaces over function fields of curves

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

Contact Info

Product

Resources

About