Terminal Fano threefolds and their smoothings

Jahnke, Priska; Radloff, Ivo

doi:10.1007/s00209-010-0780-8

Cited by 31 publications

(32 citation statements)

References 13 publications

(3 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Further, it follows from the proof of Theorem 1.4 in [18] that for every t ∈ ∆ there is a group isomorphism ϕ : Pic(X) ≃ Pic(X t ) such that ϕ(K X ) = K Xt . Let us now assume that (−K X ) 3 = 64.…”

Section: Some Auxiliary Results About Extremal Raysmentioning

confidence: 98%

Fano threefolds with canonical Gorenstein singularities and big degree

Karzhemanov

2014

Math. Ann.

View full text Add to dashboard Cite

We provide a complete classification of Fano threefolds X having canonical Gorenstein singularities and anticanonical degree (−K X ) 3 = 64.Lemma 5.9. Let Z ⊂ Y ′ be an irreducible rational curve such that dim |Z| > 0 and E Z ≃ O P 1 (d 1 ) ⊕ O P 1 (d 2 ) for some d 1 , d 2 ∈ Z. Then |d 1 − d 2 | 2 + (Z 2 ).Proof. This follows from exactly the same arguments as in the proof of [30, Lemma 10.6] (recall that the linear system | − nK Y | is basepoint-free for n large).Hence we may assume that both c i are integers. We have two cases:(1) c 1 is even. Then we may assume that c 1 = 0 up to twisting E by a line bundle (cf. [8]). In this case, from (5.7) we obtain the equalitywhich implies that c 2 = −5/4 ∈ Z, a contradiction. 9(2) c 1 is odd. Then, as above, we may assume that c 1 = −K Y ′ . In this case, from (5.5) we get the equalityIdentify D with a generic element in |D|. This is a weak del Pezzo surface with at worst Du Val singularities (see [35]). Then from the adjunction formula we obtainand since (−K Y ) 3 = 64, we get K 2 D = 8. Now, since Y ′ ≃ P 2 , one easily gets that either D ≃ P 2 or F 1 (see e.g. [2, Theorem 8.1.5]). But the former case is impossible forBut then either ext * (D) = 0 or Z is a fiber of ext, respectively, thus all being absurd. Hence we get Z = ∅. On the other hand, dim E f = 2 (see Lemma 5.2) and D = F 1 contains a fiber of ext by construction, which implies that D ∩ E f = ∅, a contradiction.Lemma 5.11. Let Y ′ = P 1 × P 1 . Then X is isomorphic to the cone from Example 1.3.Proof. Recall that Y ′ is smooth and the locus E f ′ is of pure codimension 1 for X ′ = P(6, 4, 1, 1) (see Example 3.13 and Remark 3.14). Then we have the following two results: Lemma 6.10. f ′ (ext(E)) is a point.Proof. Suppose that f ′ (ext(E)) is a curve. Then C := ext(E) is also a curve and we have E f ′ ∩C = ∅ by Lemma 6.3. Further, the map p : X ′ X is the linear projection from a subspace Λ ⊂ P 38 such that X ′ ∩ Λ = f ′ (C) (see Lemma 6.4). More precisely, since g = 33 and g ′ = 37, we find that dim Λ = 3, which gives −K X ′ · f ′ (C) 4 (for X ′ is the intersection of quadrics). But on X ′ = P(6, 4, 1, 1) we have O X ′ (−K X ′ ) ≃ O X ′ (12) (see Example 3.13), and hencewhich implies that the curve f ′ (C) passes through a singular point on X ′ . This contradicts E f ′ ∩ C = ∅. 13 Lemma 6.11. ext(E) is a point.Proof. Suppose that C := ext(E) is a curve. Then Lemmas 6.10 and 6.4 show that p is the linear projection from the point f ′ (C). In particular, we get g ′ − g = 1, which contradicts g = 33, g ′ = 37.Further, Lemmas 6.11 and 6.5 imply that p is the linear projection from the tangent space at the smooth point f ′ (ext(E)) on X ′ .Conversely, linear projection from the tangent space at a smooth point on X ′ = P (6, 4, 1, 1) is birational (see Lemma 3.16) and leads to a Fano threefold X := p(X ′ ) with (−K X ) 3 = 64 (cf. Remark 3.10 and the construction in [28, Section 7]). Proposition 6.9 is completely proved. Proposition 6.12. Let X ′ = X 70 . Then p is the linear projection from a plane Π. More precisel...

show abstract

Section: Some Auxiliary Results About Extremal Raysmentioning

confidence: 98%

Fano threefolds with canonical Gorenstein singularities and big degree

Karzhemanov

2014

Math. Ann.

View full text Add to dashboard Cite

show abstract

“…Namikawa has shown in [Na97] that a smoothing always exists if X ′ has only terminal Gorenstein singularities, not necessarily Q-factorial: In this case the Picard groups of X ′ and the general X t are isomorphic (over Z) by [JR06a].…”

Section: Preliminariesmentioning

confidence: 99%

“…, [JR06a]]. Let X ′ be a Gorenstein Fano threefold with only terminal singularities (not necessarily Q−factorial).…”

Section: Proposition [[Na97]mentioning

confidence: 99%

“…If W is Q-factorial, the claim is just Lemma 2.8 above. Case (3) in Proposition 2.7 is explicitely described in [JR06a]. The only remaining case is the quadric cone.…”

Section: Proof If Xmentioning

confidence: 99%

“…Since ψ is small, X ′ has only terminal singularities, is hence smoothable by [Na97]. Then the smoothing X t has the same index as X by [JR06a], which is 1 by assumption. Moreover, |−K X | is base point free, hence…”

Section: Proposition Supposementioning

confidence: 99%

See 2 more Smart Citations

Threefolds with big and nef anticanonical bundles II

Jahnke¹,

Peternell²,

Radloff³

2011

Open Mathematics

Self Cite

View full text Add to dashboard Cite

In a follow-up to our paper [Threefolds with big and nef anticanonical bundles I, Math. Ann., 2005, 333(3), 569–631], we classify smooth complex projective threefolds Xwith −K X big and nef but not ample, Picard number γ(X) = 2, and whose anticanonical map is small. We assume also that the Mori contraction of X and of its flop X + are not both birational.

show abstract

Birational Geometry via Moduli Spaces

Cheltsov

Katzarkov

Przyjalkowski

2013

Birational Geometry, Rational Curves, and Arithmetic

View full text Add to dashboard Cite

Summary. In this paper we connect degenerations of Fano threefolds by projections. Using Mirror Symmetry we transfer these connections to the side of Landau-Ginzburg models. Based on that we suggest a generalization of Kawamata's categorical approach to birational geometry enhancing it via geometry of moduli spaces of Landau-Ginzburg models. We suggest a conjectural application to Hasset-Kuznetsov-Tschinkel program based on new nonrationality "invariants" we consider -gaps and phantom categories. We make several conjectures for these invariants in the case of surfaces of general type and quadric bundles.

show abstract

Terminal Fano threefolds and their smoothings

Cited by 31 publications

References 13 publications

Fano threefolds with canonical Gorenstein singularities and big degree

Fano threefolds with canonical Gorenstein singularities and big degree

Threefolds with big and nef anticanonical bundles II

Birational Geometry via Moduli Spaces

Contact Info

Product

Resources

About