Fano threefolds with canonical Gorenstein singularities and big degree

Abstract

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...

On Polarized K3 Surfaces of Genus 33

“…It can be easily checked that Y 1 is a weak Fano threefold and X 1 ⊂ P 34 is an anticanonically embedded Fano threefold of genus 33 (cf. the proof of Proposition 6.12 in [10]). Moreover, we obtain Pic(Y 1 ) = Z · K Y1 ⊕ Z · E f1 , where E f1 ≃ F 4 is the f 1 -exceptional divisor, and the morphism g 1 contracts the surface f −1 1 * ( E) to a point.…”

“…It can be easily checked that Y 1 is a weak Fano threefold and X 1 ⊂ P 34 is an anticanonically embedded Fano threefold of genus 33 (cf. the proof of Proposition 6.12 in [10]). Moreover, we obtain Pic(…”

“…2.8. Let X be the Fano threefold with canonical Gorenstein singularities and anticanonical degree (−K X ) 3 = 70 (see [9], [10], [11]). Recall that divisor −K X is very ample and the linear system | − K X | gives an embedding of X into P 37 .…”

Birational Geometry via Moduli Spaces

“…Fano threefolds with canonical Gorenstein singularities are not yet classified, but first steps in this directions are already have been made by S. Mukai, P. Jahnke, I. Radloff, I. Cheltsov, C. Shramov, V. Przyjalkowski, Yu. Prokhorov, and I. Karzhemanov (see [Mu02], [JR04], [CPS04], [Pr05], [Ka08], and [Ka09]).…”