3-Dimensional Fano Varieties with Canonical Singularities

Shin, Kilho

doi:10.3836/tjm/1270133187

Cited by 55 publications

(50 citation statements)

References 3 publications

(4 reference statements)

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“…Suppose now that dim B = 1. Then, according to [20], we have B P 1 . Thus, since τ (B) = B, there are at least two τ -fixed points on B.…”

Section: Preliminariesmentioning

confidence: 99%

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On some Fano–Enriques threefolds

Karzhemanov

2011

Advg

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“…Suppose now that dim B = 1. Then, according to [20], we have B P 1 . Thus, since τ (B) = B, there are at least two τ -fixed points on B.…”

Section: Preliminariesmentioning

confidence: 99%

“…Suppose that B := Bs | − K X | = ∅. 2) If dim B = 0, then, according to [20], B is a point. We have B = τ (B) and B ∈ S. On the other hand, τ acts freely on S, a contradiction.…”

Section: Preliminariesmentioning

confidence: 99%

On some Fano–Enriques threefolds

Karzhemanov

2011

Advg

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“…Then we have Bs |H | = ∅ if d ≥ 2, and Bs |H | consists of one point on the smooth part of V * if d = 1 (cf. [33]). Lemma 1.1.…”

Section: Proof Of Main Theoremmentioning

confidence: 99%

“…It is well-known that the base point set Bs |H | = ∅ if d ≥ 2 and the degree d = (H 3 ) ≤ 9, in particular, H is very ample if d ≥ 3 (cf. [33]). By Iskovskih [16, (4.6),(4.7)]), if d = 9, then V * is the cone over P 2 , and if d = 8, then V * is the cone over P 1 × P 1 or F 1 .…”

Section: Remark 12mentioning

confidence: 99%

Singular Fano compactifications of (I)

Furushima

2004

Math. Z.

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In this paper we study the singular Fano compactifications of C 3 with small Gorenstein singularities and determine the structure of such a compactification in the case of the Fano index r ≥ 2. Mathematics Subject Classification (2000): 14J45 , 32J05NotationBs |L| : base locus of the linear system |L| b i (X) := dim R H i (X; R) ρ(X) : Picard number of X mult A X : multiplicity of X at a general point of A K X : canonical divisor of X ∼ : linear equivalence ∼ = : isomorphism F n : Hirzebruch surface of degree n Q n : smooth quadric hypersurface in P n+1 Q 2 0 : quadric cone in P 3 Q 3 0 : quadric hypersurface in P 4 with an isolated singular point V 5 : smooth Fano threefold of index two and degree 5 in P 6 (see [16]) V * 5 : singular Fano threefold of index two and degree 5 in P 6 with small Gorenstein singularities (see[3],[19]) V 22 : smooth Fano threefold of index one and degree 22 in P 13 (see [17]) V * 22 : singular Fano threefold of index one and degree 22 in P 13 with small Gorenstein singularities 710 M. Furushima

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“…[Shi89] shows that when i(Y ) = 4, Y is isomorphic to P 3 . If i(Y ) = 2, both α and ϕ are either E2 contractions,étale conic bundles or quadric bundles.…”

Section: A Geometric Motivation Of Non-factorialitymentioning

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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3-Dimensional Fano Varieties with Canonical Singularities

Abstract: In this article by a variety we mean an irreducible reduced projective variety over the field of complex numbers.

Cited by 55 publications

References 3 publications

On some Fano–Enriques threefolds

On some Fano–Enriques threefolds

Singular Fano compactifications of (I)

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

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