On the Classification of Cubic Surfaces

Bruce, J. W.; Wall, C. T. C.

doi:10.1112/jlms/s2-19.2.245

Cited by 170 publications

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“…It is easy to see that the natural mapḠ\∆ s → G\∆ is a bijection; this relies on the fact that the symmetry group of a cubic surface acts transitively on the nodes of the surface. (See the remark on p. 249 of [11].) Since it is proper, this map is a homeomorphism.…”

Section: Homeomorphism On the Nodal Divisormentioning

confidence: 99%

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The complex hyperbolic geometry of the moduli space of cubic surfaces

2002

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Section: Homeomorphism On the Nodal Divisormentioning

confidence: 99%

“…Restating Lemma 2 of [11], we define a homeomorphism φ : M nod −→ M (6) s in the following way. Let∆ s denote the set of those F ∈ ∆ s such that (0 : 0 : 0 : 1) is a node of S and such that the tangent cone to S at (0 : 0 : 0 : 1) is X 0 X 2 − X 2 1 = 0.…”

Section: Homeomorphism On the Nodal Divisormentioning

confidence: 99%

The complex hyperbolic geometry of the moduli space of cubic surfaces

2002

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“…Indeed, for the surfaces of degree three this follows from the classification given in [BW79]. This also implies uniqueness for all surfaces of degree greater than three, except for perhaps the quartic del Pezzo surface of type A 3 with four lines.…”

Section: Actions On Generalised Del Pezzo Surfacesmentioning

confidence: 79%

Equivariant Compactifications of Two-Dimensional Algebraic Groups

Derenthal¹,

Loughran²

2014

Proceedings of the Edinburgh Mathematical Society

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“…It follows from [1], that the equalities Proof of Theorem 1.5. Suppose that X is log terminal and lct(X) 1, but ρ is not an isomorphism.…”

Section: )mentioning

confidence: 99%

“…Using the classification of possible singularities of the surface S obtained in [1], we see that it follows from Lemmas 3.4, 3.5, 3.6, 3.7 and 3.8 that we may assume that Σ = A i1 , . .…”

Section: )mentioning

confidence: 99%

On Singular Cubic Surfaces

Cheltsov¹

2009

Asian Journal of Mathematics

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Abstract. We study global log canonical thresholds of singular cubic surfaces.Key words. Cubic surfaces, singularities, log canonical thresholds, del Pezzo fibrations, birational maps, Kahler-Einstein metric, alpha-invariant of Tian, orbifolds.AMS subject classifications. 14J26, 14J45, 14J70, 14Q10, 14B05, 14E05, 32Q20All varieties are assumed to be defined over C.1. Introduction. Let X be a variety with at most log terminal singularities, let Z ⊆ X be a closed subvariety, and let D be an effective Q-Cartier Q-divisor on X. Then the number lct Z X, D = sup λ ∈ Q the log pair X, λD is log canonical along Z is said to be the log canonical threshold of D along Z (see [8]).For the case Z = X we use the notation lct(X, D) instead of lct X (X, D). Then lct X, D = inf lct P X, D P ∈ X = sup λ ∈ Q the log pair X, λD is log canonical .Suppose, in addition, that X is a Fano variety. Definition 1.2. We define the global log canonical threshold of X by the numberThe number lct(X) is an algebraic counterpart of the α-invariant introduced in [11]. In this paper we prove the following result 1 .Theorem 1.4. Let X be a singular cubic surface in P 3 with canonical singularities. Then such that π andπ are flat morphisms, and ρ is a birational map that induces an isomorphismwhere X andX are scheme fibers of π andπ over a point O ∈ Z, respectively. Suppose that • the varieties V andV have terminal Q-factorial singularities,• the divisors −K V and −KV are π-ample andπ-ample, respectively, • the fibers X andX are irreducible. Then ρ is an isomorphism if one of the following conditions hold:• the varieties X andX have log terminal singularities, and lct(X)+lct(X) > 1;• the variety X has log terminal singularities, and lct(X) 1.The assertion of Theorem 1.5 is sharp (see [10, Example 5.2-5.6]).Example 1.8. Let V beV subvarieties in C 1 × P 3 given by the equations x 3 + y 3 + z 2 w + t 6 w 3 = 0 and x 3 + y 3 + z 2 w + w 3 = 0, respectively, where t is a coordinate on C 1 , and (x, y, z, w) are coordinates on P 3 . The projections π : V −→ C 1 andπ :V −→ C 1 1 A cubic surface in P 3 with isolated singularities has canonical singularities ⇐⇒ it is not a cone.

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On the Classification of Cubic Surfaces

Cited by 170 publications

References 0 publications

The complex hyperbolic geometry of the moduli space of cubic surfaces

The complex hyperbolic geometry of the moduli space of cubic surfaces

Equivariant Compactifications of Two-Dimensional Algebraic Groups

On Singular Cubic Surfaces

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