Weakly-exceptional singularities in higher dimensions

Cheltsov, Ivan; Shramov, Constantin

doi:10.1515/crelle-2012-0063

Cited by 11 publications

(23 citation statements)

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“…If C was contained in a quadric cone, we would have g = (a − 1)(a + e − 1) and d = 2a + e for some a, e ∈ N (see the last part of the proof of Proposition 3.1); but there is no such relation for (g, d) = (14,11). Thus, C must be a curve of bidegree (3,8) on a smooth quadric. The dimension of such curves is g + 2d + 8 = 44, and they form an irreducible component of H S 14,11 .…”

Section: Linkagementioning

confidence: 99%

“…Finally, we observe that we can construct C contained in an irreducible quartic by linkage. Indeed, start from a smooth curve C ∈ H S 2,5 contained in a smooth quadric Q : if we identify Q to P 1 × P 1 , C is a curve of bidegree (2,3). One can show that such a curve is 4-regular, and applying a theorem of Martin-Deschamps and Perrin [25,Theorem 3.4], we obtain that a general linkage of type [4,4] yields a smooth curve C ∈ H S 14,11 .…”

Section: Linkagementioning

confidence: 99%

“…Namely we are in one of the following cases, the details of which are discussed in Examples 3.3(i)-(iii), 3.4(i)-(v) and 3.5(i)-(ii), respectively: (a) S is a plane: (g, d) = (0, 1), (0, 2) or (1, 3); (b) S is a smooth quadric, and C is not a plane curve: (g, d) = (0, 3), (0, 4), (1,4), (2,5) or (4, 6); (c) S is a quadric cone, and C is not contained in any other quadric: (g, d) = (2, 5) or (4, 6); (ii) −K X is nef. but not ample, then the curves numerically equivalent to r = 4l − f cover the surface S, the anticanonical morphism is a divisorial contraction and we are in one of the following cases: (a) S is a plane: (g, d) = (3, 4); (b) S is a smooth quadric, and C is not a plane curve: (g, d) = (0, 5), (3,6), (6,7) or (9,8). (c) S is a quadric cone, and C is not contained in any other quadric: (g, d) = (6, 7) or (9,8).…”

Section: Curves In a Plane Or A Quadricmentioning

confidence: 99%

“…if and only if a = 4, and in this case the anticanonical morphism contracts the strict transform of Q. The possibilities for a = 4 are (a, b) = (4, 1), (4, 2), (4,3) and (4,4), corresponding to (g, d) = (0, 5), (3,6), (6,7) and (9,8). If a 5, −K X is not nef.…”

Section: Curves In a Plane Or A Quadricmentioning

confidence: 99%

“…If a 3, −K X is ample and the contraction of all curves equivalent to r yields a contraction X → Y . The possibilities are (a, b) = (2, 1), (2, 2), (3, 1), (3,2) and (3,3), corresponding to (g, d) = (0, 3), (1,4), (0, 4), (2,5) and (4, 6): see Example 3.4.…”

Section: Curves In a Plane Or A Quadricmentioning

confidence: 99%

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Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links

Blanc

Lamy

2012

Proceedings of the London Mathematical Society

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We characterize smooth curves in P 3 whose blow-up produces a threefold with anticanonical divisor big and nef.These are curves C of degree d and genus g lying on a smooth quartic, such that (i) 4d − 30 g 14 or (g, d) = (19, 12), (ii) there is no 5-secant line, 9-secant conic nor 13-secant twisted cubic to C. This generalizes the classical similar situation for the blow-up of points in P 2 .We describe then Sarkisov links constructed from these blow-ups, and are able to prove the existence of Sarkisov links which were previously only known as numerical possibilities.We obtain in fact a more precise result, which is our main theorem below. Here, we denote by H S g,d the Hilbert scheme of smooth irreducible curves of genus g and degree d in P 3 . We always work over the base field C of complex numbers.Theorem 1.1 (see Table 1). Let C ∈ H S g,d be a curve of genus g and degree d. We denote by X the blow-up of P 3 along C and by −K X its anticanonical divisor. Consider the sets A 0These pairs form a partition of all (g, d) corresponding to non-empty H S g,d with 4d − 30 g 14 or (g, d) = (19, 12), and we have the following characterizations. or (g, d) ∈ A 2 and there is no 4-secant line to C;(ii) The variety X is weak Fano, that is, −K X is big and nef., if and only if one of the following condition holds:and there is no 4-secant line to C; (c) (g, d) ∈ A 3 , there is no 5-secant line to C, and C is contained in a smooth quartic; (d) (g, d) ∈ A 4 , there is no 5-secant line, 9-secant conic, nor 13-secant twisted cubic to C, and C is contained in a smooth quartic. Moreover, Condition (i) corresponds to a non-empty open subset of H S g,d if (g, d) ∈ A 2 , and Condition (ii) corresponds to a non-empty open subset of H S g,d if (g, d) ∈ A 3 ∪ A 4 . Furthermore, for a general curve in these sets, there are finitely many irreducible curves intersecting trivially K X , except for (g, d) ∈

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Section: Linkagementioning

confidence: 99%

Section: Linkagementioning

confidence: 99%

Section: Curves In a Plane Or A Quadricmentioning

confidence: 99%

Section: Curves In a Plane Or A Quadricmentioning

confidence: 99%

Section: Curves In a Plane Or A Quadricmentioning

confidence: 99%

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Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links

Blanc

Lamy

2012

Proceedings of the London Mathematical Society

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Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces

Cheltsov

Rubinstein

Zhang

2019

Sel. Math. New Ser.

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The purpose of this article is to develop techniques for estimating basis log canonical thresholds on logarithmic surfaces. To that end, we develop new local intersection estimates that imply log canonicity. Our main motivation and application is to show the existence of Kähler-Einstein edge metrics on all but finitely many families of asymptotically log del Pezzo surfaces, partially confirming a conjecture of two of us. In an appendix we show that the basis log canonical threshold of Fujita-Odaka coincides with the greatest lower Ricci bound invariant of Tian.is also the necessary and sufficient condition for (b) if µ ≤ 0 [40, Theorem 2]; moreover, a classification (i.e., part (a)) is essentially impossible when µ ≤ 0 [32], [54, §8], and so we will restrict our attention in (a)-(b) exclusively to the case µ > 0, that we have previously called the asymptotically log Fano regime [21, Definition 1.1].Our previous work accomplished (a) in dimension 2, providing a complete classification [21, Theorem 2.1]. Furthermore, we also obtained the "necessary" portion of (b) [21,22], and this was extended to higher dimensions by Fujita [34]. The purpose of this article is to complete the "sufficient" portion of (b) in dimension 2 in all but finitely many (in fact, all but 6) of the (infinite list of) cases classified in [21]. The Calabi problem for asymptotically log Fano varietiesA special class of asymptotically log Fano varieties is as follows. This is a special case of [21, Definition 1.1].Definition 1.1. We say that a pair (X, D) consisting of a smooth projective variety X and a smooth irreducible divisor D on X is asymptotically log Fano if the divisor −K X − (1 − β)D is ample for sufficiently small β ∈ (0, 1].This definition contains the class of smooth Fano varieties (D = 0) as well as the classical notion of a smooth log Fano pair due to Maeda (β = 0) [48].One can show using a result of Kawamata-Shokurov that if (X, D) is asymptotically log Fano then |k(K X + D)| (for some k ∈ N) is free from base points and gives a morphism [21, §1] η : X → Z.The following conjecture, posed in our earlier work, gives a rather complete picture concerning (b) when D is smooth. Conjecture 1.2. [21, Conjecture 1.11] Suppose that (X, D) is an asymptotically log Fano manifold with D smooth and irreducible. There exist KEE metrics with angle 2πβ along D for all sufficiently small β if and only if η is not birational.This conjecture stipulates that the existence problem for KEE metrics in the small angle regime boils down to a simple birationality criterion. In fact, this amounts to computing a single intersection number, i.e., checking whether (K X + D) n = 0.This would be a rather far-reaching simplification as compared to checking the much harder condition of log K-stability that involves, in theory, computing the Futaki invariant of an infinite number of log test configurations, or else estimating the blct which involves, in theory, estimation of singularities of pairs that may occur after an unbounded number of blow-ups.3

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