Positivity in Algebraic Geometry II

Lazarsfeld, Robert

doi:10.1007/978-3-642-18810-7

Cited by 697 publications

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“…This argument is essentially repeated in Example 3.6 of [ELSV04], and discussed at greater length in Section 9.3.C of [Laz04]. Note that since this curve is analytically irreducible, the result also follows from [Jär06].…”

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confidence: 86%

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Jumping numbers on algebraic surfaces with rational singularities

Tucker

2009

Trans. Amer. Math. Soc.

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Abstract. In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored.

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confidence: 86%

“…A detailed account of their properties, applications, and further references, may be found in [Laz04].…”

Section: Multiplier Ideals On Rational Surface Singularitiesmentioning

confidence: 99%

Jumping numbers on algebraic surfaces with rational singularities

Tucker

2009

Trans. Amer. Math. Soc.

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“…They are periodic, completely determined by the jumping numbers less than 1, but otherwise difficult to compute in general, even if a set of candidates is easy to provide, cf. [9,Lemma 9.3.16].…”

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confidence: 99%

“…The jumping numbers are expressed in terms of the Zariski exponents of the ideal. Moreover (see [9,Proposition 9.2.8] and also [8,Theorem 9.4]) the jumping numbers < 1 of the ideal p coincide to those of the unibranch plane curve corresponding to a general element of p, and they amount to the jumping numbers given in (1).…”

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confidence: 99%

Jumping numbers of a unibranch curve on a smooth surface

Naie

2008

manuscripta math.

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A formula for the jumping numbers of a curve unibranch at a singular point is established. The jumping numbers are expressed in terms of the Enriques diagram of the log resolution of the singularity, or equivalently in terms of the canonical set of generators of the semigroup of the curve at the singular point.The jumping numbers of a curve on a smooth complex surface are a sequence of positive rational numbers revealing information about the singularities of the curve. They extend in a natural way the information given by the log-canonical threshold, the smallest jumping number (see [3] for example). They are periodic, completely determined by the jumping numbers less than 1, but otherwise difficult to compute in general, even if a set of candidates is easy to provide, cf. [9, Lemma 9.3.16].The aim of this paper is to give a formula for the jumping numbers of a curve unibranch at a singular point. A curve C will be said to be unibranch at a point P , if the analytic germ of C at P is irreducible. The formula is expressed in terms of the Enriques diagram associated to the singularity, or equivalently (see Theorem 3.1) in terms of a minimal set of generators (β 0 , β 1 , . . . , β g ) of the semigroup S(C, P ) of C at P :where the m j are defined below. HereThe semigroup is defined by S(C) = {ord P s | s ∈ O C,P }, the order of the local section s being computed using a normalization of C. It is finitely generated and a minimal set of generators (β 0 , β 1 , . . . , β g ) is constructed as follows (see [13, Theorem 4.3.5]): β 0 is the least element of S(C); set m 1 = β 0 ; β j is the least element of S(C) not divisible by m j and m j+1 = gcd(m j , β j ).To prove (1) we use the notion of relevant divisors of the minimal log resolution of C at P , notion introduced in [12], and previously in [6] from the point of view of valuations corresponding to Puiseux exponents: a relevant divisor is an irreducible exceptional divisor that intersects at least three other components of the total transform of C through the resolution. When C is unibranch at P , we show that the relevant divisors account for all the jumping numbers. This is the content of Proposition 2.5 and represents the key step of the proof. In 1

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“…The following standard result (see [La04], lemma 4.1.14) shows that this is only possible for infinite covers.…”

Section: Introductionmentioning

confidence: 94%

Vanishing theorems of negative vector bundles on projective varieties and the convexity of coverings

Bogomolov

Oliveira

2006

J. Algebraic Geom.

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Abstract. We give a new proof of the vanishing of H 1 (X, V ) for negative vector bundles V on normal projective varieties X satisfying rank V < dim X. Our proof is geometric, it uses a topological characterization of the affine bundles associated with non-trivial cocycles α ∈ H 1 (X, V ) of negative vector bundles. Following the same circle of ideas, we use the analytic characteristics of affine bundles to obtain convexity properties of coverings of projective varieties. We suggest a weakened version of the Shafarevich conjecture: the universal coveringX of a projective manifold X is holomorphically convex modulo the pre-image ρ −1 (Z) of a subvariety Z ⊂ X. We prove this conjecture for projective varieties X whose pullback map ρ * identifies a nontrivial extension of a negative vector bundle V by O with the trivial extension.

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Positivity in Algebraic Geometry II

Cited by 697 publications

References 0 publications

Jumping numbers on algebraic surfaces with rational singularities

Jumping numbers on algebraic surfaces with rational singularities

Jumping numbers of a unibranch curve on a smooth surface

Vanishing theorems of negative vector bundles on projective varieties and the convexity of coverings

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