A remark on hypersurfaces with isolated singularities

Park, Jihun; Woo, Youngho

doi:10.1007/s00229-006-0047-1

Cited by 6 publications

(9 citation statements)

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“…When n = 3 and m = 2 for instance, the bound is one worse than the Severi bound, but at least when n ≥ 5 and m ≥ 3, in many instances this improves what comes out of [PW06] or similar methods. 3 Remark 27.1.…”

Section: H Vanishing On P N and Abelian Varieties With Applicationsmentioning

confidence: 99%

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Hodge Ideals

2019

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Section: H Vanishing On P N and Abelian Varieties With Applicationsmentioning

confidence: 99%

“…Rob Lazarsfeld has shown us a different approach, based on multiplier ideals, showing that the isolated points of multiplicity m ≥ 2 impose independent conditions on hypersurfaces of degree at least n m−1 (d − 1) − n; this is often stronger than the bound in[PW06]. Since d ≥ m, it is somewhat weaker than the bound in Corollary H when m ≤ n + 1.…”

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confidence: 99%

Hodge Ideals

2019

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“…And then let us apply the similar technique as Lemma 2.2, which has evolved from the papers [4,14], to the following case.…”

Section: Lemma 42 If There Is a Set Of At Least 20 Points Of Such Thmentioning

confidence: 99%

Factorial hypersurfaces in $$\mathbb{P}^4$$ with nodes

Cheltsov

Park

2006

Geom Dedicata

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“…When r = 3 the statement was given by Severi. The general case, as well as the proof that follows, is due to Park and Woo [24].…”

Section: Vanishing Theorems For Multiplier Idealsmentioning

confidence: 99%

“…This uses that m ≥ 3: see[24, Lemma 3.2] 2. The precise statement is that the Jacobians form an irreducible component of the locus of all (A, Θ) defined by the stated condition.…”

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A short course on multiplier ideals

Lazarsfeld¹

2010

Analytic and Algebraic Geometry

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where the D i are distinct prime divisors and each a i ∈ Q is a non-negative rational number. We will attach to these data multiplier ideal sheavesThe intuition is that these ideals will measure the singularities of D or of functions f ∈ a, with "nastier" singularities being reflected in "deeper" multiplier ideals.Although we will mainly focus on algebraic constructions, it is perhaps most intuitive to start with the analytic avatars of multiplier ideals. Definition 1.1 (Analytic multiplier ideals). Given D = a i D i as above, choose local equations f i ∈ O X for each D i . Then the (analytic) multiplier ideal of D is given locally bySimilarly, if f 1 , . . . , f r ∈ a are local generators, then(One checks that these do not depend on the choice of the f i .)Equivalently, J an (D) and J an (a c ) arise as the multiplier ideal J (φ), where φ is the appropriate one of the two plurisubharmonic functionsblow-up along the support of D or the zeroes of a, and this in turn is measured by the vanishing of the multipliers h required to ensure integrability.Exercise 1.2. Suppose that D = a i D i has simple normal crossing support. Then(Hint: This boils down to the assertion that if z 1 , . . . , z d are the standard complex coordinates in C d , and if h ∈ C{z 1 , . . . , z d } is a convergent power series, thenBy separating variables, this in turn follows from the elementary computation that the function 1/|z| 2c of one variable is locally integrable if and only if c < 1.)Multiplier ideals can also be constructed algebro-geometrically. Let µ : X −→ X be a log resolution of D or of a. Recall that this means to begin with that µ is a proper morphism, with X smooth. In the first instance we require that µ * D + Exc(µ) have simple normal crossing (SNC) support, while in the second one asks thatwhere F is an effective divisor and F + Exc(µ) has SNC support. We consider also the relative canonical bundle K X /X = det(dµ), so that K X /X ≡ lin K X − µ * K X . Note that this is well-defined as an actual divisor supported on the exceptional locus of µ (and not merely as a linear equivalence class).Definition 1.3 (Algebraic multiplier ideal). The multiplier ideals associated to D and to a are defined to be:(As in the previous Exercise, the integer part of a Q-divisor is defined by taking the integer part of each of its corefficients.)We start by defining "mixed" multiplier ideals:Definition 3.10. Let a, b ⊂ O X be two ideal sheaves. Given c, e ≥ 0, the multiplier idealis defined by taking a common log resolution µ : X −→ X of a and b, withfor divisors A, B with SNC support, and settingThe subadditivity theorem compares these mixed ideals to the multiplier ideals of the two factors. Theorem 3.11 (Subadditivity Theorem). One has an inclusion J (a c · b e ) ⊆ J (a c ) · J (b e ).Similarly, J (X, D 1 + D 2 ) ⊆ J (X, D 1 ) · J (X, D 2 ) for any two effective Q-divisors D 1 , D 2 on X Sketch of Proof of Theorem 3.11. Consider the product X × X with projections p 1 , p 2 : X × X −→ X.The first step is to show via the Künneth formula that). ...

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A remark on hypersurfaces with isolated singularities

Cited by 6 publications

References 7 publications

Hodge Ideals

Hodge Ideals

Factorial hypersurfaces in $$\mathbb{P}^4$$ with nodes

A short course on multiplier ideals

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