Multiplier ideals of monomial ideals

By J. Howald. 2000

194 citations recorded by the source corpus.

In this note we calculate the multiplier ideal associated to an arbitrary monomial ideal in C^n. We discuss applications to the calculation of log canonical thresholds.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1842e8657562cdc811dfe00bde6397be7e9bddc6

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In Algebra and Number Theory

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Citations in the corpus

Contact loci in arc spaces (2003, unknown): “saying that for every q′ ∈ P◦ a , we have β· inf v∈Pb X i q′ i vi ! + X i q′ i ≥ α. Since (P◦ a ) ◦ = P a and α>0, the above condition is equivalent to β· P b + e ⊆ α·P a. Corollary 3.10. (Howald, [How]) If X= Ad and a ⊆ R= C[T 1,...,Td] is a nonzero monomial ideal, then for every α∈ Q∗ +, the mu”
F-thresholds and Bernstein–Sato Polynomials (2004, methodology): “,− 23 20 ,− 13 10 ,− 27 20 . As in Example 4.2, since the exponents of the monomials in f satisfy that genericity condition, we can compute the jumping coeﬃcients of f using Howald’s description from [Ho2]. The ones in (0,1] are 9 20 , 13 20 , 7 10 , 17 20 , 9 10 , 19 20 , 1. As pointed out in [Sa],”
Multiplicities and log canonical threshold (2002, background): “ted to J is deﬁned as the convex hull in Rn + of the points corresponding to the monomials in J. Let lc(J) be the log canonical threshold of (R,J), and set µ(J) = 1/lc(J). Then a result due to Howald [Ho] implies that µ(J) = min{α > 0 | α·e∈ P(J)}, 4 T. DE FERNEX, L. EIN, AND M. MUSTAT¸Aˇ where e”
Multiplier ideals, $b$-function, and spectrum of a hypersurface singularity (2004, methodology): “of Pn−1 calculating Hn−1(F x,C)λ, see Proposition 4.9. MULTIPLIER IDEALS, b-FUNCTION, AND SPECTRUM 3 In general it is not easy to calculate J(X,αD) explicitly except for some special cases, see [15], [16], [26], etc. In thispaper, we giveanexplicit formulaforJ(X,αD) in the case Dis a locally conical”
Birationally rigid hypersurfaces (2006, methodology): “hts a/ai along the coordinates xi, where a:= a1 ...an), and thus the multiplier ideal J(X,cZ) is nontrivial. On the other hand, using Howald’s characterization of multiplier ideals of monomial ideals [How], one sees that J(Ye,cZ|Ye) = J(Ye,ceZ|Ye) = (xe+1,...,xn) for every 1 ≤ e≤ n−1, since, in view”
Uniform approximation of Abhyankar valuation ideals in smooth function fields (2002, unknown): “We recall the formula for the multiplier ideal of a monomial ideal from [10].”
Bounds for log canonical thresholds with applications to birational rigidity (2002, methodology): “This generalizes the formula for the log canonical threshold of a monomial ideal from [ How ].”
A generalization of tight closure and multiplier ideals (2002, background): “In Section 4, we establish properties of ideals τ(a) in fixed characteristic p > 0 similar to those of multiplier ideals J(a) in characteristic zero, namely, the restriction theorem (Theorem 4.1), the subadditivity (Theorem 4.5), and a description of ideals τ(a) in toric case (Theorem 4.8); see [DEL”
Valuations and asymptotic invariants for sequences of ideals (2010, methodology): “If am = am for some monomial ideal a, then P (a•) = P (a), J (aλ•) = J (aλ) and (8.4) becomes Howald’s original formula from [26].”
Bernstein–Sato polynomials of arbitrary varieties (2004, unknown): “By [16] (see also [10]), J (X,αZ) is generated by the monomials x := ∏ i x νi i with ν + 1 ∈ (α + ε)P where ε > 0 is sufficiently small.”
Integral closure of ideals, rings, and modules (2006, background): “Howald [129] proved the precise characterization of adjoints of monomial ideals in a polynomial ring over a field of characteristic zero, the general form is due to Hübl and Swanson [135]:”
A sharp lower bound for the log canonical threshold (2012, unknown): “riction to the set Σ = n x= (x1,...,xn) ∈ Rn + : Xn j=1 xj = 1 o . We choose x0 = (x0 1,...,x 0 n) ∈ Σ such that νϕ(x0) = max{νϕ(x) : x∈ Σ} ∈ he 1(ϕ) n ,e1(ϕ) i . By Theorem 5.8 in [Kis94a] (see also [Ho01] for similar results in an algebraic context) we have the formula c(ϕ) = 1 νϕ(x0) . Set ζ(x) =”
Jumping coefficients of multiplier ideals (2003, background): “For divisors with isolated singularities, we show that the jumping numbers obey a semicontinuity property analogous to that satisfied by the spectrum of a singularity [32], and we establish a statement of Thom–Sebastiani type for divisors defined by functions in independent sets of variables.”
IMPANGA lecture notes on log canonical thresholds (2011, background): “ we write x u = xu1 1 ···x un n . Given u = (u 1,...,un) and v = (v 1,...,vn) in Rn, we put hu,vi = Pn i=1uivi. The Newton polyhedron of ais P(a) = convexhull {u ∈ Zn ≥0 | x u ∈ a} . Howald showed in [How] that lct(a) = lct 0(a) = max{λ ∈ R≥0 | (1,...,1) ∈ λ·P(a)}. This follows rather easily using s”
Toric plurisubharmonic functions and analytic adjoint ideal sheaves (2010, methodology): “orem to the psh function attached to g= minihαi,·i for some αi∈ Rn. But thanks to Proposition 1.10, P(a) and P(g) have same interiors, so we obtain: 8 HENRI GUENANCIA Corollary 1.14 (Howald’s theorem,[How01]). — Let a= (zα 1,...,zαk ) be a monomial ideal of C[z1,...,zn], and let P(a) be the Newton p”
On a non-vanishing conjecture of Kawamata and the core of an ideal (2003, unknown): “, hAr in Nd; [21].”