2024/07/18 by László Koltai, Koltai, László, Tamás László +3
Computer Science · Mathematics · #14B05 #14Fxx #32S05 #32S10 #32S25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.2407.13413
openalex publication_date 2024/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the multiplier ideals and the corresponding jumping numbers and multiplicities \m(c)\c∈ ℝ in the following context: (X,o) is a complex analytic normal surface singularity, \mathfrak a⊂ OX,o is an \mathfrak mX,o--primary ideal, ϕ:\widetildeX→ X is a log resolution of \mathfraka such that \mathfrakaO_\widetildeX=O_\widetildeX(-F), for some nonzero effective divisor F supported on ϕ-1(0). We show that \m(c)\c>0 is combinatorially computable from F and the resolution graph Γ of ϕ, and we provide several formulae. We also extend Budur's result (valid for (X,o)=(ℂ2,0)), which makes an identification of ∑c∈[0,1]m(c)tc with a certain Hodge spectrum. In our general case we use Hodge spectrum with coefficients in a mixed Hodge module. We show that \m(c)\c≤ 0 usually depends on the analytic type of (X,o). However, for some distinguished analytic types we determine it concretely. E.g., when (X,o) is weighted homogeneous (and F is associated with the central vertex), we recover ∑cm(c)tc from the Poincaré series of (X,o) and when (X,o) is a splice quotient then we recover ∑cm(c)tc from the multivariable topological Poincaré (zeta) function of Γ.