The asymptotic Samuel functions and applications to resolution of singularities techniques

2025/06/17 by Benito, A., Bravo, A., Encinas, S.
#13B22 #13H15 #14E15 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.14993

Abstract

We use the asymptotic Samuel function to define the Samuel slope of a Noetherian local ring, and we prove that it characterizes regularity in the case of local excellent rings. In addition, we introduce a second invariant that refines the Samuel slope using generic linear cuts, the refined Samuel slope. We show that both, the Samuel slope and the refined Samuel slope, are connected to invariants coming from resolution of singularities.

Citations

• On some properties of the asymptotic Samuel function
• The asymptotic Samuel function and invariants of singularities
• Invariance of Hironaka's characteristic polyhedron
• Resolution of Singularities of Arithmetical Threefolds II
• Characteristic polyhedra of singularities without completion -- Part II
• Equimultiplicity, algebraic elimination, and blowing-up
• Characteristic polyhedra of singularities without completion
• On Rees algebras and invariants for singularities over perfect fields
• On elimination of variables in the study of singularities in positive characteristic
• Monoidal transforms and invariants of singularities in positive\n characteristic
• Rees algebras on smooth schemes: integral closure and higher differential operators

Related

• The asymptotic Samuel function and invariants of singularities
• On some properties of the asymptotic Samuel function
• Constancy of the Hilbert-Samuel function
• Fonction asymptotique de Samuel des sections hyperplanes et multiplicité
• The Asymptotic Samuel Function of a Filtration