Relative Mather discrepancy on arc spaces

2025/08/17 by de Fernex, Tommaso, Mere, Zach
#Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 14E18 #Secondary 14B05

paper · doi:10.48550/arxiv.2508.12420

Abstract

Given any generically étale morphism of varieties f \colon X → Y, we define the relative Mather discrepancy function on the arc space X_∞ of the domain and show that this function computes the dimension of the kernel of the differential map of the induced morphism on arc spaces f_∞ \colon X_∞ → Y_∞. We relate this result to the change-of-variable formula in motivic integration. We introduce the notion of \widehat K-equivalence, which agrees with K-equivalence for smooth varieties, and prove that \widehat K-equivalent varieties of arbitrary characteristic define the same class in the motivic ring.

Citations

• Motivic integration for singular Artin stacks
• On arc fibers of morphisms of schemes
• A motivic change of variables formula for Artin stacks
• Jet schemes and singularities
• D-equivalence and K-equivalence
• The universal Euler characteristic for varieties of characteristic zero
• Non-Archimedian integrals and stringy Euler numbers of log terminal\n pairs

Related

• A motivic change of variables formula for Artin stacks
• Mather discrepancy and the arc spaces
• Generically finite morphisms and formal neighborhoods of arcs
• Grothendieck rings of definable subassignments and equivariant motivic measures
• On arc fibers of morphisms of schemes