On boundedness of divisors computing minimal log discrepancies for surfaces

2020/05/19 by Jingjun Han, Yujie Luo, Han, Jingjun +1
Mathematics · #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2005.09626

Abstract

Let Γ be a finite set, and X\ni x a fixed klt germ. For any lc germ (X\ni x,B:=∑_i b_iB_i) such that b_i∈ Γ, Nakamura's conjecture, which is equivalent to the ACC conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor E over X\ni x, such that a(E,X,B)=\rmmld(X\ni x,B), and a(E,X,0) is bounded from above. We extend Nakamura's conjecture to the setting that X\ni x is not necessarily fixed and Γ satisfies the DCC, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of a(E,X,0) for any such E.

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