Finite determination conjecture for Mather–Jacobian minimal log discrepancies and its applications

Abstract

In this paper we study singularities in arbitrary characteristic. We propose Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies in terms of jet schemes of a singularity. The conjecture is equivalent to the boundedness of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy. We also show that this conjecture yields some basic properties of singularities; eg., openness of Mather-Jacobian (log) canonical singularities, stability of these singularities under small deformations and lower semi-continuity of Mather-Jacobian minimal log discrepancies, which are already known in characteristic 0 and open for positive characteristic case. We show some evidences of the conjecture: for example, for non-degenerate hypersurface of any dimension in arbitrary characteristic and 2-dimensional singularities in characteristic not 2. We also give a bound of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy.Mathematical Subject Classification: 14B05,14E18, 14B07

The minimal log discrepancies on a smooth surface in positive characteristic

“…For the case mld(0; A 2 , a e ) = −∞ an upper bound of the minimal k E such that E computes the mld is 9. In this case E = E (7,3) and the ideal a is generated by x 3 and y 7 . Therefore, we obtain ℓ {1/2} = 9.…”

Characterization of two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic

“…(i) (1, 1, 1), (ii) (3, 2, 2), (iii) (2, 1, 1), (iv) (6,4,3), (v) (9,6,4), (vi) (15,10,6), (vii) (3, 2, 1), (viii) (10,5,4), (ix) (15,8,6), (x) (21, 14,6).…”

“…In [8], Ishii posed the following conjecture, which is a special case of Mustat ¸ǎ-Nakamura's conjecture ( [15]).…”

“…It is known that this conjecture bounds the number of blow-ups to obtain a prime divisor computing the minimal log discrepancy. In [8], Conjecture D 1 is proved for arbitrary characteristic and one can take M 1 = 4. On the other hand, Conjecture D 2 is also proved in [8] for characteristic p = 2 and one can take M 2 ≤ 58 in this case.…”

“…In [8], Conjecture D 1 is proved for arbitrary characteristic and one can take M 1 = 4. On the other hand, Conjecture D 2 is also proved in [8] for characteristic p = 2 and one can take M 2 ≤ 58 in this case.…”