Singularities in arbitrary characteristic via jet schemes

Abstract

This paper summarizes recent results concerning singularities with respect to the Mather-Jacobian log discrepancies over an algebraically closed field of arbitrary characteristic. The basic point is that the inversion of adjunction with respect to Mather-Jacobian discrepancies holds under arbitrary characteristic. Using this fact, we will reduce several geometric properties of the singularities to jet scheme problems and try to avoid discussions that are distinctive to characteristic 0.

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

“…In the same way as Step 1 in the proof of Lemma 3.5, we can prove that E (10,5,4) computes mld(0; A, (f )) and mld(0; A, (in (10,5,4) f )) and we have mld(0; A, (f )) = mld(0; A, (in (10,5,4)…”

“…Arc spaces and minimal log discrepancies. We briefly review in this section the results of arc spaces and minimal log discrepancies in [10]. For simplicity, we consider only the case when a scheme is Speck[[x 1 , .…”

“…, x n ]]. We remark that in [10], we assume that a scheme is of finite type over a field. However, the proofs in [10] also work for Speck…”

“…Theorem 2.17 (inversion of adjunction, [2, Theorem 8.1], [10,Theorem 3.23]). Let f be a non-zero element of (x 1 , .…”

“…(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).…”