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This paper characterizes singularities with Mather minimal log discrepancies in the highest unit interval, i.e., the interval between $d-1$ and $d$, where $d$ is the dimension of the scheme. The class of these singularities coincides with one of the classes of (1) compound Du Val singularities, (2) normal crossing double singularities, (3) pinch points, and (4) pairs of non-singular varieties and boundaries with multiplicities less than or equal to 1 at the point. As a corollary, we also obtain one implication of an equivalence conjectured by Shokurov for the usual minimal log discrepancies.
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As is well known, the usual discrepancy is defined for a normal Q-Gorenstein variety. By using this discrepancy we can define a canonical singularity and a log canonical singularity. In the same way, by using a new notion, Mather-Jacobian discrepancy introduced in recent papers we can define a canonical singularity and a log canonical singularity for not necessarily normal or Q-Gorenstein varieties. In this paper, we show basic properties of these singularities, behavior of these singularities under deformations and determine all these singularities of dimension up to 2.
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; e.g. 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 hypersurfaces of any dimension in arbitrary characteristic and 2-dimensional singularities in characteristic not 2. We aslo give a bound of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy.
We prove the existence of $n$-complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities. In order to prove these results, we develop the theory of complements for real coefficients. We introduce $(n,Gamma_0)$-decomposable $mathbb{R}$-complements, and show its existence for pairs with DCC coefficients.
We study a pair consisting of a smooth 3-fold defined over an algebraically closed field and a general real ideal. We show that the minimal log discrepancy of every such a pair is computed by a prime divisor obtained by at most two weighted blow-ups. This bound is regarded as a weighted blow-up version of Mustata-Nakamura Conjecture. We also show that if the mld of such a pair is not less than 1, then it is computed by at most one weighted blow-up. As a consequence, ACC of mld holds for such pairs.
Let $X$ be a smooth projective surface and $Delta$ is a normal crossing curve on $X$ such that $K_X+Delta$ is big. We show that the minimal possible volume of the pair $(X, Delta)$ is $frac{1}{143}$ if its (log) geometric genus is positive. Based on this, we establish a Noether type inequality for stable log surfaces, be they normal or non-normal. In the other direction, we show that, if the volume of $(X, Delta)$ is less than $frac{1}{143}$ then $X$ must be a rational surface and the connected components of $Delta$ are trees of smooth rational curves.
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