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Register a new userSingularities with respect to Mather-Jacobian 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.
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Inspired by several works on jet schemes and motivic integration, we consider an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. The resulting invariant, which we call Jacobian discrepancy, is closely related to the jet schemes and the Nash blow-up of the variety. This notion leads to a framework in which adjunction and inversion of adjunction hold in full generality, and several consequences are drawn from these properties. The main result of the paper is a formula measuring the gap between the dualizing sheaf and the Grauert-Riemenschneider canonical sheaf of a normal variety. As an application, we give characterizations for rational and Du Bois singularities on normal Cohen-Macaulay varieties in terms of Jacobian discrepancies. In the case when the canonical class of the variety is Q-Cartier, our result provides the necessary corrections for the converses to hold in theorems of Elkik, of Kovacs, Schwede and Smith, and of Kollar and Kovacs on rational and Du Bois singularities.
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.
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.
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.
L. Moret-Bailly constructed families $mathfrak{C}rightarrow mathbb{P}^1$ of genus 2 curves with supersingular jacobian. In this paper we first classify the reducible fibers of a Moret-Bailly family using linear algebra over a quaternion algebra. The main result is an algorithm that exploits properties of two reducible fibers to compute a hyperelliptic model for any irreducible fiber of a Moret-Bailly family.
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