Singularities in arbitrary characteristic via jet schemes
Shihoko Ishii,
Ana Reguera
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.
“…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)…”
Section: Note That A(ementioning
confidence: 70%
“…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 , .…”
Section: 4mentioning
confidence: 99%
“…, 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…”
Section: 4mentioning
confidence: 99%
“…Theorem 2.17 (inversion of adjunction, [2, Theorem 8.1], [10,Theorem 3.23]). Let f be a non-zero element of (x 1 , .…”
In this paper we characterize two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic from the viewpoint of the initial term of the defining equation. As an application, we prove a conjecture about a uniform bound of divisors computing minimal log discrepancies for two dimensional varieties, which is a conjecture by Ishii and also a special case of the conjecture by Mustat ¸ǎ-Nakamura.
“…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)…”
Section: Note That A(ementioning
confidence: 70%
“…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 , .…”
Section: 4mentioning
confidence: 99%
“…, 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…”
Section: 4mentioning
confidence: 99%
“…Theorem 2.17 (inversion of adjunction, [2, Theorem 8.1], [10,Theorem 3.23]). Let f be a non-zero element of (x 1 , .…”
In this paper we characterize two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic from the viewpoint of the initial term of the defining equation. As an application, we prove a conjecture about a uniform bound of divisors computing minimal log discrepancies for two dimensional varieties, which is a conjecture by Ishii and also a special case of the conjecture by Mustat ¸ǎ-Nakamura.
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
In this paper, we study the singularities of a general hyperplane section H of a three-dimensional quasi-projective variety X over an algebraically closed field of characteristic p > 0. We prove that if X has only canonical singularities, then H has only rational double points. We also prove, under the assumption that p > 5, that if X has only klt singularities, then so does H.
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