Mather discrepancy and the arc spaces

Abstract

A goal of this paper is a characterization of singularities according to a new invariant, Mather discrepancy. We also show some evidences convincing us that Mather discrepancy is a reasonable invariant in a view point of birational geometry.

Corrigendum to “Mather discrepancy and the arc spaces”

“…Then, by the same argument in the corresponding part of the proof in the paper [2], we obtain dim g(…”

“…-The proof in [2] of equivalence among (i), (ii) and (iii) is not affected by the change in (v). The implication (iii) ⇒ (iv) is obvious.…”

“…In this paper, we make a correction of the statement of Theorem 4.7 in [2], where (v) was misstated as:…”

Towards a link theoretic characterization of smoothness

“…The proof of this theorem relies on a theorem of Ishii on minimal Mather log discrepancies [Ish13]. The bound mld x (X) ≤ n in the setting of the theorem is a direct application of Ishii's result, and our contribution is to observe that equality holds only if X is smooth at x, a property we deduce by looking at the Nash blow-up of X.…”

“…However, the theorem is stated incorrectly, and the corrected statement does apply. We have been informed by the author of [Ish13] that an Erratum is in the process of being submitted.…”