Erratum to “Resolution except for minimal singularities I” [Adv. Math. 231 (2012) 3022–3053]

Supplemental Notes for Math 704 . Spring 2012

1.1. Normed spaces. Recall that a (real) vector space V is called a normed space if there exists a function ‖ · ‖ : V → R such that (1) ‖f‖ ≥ 0 for all f ∈ V and ‖f‖ = 0 if and only if f = 0. (2) ‖af‖ = |a| ‖f‖ for all f ∈ V and all scalars a. (3) (Triangle inequality) ‖f + g‖ ≤ ‖f ||+ ‖g‖ for all f, g ∈ V . If V is a normed space, then d(f, g) = ‖f−g‖ defines a metric on V . Convergence w.r.t ...

Homework for Math 6170 §1, Spring 2012

Singularities of Minimal. Surfaces

is interpreted as the potential equation of a hypothetical gas, which yields flows closely approximating adiabatic flows of low Mach number. This interpretation, due to Chaplygin, has been used extensively in recent aerodynamical literature. In the general theory of partial differential equations, finally, equation (1) appears as the simplest nonlinear equation of elliptic type. From the point ...

Resolution Graphs of Some Surface Singularities , I . ( Cyclic Coverings )

The article starts with some introductory material about resolution graphs of normal surface singularities (definitions, topological/homological properties, etc). Then we discuss the problem of N-cyclic coverings (Xf,N , 0) → (X, 0) of (X, 0), branched along ({f = 0}, 0), where f : (X, 0) → (C, 0) is the germ of an analytic function. We present non–trivial examples in order to show that from th...

Minimal Discrepancies of Hypersurface Singularities

We give an upper bound for the minimal discrepancies of hypersurface singularities. As an application, we show that Shokurov’s conjecture is true for log-terminal threefolds.