Singularities — Kagoshima 2017

The following sections are included:

• Cover Photo
• Preface
• Editorial board
• Organization
• Contents

The classical Łojasiewicz inequality says that if f is a germ of an analytic function at $0\in {\mathbb{\text{R}}}^n$, then there exist constants c, α > 0 and a neighborhood U of 0 such that, for all x ∈ U, we have

In this paper we report on some recent results of [DHT], [H], [HNP] giving some analogs of this inequality for smooth functions, both of local and global cases.

This text is based on the notes of the lectures given by the author at the 5th Franco-Japanese-Vietnamese Symposium on Singularities in Kagoshima.

The aim of this article is to present some recent developments in real algebraic geometry. We focus on continuous functions that admit a rational representation. Such functions have some remarkable algebraic and geometric properties.

We study the substitution property for the ring ${\mathcal{K}}^0\left(V\right)$ of continuous rational functions on a real algebraic affine variety V. We show that ${\mathcal{K}}^0\left(V\right)$ satisfies a substitution property along points; moreover, when V is non-singular, it satisfies also a substitution property along Puiseux arcs, which characterizes ${\mathcal{K}}^0\left(V\right)$.

We give a method for constructing a shadowed polyhedron from a divide. The 4-manifold reconstructed from a shadowed polyhedron admits the structure of a Lefschetz fibration if it satisfies a certain property, which is formulated as an LF-structure on a shadowed polyhedron. We will show that the shadowed polyhedron constructed from a divide satisfies this property and the Lefschetz fibration of this polyhedron is isomorphic to the Lefschetz fibration of the divide. Furthermore, applying the same technique to certain free divides we will show that the links of those free divides are fibered with positive monodromy.

In this paper, we construct pairs of plane curves such that they have the same degree and the same singularities. One is a quasi torus curve of type (p, 2) and the other is a torus curve of type (p, 2). For the construction, we use Nagata transformations. If p is even, their complements in ${\mathbb{\text{P}}}^2$ have different topologies. Thus they are a weak Zariski pair.

The following sections are included:

• Introduction
• Basics
• Valuations and blowing ups
• A Complete Set of Key Polynomials
• The order relation on $\mathcal{V}$
• Nonmetric Tree Structure on $\mathcal{V}$

For each n = 2, 3, ..., the higher topological complexity TC_n(X) of a topological space X is a numerical homotopy invariant of the topological space X, defined by Y. Rudyak in [12]. It is a generalization of the topological complexity TC(X) defined by M. Farber in [5] in relation to the problem of motion planning algorithm. In this paper, we compute the higher topological complexity TC_n of configuration spaces of spheres of odd dimension. Our main tool is a method developed in [9] giving the upper bound for the higher topological complexity TC_n. The same method will also be applied to compute TCn of the 2-dimensional torus $\mathbb{\text{T}}$.

In this paper, the tropical delta number δ^trop(T, V) of a vertex V of a tropical plane curve T is defined by the maximal number of locally irreducible components of T at V minus the number of interior lattice points in the polygon Ξ(V) dual to V. It will be proved that if Ξ(V) is prime then the delta invariants of singularities of reduced polynomial maps with Newton polygon Ξ(V) is bounded above by δ^trop(T, V). We then show that a tropical version of the degree-genus formula holds for tropical plane curves that are irreducible and unweighted. We will verify these results using several examples, including our previous works concerning tropical tacnodal curves.

Let (X, o) be a normal complex surface singularity and h a reduced element of the maximal ideal $m_{X,o}$ of ${\mathcal{O}}_{X,o}$. Let $\pi :\left(\tilde{X},E\right)\to \left(X,o\right)$ be the h-minimal good resolution (see Definition 3.1). Let (X_n, o) be the n-fold cyclic covering over (X, o) defined by zⁿ = h. Let ${\pi }_n:\left({\tilde{X}}_n,E\left(n\right)\right)\to \left(X_n,o\right)$ be the h-minimal good resolution. In this paper, we study the maximal ideal cycle on E(n) and embedded points of the pull-back of $m_{X,o}$ onto ${\tilde{X}}_n$. From them, we study the value of mult(X_n, o) (:=the multiplicity of (X_n, o)). Under a condition about h and a defining function of the maximal ideal cycle on E, we prove that mult(X_n, o) takes a constant value for any n with n ≧ N₁(h), where N₁(h) is a positive integer determined by (X, o) and h (see Theorem 3.6 (iv)).

We formulate a conjecture on the motivic McKay correspondence for the group scheme α_p in characteristic p > 0 and give a few evidences. The conjecture especially claims that there would be a close relation between quotient varieties by α_p and ones by the cyclic group of order p.

For generic maps from compact surfaces with boundary into the plane we develop an explicit algorithm for minimizing both the number of cusps and the number of components of the singular locus. More precisely, we minimize among maps with fixed boundary conditions and prescribed singular pattern, by which we mean the combinatorial information of how the 1-dimensional singular locus meets the boundary. Each step of our algorithm modifies the given map only locally by either creating or eliminating a pair of cusps. We show that the number of cusps is an invariant modulo 2 and can be reduced to at most one, and we compute the minimal number of components of the singular locus in terms of the prescribed data. Applications include a discussion of pseudo-immersions as well as the computation of state sums in Banagl’s positive topological field theory.

In this paper we give a survey about what is called Milnor–Hirzebruch class, which is the difference of the motivic Hirzebruch class (of a local complete intersection X in a smooth variety) and the Poincar´e dual of the Hirzebruch cohomology class of the virtual tangent bundle. For that we give a quick review of the motivic Hirzebruch class, and Verdier–Riemann–Roch formula, which is a motivation behind defining the so-called Milnor class. In the final section we discuss how to capture the Milnor–Hirzebruch class as a special value of a natural transformation.

The following section is included:

• Participants