Resolution of Singularities of Germs in Characteristic Positive Associated with Valuation Rings of Iterated Divisor Type

Equivariant Poincaré series of filtrations and topology

Earlier, for an action of a finite group $G$ on a germ of an analytic variety, an equivariant $G$-Poincaré series of a multi-index filtration in the ring of germs of functions on the variety was defined as an element of the Grothendieck ring of $G$-sets with an additional structure. We discuss to which extend the $G$-Poincaré series of a filtration defined by a set of curve or divisorial valuations on the ring of germs of analytic functions in two variables determines the (equivariant) topology of the curve or of the set of divisors.

On the topological type of a set of plane valuations with symmetries

Let $\{C_i : i=1,\ldots,r\}$ be a set of irreducible plane curve singularities. For an action of a finite group $G$, let $Δ^{L}(\{t_{a i}\})$ be the Alexander polynomial in $r\vert G\vert$ variables of the algebraic link $(\bigcup\limits_{i=1}^{r}\bigcup\limits_{a\in G}a C_i )\cap S^3_{\varepsilon}$ and let $ζ(t_1,\ldots, t_r) = Δ^{L}(t_1,\ldots,t_1,t_2,\ldots,t_2, \ldots,t_r,\ldots,t_r)$ with $\vert G\vert$ identical variables in each group. (If $r=1$, $ζ(t)$ is the monodromy zeta function of the function germ $\prod\limits_{a\in G} a^*f$, where $f=0$ is an equation defining the curve $C_1$.)…

驯服的最大权、相对类型与赋值

Tame maximal weights, relative types and valuations

本文引入驯服最大权（Zhou 权），利用 Tian 函数建立相对类型的积分表达式，证明其满足热带性质，从而构成赋值，并用于度量奇异性及推广多重复 Green 函数。

Computing the equisingularity type of a pseudo-irreducible polynomial

Germs of plane curve singularities can be classified accordingly to their equisingularity type. For singularities over C, this important data coincides with the topological class. In this paper, we characterise a family of singularities, containing irreducible ones, whose equisingularity type can be computed in quasi-linear time with respect to the discriminant valuation of a Weierstrass equation.

Families of higher dimensional germs with bijective Nash map

Let (X,0) be a germ of complex analytic normal variety, non-singular outside 0. An essential divisor over (X,0) is a divisorial valuation of the field of meromorphic functions on (X,0), whose center on any resolution of the germ is an irreducible component of the exceptional locus. The Nash map associates to each irreducible component of the space of arcs through 0 on X the unique essential divisor intersected by the strict transform of the generic arc in the component. Nash proved its injectivity and asked if it was bijective. We prove that this is the case if there exists a divisorial resolu…

Poincare series of a toric variety

For an affine toric variety X we compute the Poincare series of the multi-index filtration defined by a finite number of monomial divisorial valuations on the ring O_{X,0}. We give an alternative description of the Poincare series as an integral with respect to the Euler characteristic over the projectivization of the space of germs O_{X,0}. In particular we study divisorial valuations on the ring O_{C^d,0} that arise by considering toric constellations. We give an explicit formula for the Poincare series and a nice geometric description. This generalizes an expression of the Poincare series f…

Ultrametric properties for valuation spaces of normal surface singularities

Let $L$ be a fixed branch -- that is, an irreducible germ of curve -- on a normal surface singularity $X$. If $A,B$ are two other branches, define $u_L(A,B) := \dfrac{(L \cdot A) \: (L \cdot B)}{A \cdot B}$, where $A \cdot B$ denotes the intersection number of $A$ and $B$. Call $X$ arborescent if all the dual graphs of its resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of Płoski by proving that whenever $X$ is arborescent, the function $u_L$ is an ultrametric on the set of branches on $X$ different from $L$. In the present paper we prove that, conve…

Local Fourier transform and blowing up

We consider a resolution of ramified irregular singularities of meromorphic connections on a formal disk via local Fourier transforms. A necessary and sufficient condition for an irreducible connection to have a resolution of the ramified singularity is determined as an analogy of the blowing up of plane curve singularities. We also relate the irregularity of Komatsu and Malgrange of connections to the intersection numbers and the Milnor numbers of plane curve germs. Finally, we shall define an analogue of Puiseux characteristics for connections and find an invariant of the family of connectio…

旗簇的Newton-Okounkov体与组合突变

Newton-Okounkov bodies of flag varieties and combinatorial mutations

本文利用源自Fano流形镜像对称的组合突变框架，通过迭代组合突变建立了旗簇的string polytopes、Nakashima-Zelevinsky polytopes与FFLV polytopes等Newton-Okounkov体之间的联系。

Valuations and plurisubharmonic singularities

We extend to higher dimensions some of the valuative analysis of singularities of plurisubharmonic (psh) functions developed by the last two authors. Following Kontsevich and Soibelman we describe the geometry of the space V of all normalized valuations on C[x_1,...,x_n] centered at the origin. It is a union of simplices naturally endowed with an affine structure. Using relative positivity properties of divisors living on modifications of C^n above the origin, we define formal psh functions on V, designed to be analogues of the usual psh functions. For bounded formal psh functions on V, we def…

New Ideas for Resolution of Singularities in Arbitrary Characteristic

This is the manuscript for Proceedings of International Conference and Workshop on Valuation Theory held at University of Saskachewan, Canada in 1999. I have succeeded in showing that any two-dimensional hypersurface singularities of germs of varieties in any characteristic can be resolved by iterated monoidal transformations with centers in smooth subvarieties. The new proof for the two-dimensional case depends on new ideas. Ideas are essentially different from Abhyankar's one in 1956 and Lipman's one in 1978. It seems to be possible to generalize the new proof into higher dimensional cases, …