Let $V$ be a finite set of divisorial valuations centered at a 2-dimensional regular local ring $R$. In this paper we study its structure by means of the semigroup of values, $S_V$, and the multi-index graded algebra defined by $V$, $\gr_V R$. We prove that $S_V$ is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup.
The comparison between valuations in $V$, the approximation of a reduced plane curve singularity $C$ by families of sets $V^{(k)}$ of divisorial valuations, and the relationship between the value semigroup of $C$ and the semigroups of the sets $V^{(k)}$, allow us to obtain the (finite) minimal generating sequences for $C$ as well as for $V$.
We also analyze the structure of the homogeneous components of $\gr_V R$. The study of their dimensions allows us to relate the Poincaré series for $V$ and for a general curve $C$ of $V$. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A'Campo type for the Poincaré series of $V$. Moreover, the Poincaré series of $C$ could be seen as the limit of the series of $V^{(k)}$, $k\ge 0$.
This paper shows a finiteness property of a divisorial valuation in terms of arcs. First we show that every divisorial valuation over an algebraic variety corresponds to an irreducible closed subset of the arc space. Then we define the codimension for this subset and give a formula of the codimension in terms of "relative Mather canonical class". By using this subset, we prove that a divisorial valuation is determined by assigning the values of finite functions. We also have a criterion for a divisorial valuation to be a monomial valuation by assigning the values of finite functions.
Using the theory of jet schemes, we give a new approach to the description of a minimal generating sequence of a divisorial valuations on $\textbf{A}^2.$ For this purpose, we show how one can recover the approximate roots of an analytically irreducible plane curve from the equations of its jet schemes. As an application, for a given divisorial valuation $v$ centered at the origin of $\textbf{A}^2,$ we construct an algebraic embedding $\textbf{A}^2\hookrightarrow \textbf{A}^N,N\geq 2$ such that $v$ is the trace of a monomial valuation on $\textbf{A}^N.$ We explain how results in this direction …
We study an analytically irreducible algebroid germ (X, 0) of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the divisorial valuations associated to the irreducible components of the exceptional divisor of the normalized blow-up of the normalization (Y, 0) of (X, 0), centered at the point 0 of Y. If (X, 0) is a quasi-ordinary hypersurface singularity, we obtain that the associated graded ring is an algebra of finite type over the field of complex numbers, namely the coordinate ring of a non necessarily normal affine tor…
We consider surfaces $X$ defined by plane divisorial valuations $ν$ of the quotient field of the local ring $R$ at a closed point $p$ of the projective plane $\mathbb{P}^2$ over an arbitrary algebraically closed field $k$ and centered at $R$. We prove that the regularity of the cone of curves of $X$ is equivalent to the fact that $ν$ is non positive on ${\mathcal O}_{\mathbb{P}^2}(\mathbb{P}^2\setminus L)$, where $L$ is a certain line containing $p$. Under these conditions, we characterize when the characteristic cone of $X$ is closed and its Cox ring finitely generated. Equivalent conditions …
To a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincare series of this filtration turnes out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincare series of the corresponding multi-index filtration similar to the one associated …
Given a divisorial discrete valuation 'centered at infinity' on C[x,y], we show that its sign on C[x,y] (i.e. whether it is negative or non-positive on non-constant polynomials) is completely determined by the sign of its value on the 'last key form' (key forms being the avatar of 'key polynomials' of valuations (introduced by [Maclane, 1936]) in 'global coordinates'). The proof involves computations related to the cone of curves on certain compactifications of C^2 and gives a characterization of the divisorial valuations centered at infinity whose 'skewness' can be interpreted in terms of the…
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…
We consider the last value $\hatμ (ν)$ of the vanishing sequence of $H^0(L)$ along a divisorial or irrational valuation $ν$ centered at $\mathcal{O}_{\mathbb{P}^2,p}$, where $L$ resp. $p$ is a line resp. a point of the projective plane $\mathbb{P}^2$ over an algebraically closed field. This value contains, for valuations, similar information as that given by Seshadri constants for points. It is always true that $\hatμ (ν) \geq \sqrt{1 / \mathrm{vol}(ν)}$ and minimal valuations are those satisfying the equality. In this paper, we prove that the Greuel-Lossen-Shustin Conjecture implies a variati…
A notion of Poincaré series was introduced by A.Campillo et al. It was developed by Delgado and Gusein-Zade for a multi-index filtration corresponding to the sequence of blow-ups. The present paper suggests the way to generalize the notion of Poincaré series to the case of arbitrary locally free sheaf on the modification of complex plane $\mathbb{C}^2$. This series is expressed through the topological invariants of the sheaf. For the sheaf of holomorphic functions the answer coincides with the Poincaré series found before.
We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano variety degenerates to a uniquely determined twisted K-polystable Fano variety. We also show that the stability threshold can be approximated by divisorial valuations induced by special test configurations. As an application of the above results and the analytic work of Datar, Székelyhidi, and Ross, we deduce that greatest Ricci lower bounds of Fa…
We prove that the Newton-Okounkov body of the flag $E_{\bullet}:= \left\{ X=X_r \supset E_r \supset \{q\} \right\}$, defined by the surface $X$ and the exceptional divisor $E_r$ given by any divisorial valuation of the complex projective plane $\mathbb{P}^2$, with respect to the pull-back of the line-bundle $\mathcal{O}_{\mathbb{P}^2} (1)$ is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton-Okounkov bodies which…
The equivariant with respect to a finite group action Poincaré series of a collection of $r$ valuations was defined earlier as a power series in $r$ variables with the coefficients from a modification of the Burnside ring of the group. Here we show that (modulo simple exceptions) the equivariant Poincaré series determines the equivariant topology of the collection of valuations.
In this paper, we study the singularities of a pair (X,Y) in arbitrary characteristic via jet schemes. For a smooth variety X in characteristic 0, Ein, Lazarsfeld and Mustata showed that there is a correspondence between irreducible closed cylinders and divisorial valuations on X. Via this correspondence, one can relate the codimension of a cylinder to the log discrepancy of the corresponding divisorial valuation. We now extend this result to positive characteristic. In particular, we prove Mustata's log canonical threshold formula avoiding the use of log resolutions, making the formula availa…
To study singularities on complex varieties we study Poincaré series of filtrations that are defined by discrete valuations on the local ring at the singularity. In all previous papers on this topic one poses restrictions on the centers of these valuations and often one uses several definitions for Poincaré series. In this article we show that these definitions can differ when the centers of the valuations are not zero-dimensional, i.e. do not have the maximal ideal as center. We give a unifying definition for Poincaré series which also allows filtrations defined by valuations that are all non…
To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincaré series, the generalized Poincaré series, and the generalized semigroup Poincaré series. The Hilbert function and the generalized Poincaré series are equivalent in the sense that each of them determines the other one. We show that for a filtration on the ring of germs of holomorphic functions in two variables defined by a collection of plane valuations both of them are equivalent to the generalized semigroup Poincaré s…
We show that there exists a strong connection between the generic formal neighborhood at a rational arc lying in the Nash set associated with a toric divisorial valuation on a toric variety and the formal neighborhood at the generic point of the same Nash set. This may be interpreted as the fact that, analytically along such a Nash set, the arc scheme of a toric variety is a product of a finite dimensional singularity and an infinite dimensional affine space.
This paper gives a map from the set of families of arcs on a variety to the set of valuations on the rational function field of the variety We characterize a family of arcs which corresponds to a divisorial valuation by this map. We can see that both the Nash map and a certain McKay correspondence are the restrictions of this map. This paper also gives the affirmative answer to the Nash problem for a non-normal variety in a certain category. As a corollary, we get the affirmative answer for a non-normal toric variety.
The purpose of this paper is to extend the notions of generalised Poincaré series and divisorial generalised Poincaré series (of motivic nature) introduced by Campillo, Delgado and Gusein-Zade for complex curve singularities to curves defined over perfect fields, as well as to express them in terms of an embedded resolution of curves.
Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It was shown that, except some explicitly described cases, the Alexander polynomial of an algebraic link determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere coincides with the Poincaré series of the corresponding set of curve valuations. The latter one can be defined as an integral over …
For a normalized transcendence degree zero arc valuation v on a nonsingular variety X (with dim X > 1), we describe the maximal irreducible subset C(v) of the arc space of X such that the valuation given by the order of vanishing along a general arc of C(v) equals v. We describe C(v) both algebraically, in terms of the sequence of valuation ideals of v, and geometrically, in terms of the sequence of infinitely near points associated to v. When X is a surface, our construction also applies to any divisorial valuation v, and in this case C(v) coincides with a subset Ein, Lazarsfeld, and Mustata …
We define and study the vanishing sequence along a real valuation of sections of a line bundle on a projective variety. Building on previous work of the first author with Huayi Chen, we prove an equidistribution result for vanishing sequences of large powers of a big line bundle, and study the limit measure. In particular, the latter is described in terms of restricted volumes for divisorial valuations. We also show on an example that the associated concave function on the Okounkov body can be discontinuous at boundary points.
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.
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 $C_1$.)…
In a previous paper, there was defined a multi-index filtration on the ring of functions on a hypersurface singularity corresponding to its Newton diagram generalizing (for a curve singularity) the divisorial one. Its Poincaré series was computed for plane curve singularities non-degenerate with respect to their Newton diagrams. Here we use another technique to compute the Poincaré series for plane curve singularities without the assumption that they are non-degenerate with respect to their Newton diagrams. We show that the Poincaré series only depends on the Newton diagram and not on the defi…
We define a new equivariant (with respect to a finite group $G$ action) version of the Poincaré series of a multi-index filtration as an element of the power series ring ${\widetilde{A}}(G)[[t_1, \ldots, t_r]]$ for a certain modification ${\widetilde{A}}(G)$ of the Burnside ring of the group $G$. We give a formula for this Poincaré series of a collection of plane valuations in terms of a $G$-resolution of the collection. We show that, for filtrations on the ring of germs of functions in two variables defined by the curve valuations corresponding to the irreducible components of a plane curve s…
Let $(X,B)$ be a log canonical pair and $\mathcal{V}$ be a finite set of divisorial valuations with log discrepancy in $[0,1)$. We prove that there exists a projective birational morphism $π\colon Y\rightarrow X$ so that the exceptional divisors are $\mathbb{Q}$-Cartier and correspond to elements of $\mathcal{V}$. We study how two such models are related. Moreover, we provide an application to the study of deformations of log canonical singularities.
For a certain class of varieties X, we derive a formula for the valuation d_{X} on the arc space L(Y) of a smooth ambient space Y, in terms of an embedded resolution of singularities. A simple transformation rule yields a formula for the geometric Poincare series. Furthermore, we prove that for this class of varieties, the arithmetic and the geometric Poincare series coincide. We also study the geometric valuation for plane curves.
We study normal analytic compactifications of C^2 and describe their singularities and configuration of curves at infinity, in particular improving and generalizing results of (Brenton, Math. Ann. 206:303--310, 1973). As a by product we give new proofs of Jung's theorem on polynomial automorphisms of C^2 and Remmert and Van de Ven's result that CP^2 is the only smooth analytic compactification of C^2 for which the curve at infinity is irreducible. We also give a complete answer to the question of existence of compactifications of C^2 with prescribed divisorial valuations at infinity. In partic…
The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincaré sphere: the intersection of the surface $S=\{(z_1,z_2,z_3)\in {\mathbb C}^3: z_1^5+z_2^3+z_3^2=0\}$ with the 5-dimensional sphere ${\mathbb S}_{\varepsilon}^5=\{(z_1,z_2,z_3)\in {\mathbb C}^3: \vert z_1\vert^2+\vert z_2\vert^2+\vert z_3\vert^2=\varepsilon^2\}$. An algebraic link in the Poincaré sphere is the intersection of a germ $(C,0)\subset (S,0)$ of a complex analytic curve in $(S,0)$ with the sphere ${\mathbb S}_{\varepsilon}^3$ of radius $\varepsilon$ …
Given a finite morphism $φ:Y\to X$ of quasi-smooth Berkovich curves over a complete, algebraically closed field $k$ of characteristic $0$, we prove a Riemann-Hurwitz formula relating their Euler-Poincaré characteristics (calculated using De Rham cohomology of their overconvergent structure sheaf). The main tools are $p$-adic Runge's theorem together with valuation polygons of analytic functions. Using the results obtained, we provide another point of view on Riemann-Hurwitz formula for finite morphisms of curves over algebraically closed fields of positive characteristic.
