Searcharxiv⌕ Search

arXiv subjects

D. A. Stepanov

Publications and source records attributed to D. A. Stepanov.

10 recordsLinked to original sources
arXiv2020

Elimination of parasitic solutions in theory of flexible polyhedra

The action of the rotation group $SO(3)$ on systems of $n$ points in the $3$-dimensional Euclidean space $\mathbf{R}^3$ induces naturally an action of $SO(3)$ on $\mathbf{R}^{3n}$. In the present paper we consider the following question: do there exist $3$ polynomial functions $f_1$, $f_2$, $f_3$ on $\mathbf{R}^{3n}$ such that the intersection of the set of common zeros of $f_1$, $f_2$, and $f_3$ with each orbit of $SO(3)$ in $R^{3n}$ is nonempty and finite? Questions of this kind arise when one is interested in relative motions of a given set of $n$ points, i.e., when one wants to exclude the local motions of the system of points as a rigid body. An example is the problem of deciding whether a given polyhedron is non-trivially flexible. We prove that such functions do exist. To get a necessary system of equations $f_1=0$, $f_2=0$, $f_3=0$, we show how starting by choice of a hypersurface in $\mathbf{CP}^{n-1}$ containing no conics, no lines, and no real points one can find such a system.

math.MG↗
arXiv2013

Three-dimensional isolated quotient singularities in odd characteristic

Let a finite group G act linearly on a finite dimensional vector space V over an algebraically closed field k of characteristic p>2. Assume that the quotient V/G is an isolated singularity. In the case when p does not divide the order of G, isolated singularities V/G are completely classified and their classification reduces to Zassenhaus-Vincent-Wolf classification of isolated quotient singularities over the field of complex numbers. In the present paper we show that if dimension of V is 3, then also in the modular case (p divides the order of G) classification of isolated quotient singularities reduces to Zassenhaus-Vincent-Wolf classification. Some remarks on modular quotient singularities in other dimensions and in even characteristic are also given.

math.AG↗
arXiv2013

Universal valued fields and lifting points in local tropical varieties

Let $k$ be a field with a real valuation $ν$ and $R$ a $k$-algebra. We show that there exist a $k$-algebra $K$ and a real valuation $μ$ on $K$ extending $ν$ such that any real ring valuation of $R$ is induced by $μ$ via some homomorphism from $R$ to $K$; $K$ can be chosen to be a field. Then we study the case when $ν$ is trivial and $R$ a complete local Noetherian ring with the residue field $k$. Let $K$ be the ring $\bar{k}[[t^\R]]$ of Hahn series with its natural valuation $μ$; $\bar{k}$ is an algebraic closure of $k$. Despite $K$ is not universal in the strong sense defined above, it has the following weak universality property: for any local valuation $v$ and a finite set of elements $x_1,...,x_n$ of $R$ there exists a homomorphism $f\colon R\to K$ such that $v(x_i)=μ(f(x_i))$, $i=1,...,n$. If $R=k[[x_1,...,x_n]]/I$ for an ideal $I$, this property implies that every point of the local tropical variety of $I$ lifts to a $K$-point of $R$. Similarly, if $R=k[x_1,...,x_n]/I$ is a finitely generated algebra over $k$, lifting points in the tropical variety of $I$ can be interpreted as the weak universality property of the field $\bar{k}((t^\R))$ of Hahn series.

math.AG↗
arXiv2010

Gorenstein isolated quotient singularities over C

In this paper we review the classification of isolated quotient singularities over the field of complex numbers due to H. Zassenhaus, G. Vincent, and G. A. Wolf. As an application we describe Gorenstein isolated quotient singularities over C, generalizing a result of K. Kurano and S. Nishi.

math.AG↗
arXiv2010

Smooth 3-dimensional canonical thresholds

If $X$ is an algebraic variety with at worst canonical singularities and $S$ is a $\Q$-Cartier hypersurface in $X$, the canonical threshold of the pair $(X,S)$ is the supremum of $c\in\R$ such that the pair $(X,cS)$ is canonical. We show that the set of all possible canonical thresholds of the pairs $(X,S)$, where $X$ is a germ of smooth 3-dimensional variety, satisfies the ascending chain condition. We also deduce a formula for the canonical threshold of $(\C^3,S)$, where S is a Brieskorn singularity.

math.AG↗
arXiv2009

A note on resolution of rational and hypersurface singularities

It is well known that the exceptional set in a resolution of a rational surface singularity is a tree of rational curves. We generalize the combinatoric part of this statement to higher dimensions and show that the highest cohomologies of the dual complex associated to a resolution of an isolated rational singularity vanish. We also prove that the dual complex associated to a resolution of an isolated hypersurface singularity is simply connected. As a consequence, we show that the dual complex associated to a resolution of a 3-dimensional Gorenstein terminal singularity has the homotopy type of a point.

math.AG↗
arXiv2006

Combinatorial structure of exceptional sets in resolutions of singularities

The dual complex can be associated to any resolution of singularities whose exceptional set is a divisor with simple normal crossings. It generalizes to higher dimensions the notion of the dual graph of a resolution of surface singularity. The homotopy type of the dual complex does not depend on the choice of a resolution and thus can be considered as an invariant of singularity. In this preprint we show that the dual complex is homotopy trivial for resolutions of 3-dimensional terminal singularities and for resolutions of Brieskorn singularities. We also review our earlier results on resolutions of rational and hypersurface singularities.

math.AG↗
arXiv2004

Non-rational divisors over non-Gorenstein terminal singularities

Let $(X,o)$ be a germ of a 3-dimensional terminal singularity of index $m\geq 2$. If $(X,o)$ has type cAx/4, cD/3-3, cD/2-2, or cE/2, then assume that the standard equation of $X$ in $\mathbb{C}^4/\mathbb{Z}_m$ is non-degenerate with respect to its Newton diagram. Let $π\colon Y\to X$ be a resolution. We show that there are not more than 2 non-rational divisors $E_i$, $i=1,2$, on $Y$ such that $π(E_i)=o$ and discrepancy $a(E_i,X)\leq 1$. When such divisors exist, we describe them as exceptional divisors of certain blowups of $X$ and study their birational type.

math.AG↗
arXiv2004

Non-rational divisors over non-degenerate cDV-points

Let $(X,o)$ be a 3-dimensional terminal singularity of type $cD$ or $cE$ defined in $\mathbb{C}^4$ by an equation non-degenerate with respect to its Newton diagram. We show that there is not more than 1 non-rational divisor $E$ over $(X,o)$ with discrepancy $a(E,X)=1$. We also describe all blowups $σ$ of $(X,o)$ such that $E=\Exc(σ)$ is non-rational and $a(E,X)=1$.

math.AG↗