Related papers: $(d,\textbf{\sigma})$-Veronese variety and some ap…

Let $K/F$ be a quadratic extension of $p$-adic fields, $\sigma$ the nontrivial element of the Galois group of $K$ over $F$, and $\pi$ a quasi-square-integrable representation of $GL(n,K)$. Denoting by $\pi^{\vee}$ the smooth contragredient…

A set $\mathcal{S}$ of points in $\mathbb{R}^n$ is called a rationally parameterisable hypersurface if $\mathcal{S}=\{\boldsymbol{\sigma}(\mathbf{t}): \mathbf{t} \in D\}$, where $\boldsymbol{\sigma}: \mathbb{R}^{n-1} \rightarrow…

The versal deformation space of a smooth rational curve in a smooth complex threefold is explicitly computed under certain hypotheses. Under an additional hypothesis, the versal deformation space is then shown to be the variety of critical…

In this paper, we prove three related results; (1) Extension of our result in [10] to all generic hypersurfaces. More precisely, the normal sheaf of a generic rational map $c_0$ to a generic hypersurface $X_0$ of $\mathbf P^n, n\geq 4$ has…

We show that all the automorphisms of the symmetric product Sym^d(X), d>2g-2, of a smooth projective curve X of genus g>2 are induced by automorphisms of X.

In this paper, we study the geometry of $GT-$varieties $X_{d}$ with group a finite cyclic group $\Gamma \subset \mathrm{GL}(n+1,\mathbb{K})$ of order $d$. We prove that the homogeneous ideal $\mathrm{I}(X_{d})$ of $X_{d}$ is generated by…

We consider smooth plane curves $\mathcal{X}$ of degree $d\geq4$, defined over an algebraically closed field of characteristic $0$, that possess a unique outer Galois point. This geometric condition forces the curve to be a cyclic covering…

We characterize the double Veronese embedding of P^n as the only variety that, under certain general conditions, can be isomorphically projected from the Grassmannian of lines in P^{2n+1} to the Grassmannian of lines in P^{n+1}.

We study the elliptic curve E given by y^2=x(x+1)(x+t) over the rational function field k(t) and its extensions K_d=k(\mu_d,t^{1/d}). When k is finite of characteristic p and d=p^f+1, we write down explicit points on E and show by…

An automorphism of a graph $G$ with $n$ vertices is a bijective map $\phi$ from $V(G)$ to itself such that $\phi(v_i)\phi(v_j)\in E(G)$ $\Leftrightarrow$ $v_i v_j\in E(G)$ for any two vertices $v_i$ and $v_j$ of $G$. Denote by…

In this paper we introduce generalized pseudo-quadratic forms and develope some theory for them. Recall that the codomain of a $(\sigma,\varepsilon)$-quadratic form is the group $\overline{K} := K/K_{\sigma,\varepsilon}$, where $K$ is the…

We prove and organize some results on the normal forms of Hermitian operators composed with the Veronese map. We apply this general framework to prove two specific theorems in CR geometry. First, extending a theorem of Faran, we classify…

In this paper we develop a bridge between model theory, geometric topology, and geometric group theory. In particular, we investigate the Ivanov Metaconjecture from the point of view of model theory, and more broadly we seek to answer the…

Given a knot K in an Euclidean space E and a finite dimensional space V of smooth functions on K, we express the expected number of critical points of a random function in V in terms of an integral-geometric invariant of K and V. When V…

Given integers $d\ge 3$ and $N\ge 3$. Let $G$ be a finite abelian group acting faithfully and linearly on a smooth hypersurface of degree $d$ in the complex projective space $\mathbb{P}^{N-1}$. Suppose $G\subset PGL(N, \mathbb{C})$ can be…

We study vector bundles on flag varieties over an algebraically closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(2\le d\leq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where…

The orientation morphism $Or(\cdot)(P)\colon \gamma\mapsto\dot{P}$ associates differential-polynomial flows $\dot{P}=Q(P)$ on spaces of bi-vectors $P$ on finite-dimensional affine manifolds $N^d$ with (sums of) finite unoriented graphs…

The shadow of an abstract simplicial complex $K$ with vertices in $\mathbb{R}^N$ is a subset of $\mathbb{R}^N$ defined as the union of the convex hulls of simplices of $K$. The Vietoris--Rips complex of a metric space $(S,d)$ at scale…

Let $ k $ be a field, $ G $ a totally ordered abelian group and $ \mathbb K = k((G)) $ the maximal field of generalised power series, endowed with the canonical valuation $ v $. We study the group $ v \mathrm{-Aut} K $ of valuation…

In this paper we generalize some of these results for loop algebras and groups as well as for the Virasoro algebra to the two-dimensional case. We define and study a class of infinite dimensional complex Lie groups which are central…