相关论文: Diffeomorphisms, Isotopoies, and Braid Monodromy F…

Two quasi-projective varieties are called piecewise isomorphic if they can be stratified into pairwise isomorphic strata. We show that the m-th symmetric power $S^m(C^n)$ of the complex affine space $C^n$ is piecewise isomorphic to $C^{mn}$…

Let $C_{s,t}$ be the complete bipartite geometric graph, with $s$ and $t$ vertices on two distinct parallel lines respectively, and all $s t$ straight-line edges drawn between them. In this paper, we show that every complete bipartite…

We consider Poncelet pairs $(S,C)$, where $S$ is a smooth conic and $C$ is a degree$-c$ plane curve having the Poncelet property with respect to $S$. We prove that for $c>4$ the projection $(S,C)\mapsto C$ is generically one-to-one and use…

Let $\C$ be a genus 2 curve defined over $k$, $char (k) =0$. If $\C$ has a $(3,3)$-split Jacobian then we show that the automorphism group $Aut(\C)$ is isomorphic to one of the following: $\bZ_2, V_4, D_8$, or $D_{12}$. There are exactly…

Let $C_1,C_2\subseteq\mathbb{G}_m^N(\mathbb{C})$ be irreducible closed algebraic curves, with $N\geq 3$. Suppose $C_1$ is not contained in an algebraic subgroup of $\mathbb{G}_m^N(\mathbb{C})$ of dimension $1$ and $C_1\cup C_2$ is not…

A pair $(S,C)$ is called a singular $\mathbb{Q}$-homology plane pair if $S$ is a singular projective surface with only quotient singularities having the same rational homology as $\mathbb{p}^2$ and $C \subset S$ has the same rational…

In the previous paper [E-print alg-geom/9507004] we classified the rational cuspidal plane curves C with a cusp of multiplicity deg C - 2. In particular, we showed that any such curve can be transformed into a line by Cremona…

It follows from a general theorem of Bonk and Eremenko that closed plane curves which are contractible in the complement to the integral lattice satisfy a linear isoperimetric inequality. We give an alternative proof of this fact. Our…

We show that all the possible pairs of integers occur as exponents for free or nearly free irreducible plane curves and line arrangements, by producing only two types of simple families of examples. The topology of the complements of these…

Let $\mathcal{C}_d\subset \mathbb{C}^{d+1}$ be the space of non-singular, univariate polynomials of degree $d$. The Vi\`{e}te map $\mathscr{V} : \mathcal{C}_d \rightarrow Sym_d(\mathbb{C})$ sends a polynomial to its unordered set of roots.…

We prove that any two irreducible cuspidal Hurwitz curves $C_0$ and $C_1$ (or more generally, curves with A-type singularities) in the Hirzebruch surface $F_N$ with coinciding homology classes and sets of singularities are regular…

We show that, up to a natural equivalence relation, the only non-trivial, non-identity holomorphic maps $\mathrm{Conf}_n\mathbb{C}\to\mathrm{Conf}_m\mathbb{C}$ between unordered configuration spaces, where $m\in\{3,4\}$, are the resolving…

Let A^2 denote the affine plane over an algebraically closed field of arbitrary characteristic. Besides contributing several new results in the general theory of birational endomorphisms of A^2, this article describes certain classes of…

We develop a general theory of `quantum' diffeomorphism groups based on the universal comeasuring quantum group $M(A)$ associated to an algebra $A$ and its various quotients. Explicit formulae are introduced for this construction, as well…

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…

Let $n,m\in \mathbb{N}$, and let $B_{n,m}(\mathbb{R}P^2)$ be the set of $(n + m)$-braids of the projective plane whose associated permutation lies in the subgroup $S_n\times S_m$ of the symmetric group $S_{n+m}$. We study the splitting…

This work presents the conjugacy classes of finite abelian subgroups of the Cremona group of the plane. Using a well-known theory, this problem amounts to the study of automorphism groups of some Del Pezzo surfaces and conic bundles. We…

In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex…

We introduce (weak) oddomorphisms of graphs which are homomorphisms with additional constraints based on parity. These maps turn out to have interesting properties (e.g., they preserve planarity), particularly in relation to homomorphism…

In this article we consider smooth projective curves $C$ of genus two described by integral equations of the form $y^2=xh(x)$, where $h(x)\in\mathbb{Z}[x]$ is monic of degree $4$. It turns out that if $h(x)$ is reducible, then the absolute…