Related papers: M-curves of degree 9 with three nests

We introduce and study a likely condition that implies the following form of Clemens' conjecture in degrees $d$ between 10 and 24: given a general quintic threefold $F$ in complex $\IP^4$, the Hilbert scheme of rational, smooth and…

We find that the first Hurwitz triplet possesses two distinct arithmetic structures. As Shimura curves $X_1$, $X_2$, $X_3$, whose levels are with norm 13. As non-congruence modular curves $Y_1$, $Y_2$, $Y_3$, whose levels are 7. Both of…

Consider the elliptic curve $E$ given by the Weierstrass equation $y^2 = x^3 - 11x - 14$, which has complex multiplication by the order of conductor $2$ inside $\mathbb{Z}[i]$. It was recently observed in a paper of Daniels and…

The aim of this paper is to give some properties of Hilbert genus fields and construct the Hilbert genus fields of the fields $L_{m,d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$, where $m\geq 3$ is a positive integer and $d$ is a square-free…

A self-avoiding plane-filling curve cannot be periodic, but we show that it can satisfy the local isomorphism property. We investigate three families of coverings of the plane by finite sets of nonoverlapping self-avoiding curves which…

We introduce a class of left ideals (and subalgebras) of nest algebras determined by totally ordered families of partial isometries on a complex Hilbert space $H$. Let $\mathcal{E}$ be a family of partial isometries that is totally ordered…

We develop the theory of resolvent degree, introduced by Brauer \cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this…

We prove that if two germs of irreducible complex analytic curves at $0\in\mathbb{C}^2$ have different sequence of characteristic exponents, then there exists $0<\alpha<1$ such that those germs are not $\alpha$-H\"older homeomorphic. For…

In this paper, an effort is made to classify which prime character degree graphs having eight vertices occur for some finite solvable group. To approach this, we compile known results and constructions from the literature which are used to…

The square peg problem asks whether every continuous curve in the plane that starts and ends at the same point without self-intersecting contains four distinct corners of some square. Toeplitz conjectured in 1911 that this is indeed the…

We prove that a set $\mathcal X\subset \mathbb{C}^2,\ \#{\mathcal X}=mn,\ m\le n, $ is the set of intersection points of some two plane algebraic curves of degrees $m$ and $n,$ respectively, if and only if the following conditions are…

I construct "fake algebraic curves" in $Cp^2$. More precisely, for any k>2, I construct infinitely many pairwise smoothly non-isotopic (and moreover not ambient diffeomorphic) smooth surfaces $F\subset Cp^2$ homeomorphic to a non-singular…

For real irreducible algebraic curves of the seventh degree, there are 22 types of singular points of multiplicity six, 174 types of singular points of multiplicity five, and at least 182 types of singular points of multiplicity four. For…

We consider the question: ``How bad can the deformation space of an object be?'' The answer seems to be: ``Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.'' We show this for a number of…

We consider the problem of classifying quadruples $(K,E,m_1,m_2)$ where $K$ is a number field, $E$ is an elliptic curve defined over $K$ and $(m_1,m_2)$ is a pair of relatively prime positive integers for which the intersection $K(E[m_1])…

We consider unbounded curves without endpoints. Isomorphism is equivalence up to translation. Self-avoiding plane-filling curves cannot be periodic, but they can satisfy the local isomorphism property: We obtain a set $\Omega $ of coverings…

Given positive integers $m_1, m_2, ..., m_n$, and $n$ general points $p_i$ of ${\bf CP}^2$, bounds are given for the least degree $t$ among plane curves passing through each point $p_i$ with multiplicity at least $m_i$, and for the least…

Let n_\delta be the number of \delta-nodal curves lying in a suitably ample complete linear system |L| and passing through appropriately many points on a smooth projective complex algebraic surface. A major open problem is to understand the…

We consider the class $S^m_\perp(\Omega)$ of $m$-dimensional surfaces in $\bar{\Omega} \subset {\mathbb R}^n$ which intersect $S = \partial \Omega$ orthogonally along the boundary. A piece of an affine $m$-plane in $S^m_\perp(\Omega)$ is…

Let F be the cubic field of discriminant -23 and O its ring of integers. Let Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of us (PG and…