Related papers: The Artinian Berger Conjecture

A conjecture of Berger states that, for any simply connected Riemannian manifold all of whose geodesics are closed, all prime geodesics have the same length. We firstly show that the energy function on the free loop space of such a manifold…

We introduce a new approach to the geometric Bombieri--Lang conjecture for hyperbolic varieties in characteristic 0. The main idea is to construct an entire curve on a special fiber of a variety over a complex function field from an…

Circular arc graphs are graphs whose vertices can be represented as arcs on a circle such that any two vertices are adjacent if and only if their corresponding arcs intersect. Proper circular arc graphs are graphs which have a circular arc…

We introduce and motivate a conjecture about the existence of complete, 1-dimensional families of covers of an elliptic curve. If the conjecture holds, then it would imply a uniform lower bound of 5 for slope of the moduli space of curves.…

I give a conjectural generating function for the numbers of $\delta$-nodal curves in a linear system of dimension $\delta$ on an algebraic surface. It reproduces the results of Vainsencher for the case $\delta\le 6$ and Kleiman-Piene for…

We describe applications of Koszul cohomology to the Brill-Noether theory of rank 2 vector bundles. Among other things, we show that in every genus g>10, there exist curves invalidating Mercat's Conjecture for rank 2 bundles. On the other…

In this work, we refine a formula for the Tjurina number of a reducible algebroid plane curve defined over $\mathbb C$ obtained in the more general case of complete intersection curves in [1]. As a byproduct, we answer the affirmative to a…

The long-standing Auslander and Reiten Conjecture states that a finitely generated module over a finite-dimensional algebra is projective if certain Ext-groups vanish. Several authors, including Avramov, Buchweitz, Iyengar, Jorgensen,…

We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, any product of Jacobians of smooth projective curves over the complex numbers…

An extension $B\subset A$ of finite dimensional algebras is bounded if the $B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is finite and $\mathrm{Tor}_i^B(A/B, (A/B)^{\otimes_B j})=0$ for all $i, j\geq 1$. We show…

We investigate the birational section conjecture for curves over function fields of characteristic zero and prove that the conjecture holds over finitely generated fields over Q if it holds over number fields.

Suppose $E$ is an elliptic curve over $\mathbb{Q}$ and $\chi$ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that $L(E,\chi,1) = 0$. Via the Birch and Swinnerton-Dyer…

In a recent work, Kai Tang conjectured that any compact Hermitian manifold with non-zero constant mixed curvature must be K\"ahler. He confirmed the conjecture in complex dimension $2$ and for Chern K\"ahler-like manifolds in general…

The Jacobian conjecture over a field of characteristic zero is considered directly in view of the nonlinear partial differential equations it is associated with. Exploring the integrals of such partial differential equations, this work…

A purity conjecture due to Grothendieck and Auslander--Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension $\ge 2$. The combination of several works of Gabber settles…

We give an explicit uniform result on the Mordell conjecture for non-isotrivial curves over function field of characteristic 0. The proof is based on Vojta's method, and make use of Zhang's admissible adelic line bundles and a quantitative…

The Breuil-M\'{e}zard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" in the moduli space of mod $p$ Galois representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_q/\mathbb{Q}_q)$ that should govern congruences…

Let $k$ be an arbitrary field. The main aim of this paper is to prove the Tits-Weiss conjecture for Albert division algebras over $k$ which are pure first Tits constructions. This conjecture asserts that for an Albert division algebra $A$…

Let $C$ be a curve of genus $g$. A fundamental problem in the theory of algebraic curves is to understand maps $C \to \mathbb{P}^r$ of specified degree $d$. When $C$ is general, the moduli space of such maps is well-understood by the main…

Rational points in the boundary of a hyperbolic curve over a field with sufficiently nontrivial Kummer theory are the source for an abundance of sections of the fundamental group exact sequence. We follow and refine Nakamura's approach…