相关论文: $\Cal M_{15}$ is rationally connected

Any finite conversation can be rationalized.

One of the central aims of the Minimal Model Program is to show that a projective log canonical pair $(X,\Delta)$ with $K_X+\Delta$ pseudoeffective has a good model, i.e.\ a minimal model $(Y,\Delta_Y)$ such that $K_Y+\Delta_Y$ is…

For a quadratic form $\varphi$ over a field of characteristic different from $2$, we study whether its group of proper projective similitudes ${\bf PSim}^+(\varphi)$ is rationally connected (i.e. $R$-trivial). We obtain new sufficient…

We study the following question, asked to us By Pandharipande and Starr: Let $X$ be a rationally connected $3$-fold, and $Y$ be a compact Kaehler $3$-fold symplectically equivalent to it. Is $Y$ rationally connected? We show that the answer…

These are lectures notes on rationally connected varieties, written for the "Etats de la Recherche" of the French Mathematical Society held in Strasbourg (May 2008). We focus on geometric aspects. These notes have been written in order that…

In this paper, combining the works of Miyanishi-Tsunoda and Keel-McKernan, we prove the log Castelnuovo's rationality criterion for smooth quasiprojective surfaces over complex numbers.

We propose an approach for showing rationality of an algebraic variety $X$. We try to cover $X$ by rational curves of certain type and count how many curves pass through a generic point. If the answer is $1$, then we can sometimes reduce…

We show that the intersection of the rational derived series of a one-relator group is rationally perfect and is normally generated by a single element. As a corollary, we characterise precisely when a one-relator group is residually…

Given a proper, rational map of balls, D'Angelo and Xiao introduced five natural groups encoding properties of the map. We study these groups using a recently discovered normal form for rational maps of balls. Using this normal form, we…

The works of Hassett and Kuznetsov identify countably many divisors $C_d$ in the open subset of $\mathbb{P}^{55}=\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^5}(3)))$ parametrizing all cubic 4-folds and conjecture that the cubics corresponding to…

It is shown that the rational convexity of any finite union of disjoint closed balls in C^n follows easily from the results of Duval and Sibony.

In the present article, modeling certain rational numbers, that are represented in terms of Cantor series, are described. The statements on relations between digits in the representations of rational numbers by Cantor series (for the case…

We prove boundedness of rationally-connected threefolds in $\mathbb P^6$ under some extra-assumptions.

We prove the conjecture of Grosse-Kunstleve et al. that coordination sequences of periodic structures in n-dimensional Euclidean space are rational. This has been recently proven by Nakamura et al.; however, our proof is a straightforward…

We define, for smooth projective orbifold pairs $(X,D)$ notions of `slope Rational connectedness', and of orbifold `slope Rational quotient' . These notions extend to this larger context the classical notions of rationally connected…

We prove that the group of homotopy classes of relative homotopy automorphisms of a simply connected finite CW-complex is finitely presented and that the rationalization map from this group to its rational analogue has a finite kernel.

We investigate the possible homological classes of rational curves on the moduli space $X_n=\bar{\mathcal{M}_{0,n}}$ of rational nodal curves with $n$ marked points. In the case of $X_5$ and $X_6$ the relevant homology classes belong to…

It was conjectured by Bott-Grove-Halperin that a compact simply connected Riemannian manifold $M$ with nonnegative sectional curvature is rationally elliptic. We confirm this conjecture under the stronger assumption that $M$ has entire…

The main objective of this paper is to show that the complement of a rational convex set in $\mathbb{C}^n$ is (n-2)-connected for n>2.

Recent results of Hassett, Kuznetsov and others pointed out countably many divisors $C_d$ in the open subset of $\mathbb{P}^{55}=\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^5}(3)))$ parametrizing all cubic 4-folds and lead to the conjecture that…