Related papers: $\Cal M_{15}$ is rationally connected

In this paper, we show that projective globally $F$-regular threefolds, defined over an algebraically closed field of characteristic $p\geq 11$, are rationally chain connected.

We establish a conjecture of Mumford characterizing rationally connected complex projective manifolds in several cases.

We show that an $n$-dimensional compact K\"ahler manifold $X$ admitting a non-degenerate meromorphic map $f:{\bf C}^n\to X$ of order $\rho_f<2$ is rationally connected.

Motivated by several recent results on the geometry of the moduli spaces $\bar{\Cal M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points, here we determine their birational structure for small values of $g$ and $n$ by exploiting…

We prove a conjecture of V. V. Shokurov which in particular implies that the fibers of a resolution of a variety with divisorial log terminal singularities are rationally chain connected.

We extend Hacon--M\textsuperscript{c}Kernan's rational chain connectedness theorem to the complex analytic setting. As a consequence, we prove that the fibers of any resolution of singularities of complex analytic kawamata log terminal…

We show that when $m>n$, the space of $m\times n$-matrix-valued rational inner functions in the disk is path connected.

These expository notes discuss the arithmetic of rationally connected varieties. Detailed proofs of theorems of Koll\'ar, of Koll\'ar and Szab\'o and of Esnault about rationally connected varieties over finite fields and local fields are…

In this paper, we give an affirmative answer to a conjecture in the Minimal Model Program. We prove that log $Q$-Fano varieties of dim $n$ are rationally connected. We also study the behavior of the canonical bundles under projective…

We prove that every curve on a rationally connected variety is algebraically equivalent to a (non-effective) integral sum of rational curves.

The notion of 'slope rational connectedness' is introduced in the context of smooth orbifold pairs. The main result parallels the characterization of the rational connectedness of projective manifolds in terms of either the non-existence of…

In this short note we prove that in many cases the failure of a variety to be separably rationally connected is caused by the instability of the tangent sheaf (if there are no other obvious reasons). A simple application of the results…

We consider rationally connected complex projective manifolds M and show that their loop spaces--infinite dimensional complex manifolds--have properties similar to those of M. Furthermore, we give a finite dimensional application concerning…

Let U be an open subset of a unirational variety (or more generally of a separably rationally connected variety). We prove that there is rational curve C in U such that the fundamental group of C surjects onto the fundamental group of U.…

In this paper the notion of rational simple connectedness for the quintic Fano threefold $V_5\subset \mathbb{P}^6$ is studied and unirationality of the moduli spaces $\overline{M}_{0,0}^{\text{bir}}(V_5,d)$, with $d \ge 1$, is proved. Many…

We study the connectedness of the real locus of smooth geometrically rational Fano threefolds and prove a sufficient criterion of $\mathbb{R}$-rationality.

Based on M. Hall's theorem we prove a simple result dealing with real numbers which admit exact approximations by rationals.

We give another proof of a result of Adamczewski and Bell concerning Mahler equations: A formal power series satisfying a $p-$ and a $q-$Mahler equation over ${\mathbb C}(x)$ with multiplicatively independent positive integers $p$ and $q$…

We prove that globally $+$-regular varieties are rationally chain connected in dimension three and mixed characteristic with residue field characteristic $p>5$. We also introduce a notion of strongly globally $+$-regular, and show that…

We provide an alternative proof that the finite rational linear combination of radicals, under certain constraint, are linearly independent over $\mathbb{Q}$.