Related papers: Separable rational connectedness and stability

We study some symplectic geometric aspects of rationally connected 4-folds. As a corollary, we prove that any smooth projective 4-fold symplectic deformation equivalent to a Fano 4-fold of pseudo-index at least 2 or a rationally connected…

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…

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.

Let X be a smooth, projective variety defined over a local field K. Following Manin, two K-points of X are called R-equivalent if they can be joined by a rational curve defined over K. The main result of this note shows that if there are…

Let $X$ be a mildly singular Fano variety such that the tangent sheaf is a direct sum. We show that the direct factors are algebraically integrable, so the infinitesimal decomposition induces a product structure on a quasi-\'etale cover of…

We find a relation between a cubic hypersurface $Y$ and its Fano variety of lines $F(Y)$ in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then…

We show that deformations of a surjective morphism onto a Fano manifold of Picard number 1 are unobstructed and rigid modulo the automorphisms of the target, if the variety of minimal rational tangents of the Fano manifold is non-linear or…

We address the problem of classification of contact Fano manifolds. It is conjectured that every such manifold is necessarily homogeneous. We prove that the Killing form, the Lie algebra grading and parts of the Lie bracket can be read from…

Given a morphism between smooth projective varieties $f: W \to X$, we study whether $f$-relatively free rational curves imply the existence of $f$-relatively very free rational curves. The answer is shown to be positive when the fibers of…

A smooth, proper, retract rational variety over a field $k$ is known to be $\mathbb{A}^1$-connected. We improve on this result, in the case when $k$ is infinite, showing that such varieties are naively $\mathbb{A}^1$-connected.

This paper obtains criteria for a Fano variety X with normal crossing singularities defined over an algebraically closed field of characteristic zero, to be smoothable. The difference with the original version is that the theory of…

Chains of minimal degree rational curves have been used as an important tool in the study of Fano manifolds. Their own geometric properties, however, have not been studied much. The goal of the paper is to introduce an infinitesimal method…

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 spaces of rational curves on Fano threefolds with Gorenstein terminal singularities. We generalize the results regarding Geometric Manin's Conjecture for smooth Fano threefolds, including the classification of subvarieties with…

We prove the existence of a family $\mathcal{X}\rightarrow B$ of smooth projective fourfolds, such that the very general fiber $\mathcal{X}_t$ is not stably rational (a fortiori not rational), but some special fibers $\mathcal{X}_t$ are…

The separability problem is formulated in terms of a characterization of a single entanglement witness. More specifically, we show that any (in general multipartite) state \varrho is separable if and only if a specially constructed…

We describe the families of minimal rational curves on any complete symmetric variety, and the corresponding varieties of minimal rational tangents (VMRT). In particular, we prove that these varieties are homogeneous and that for…

We give a necessary and sufficient condition for the nonsingular projective toric variety associated to a finite simple graph to be Fano or weak Fano in terms of the graph.

We show that the degrees of rational endomorphisms of very general complex Fano and Calabi-Yau hypersurfaces satisfy certain congruence conditions by specializing to characteristic p. As a corollary we show that very general n-dimensional…

We prove that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one are rational homogeneous manifolds.