Related papers: Fano Hypersurfaces in Positive Characteristic

We develop the quadratic technique of proving birational rigidity of Fano-Mori fibre spaces over a higher-dimensional base. As an application, we prove birational rigidity of generic fibrations into Fano double spaces of dimension…

Iterating the procedure of making a double cover over a given variety, we construct large families of smooth higher-dimensional Fano varieties of index 1. These varieties can be realized as complete intersections in various weighted…

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…

In this work, we show that for a certain class of threefolds in positive characteristics, rational-chain-connectivity is equivalent to supersingularity. The same result is known for K3 surfaces with elliptic fibrations. And there are…

We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain pro-jective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete…

In this paper, we prove that given a flat generically smooth morphism between smooth projective varieties with $F$-pure closed fibers, if the source space is Fano, weak Fano or a variety with the nef anti-canonical divisor, then so is the…

We give a sufficient condition in order that $n$ closed connected subsets in the $n$-dimensional real projective space admit a common multitangent hyperplane.

Complete intersections may be unexpectedly simple over fields of positive characteristic: for instance, they may be unirational despite being of general type. One explanation is given by profiles, structure that tracks the special shape of…

Let X be a Fano manifold such that every rational curve in X has anticanonical degree at least the dimension of X. We prove that X is a projective space or a quadric.

Koll\'ar proved that a very general $n$-dimensional complex hypersurface of degree at least $3\lceil (n+3)/4\rceil$ is not birational to a fibration in rational curves. This is most interesting when the hypersurface is Fano, in which case…

We give a generalization of a result of R. Pandharipande to arbitrary characteristic: We prove that, if $X$ is a convex, separably rationally connected, smooth complete intersection in $\mathbb{P}^N$ over an algebraically closed field of…

This paper works on the structure of infinitely connected Fatou damains of rational maps in terms of Koebe uniformization. Due to the complicated boundary behavior, the existing uniformization results are failed to apply in general. We…

In this paper, the technique of foliations in characteristic $p$ is used to investigate the difference between rational connectedness and separable rational connectedness in positive characteristic. The notion of being freely rationally…

We investigate geometry of D-affine varieties. Our main result is that a D-affine rational projective surface over an algebraically closed field is a generalised flag variety of a reductive group.

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…

This paper is concerned with projective rationally connected surfaces $X$ with canonical singularities and having non-zero pluri-forms, i.e. $(\Omega_X^1)^{[\otimes m]}$ has non-zero global sections for some m > 0, where…

In this paper we prove that a regular foliation on a complex weak Fano manifold is algebraically integrable.

We prove that Fano complete intersections in projective spaces satisfy Conjecture $\mathcal O$ proposed by Galkin-Golyshev-Iritani.

We prove birational superrigidity of generic Fano fiber spaces $V/{\mathbb P}^1$, the fibers of which are Fano complete intersections of index 1 and dimension $M$ in ${\mathbb P}^{M+k}$, provided that $M\geq 2k+1$. The proof combines the…

We provide supplements and open problems related to structure theorems for maximal rationally connected fibrations of certain positively curved projective varieties, including smooth projective varieties with semi-positive holomorphic…