相关论文: Rationality of Fano threefolds over non-closed fie…

We study cohomological obstructions to equivariant unirationality, with special regard to actions of finite groups on del Pezzo surfaces and Fano threefolds.

Let $V_1$ be the Fano threefold given as a hypersurface of degree 6 in $P(1,1,1,2,3)$ (over a number field $K$). Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points of $X$ is Zariski dense.

We prove that a smooth Fano hypersurface $V=V_M\subset{\Bbb P}^M$, $M\geq 6$, is birationally superrigid. In particular, it cannot be fibered into uniruled varieties by a non-trivial rational map and each birational map onto a minimal Fano…

The purpose of this paper is to show that the third unramified cohomology with divisible coefficients of a smooth projective geometrically rational threefold over a finite field must vanish under $\Z_{\ell}$-exactness Hard Lefschetz…

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…

We study rational functions over finite fields under PGL-equivalence. We say that $f, g \in \Bbb F_q(X)$ are \emph{equivalent} if there exist $\psi, \phi \in \Bbb F_q(X)$ of degree one such that $g = \psi \circ f \circ \phi$. Most…

As a generalization of the Mukai conjecture, we conjecture that the Fano manifolds $X$ which satisfy the property $\rho_X(r_X-1)\geq\dim X-1$ have very special structure, where $\rho_X$ is the Picard number of $X$ and $r_X$ is the index of…

In this paper, we prove the canonical bundle formula for Fano type fibrations and Shokurov's conjecture on boundedness of complements for Fano type threefold pairs $(X,B)$ with fibration structures in large characteristics. In particular,…

The goal of this work is to study geometric properties of geometrically irreducible subschemes on degenerations of Fano varieties (more generally, of separably rationally connected varieties). It is known that these geometrically…

It is proved that a general Fano hypersurface of index 1 (in the projective space) with isolated singularities of general position is birationally rigid. Therefore it cannot be fibered into uniruled varieties of a smaller dimension by a…

We investigate birationality of the anti-pluricanonical map $\phi_{-m}$, the rational map defined by the anti-pluricanonical system $|-mK|$, on $\mathbb{Q}$-Fano threefolds.

In this paper, we give a partial affirmative answer to the BAB conjecture for $3$-folds in characteristic $p>5$. Specifically, we prove that a set $\mathcal{D}$ of weak Fano $3$-folds over an uncountable algebraically closed field is…

We give a classification of Fano threefolds $X$ with canonical Gorenstein singularities such that $X$ possess a regular involution, which acts freely on some smooth surface in $|-K_X|$, and the linear system $|-K_X|$ gives a morphism which…

The generating series of descendent invariants of stable pairs on 3-folds is conjectured to be rational and to satisfy a $q\leftrightarrow q^{-1}$ symmetry. We prove this conjecture for Fano 3-folds. We utilize the same path of stability…

We present a simple proof that finding a rank-$R$ canonical polyadic decomposition of a 3-dimensional tensor over a finite field $\mathbb{F}$ is fixed-parameter tractable with respect to $R$ and $\mathbb{F}$. We also show a nontrivial upper…

We classify three-dimensional nodal Fano varieties that are double covers of smooth quadrics branched over intersections with quartics acted on by finite simple non-abelian groups, and study their rationality.

We prove that a general Fano fibration $\pi\colon V\to {\mathbb P}^1$, the fiber of which is a double Fano hypersurface of index 1, is birationally superrigid provided it is sufficiently twisted over the base. In particular, on $V$ there…

It is conjectured that a Fano manifold of Picard number 1 which is not a projective space admits no endomorphisms of degree bigger than 1. Beauville confirmed this for hypersurfaces of projective space. We study this problem for…

We compute global log canonical thresholds of certain birationally bi-rigid Fano 3-folds embedded in weighted projective spaces as complete intersections of codimension 2 and prove that they admit an orbifold K\"{a}hler-Einstein metric and…

We study K-stability of smooth Fano threefolds of Picard rank $2$ and degree $22$ which can be obtained by blowing up a smooth complete intersection of two quadrics in $\mathbb{P}^5$ along a conic. We also describe the automorphism groups…