Related papers: Boundedness of non-birational extremal contraction…

Let $f:X\to S$ be an extremal contraction from a threefolds with terminal singularities onto a surface (so called Mori conic bundle). We study some particular cases of such contractions: quotients of usual conic bundles and index two…

A ring has bounded factorizations if every cancellative nonunit $a \in R$ can be written as a product of atoms and there is a bound $\lambda(a)$ on the lengths of such factorizations. The bounded factorization property is one of the most…

A threefold extremal transition $Y \searrow X$ consists of a crepant extremal contraction $\phi \colon Y \to \bar Y$ with curve class $\ell \in \operatorname{NE}(Y)$, followed by a smoothing $\bar Y\rightsquigarrow X$. We consider the Type…

We classify extremal divisorial contraction which contracts a divisor to a curve from a smooth fourfold. We prove the exceptional divisor is $\Bbb P^2$bundle or quadric bundle over a smooth curve and the contraction is the blowing up along…

We study the birational geometry of a Fano 4-fold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map f: X-->Y, where Y is normal and projective,…

In this article we study conjectures regarding normalized volume and boundedness of singularities. We focus on singularities with a torus action of complexity 1, threefold singularities, and hypersurface singularities. Given a real value…

We answer an open question concerning the boundedness of canonical fiber spaces in high dimensions and prove the following: for any set of integers $n\geq 3$, $0<d<n$ and $N>0$, there exists a nonsingular projective $n$-fold $X$ of general…

We consider a separable compact line $K$ and its extension $L$ consisting of $K$ and a countable number of isolated points. The main object of study is the existence of a bounded extension operator $E: C(K)\to C(L)$. We show that if such an…

We develop a new method for establishing the extremality in the closed cone of effective curves on the moduli space of curves and determine the extremality of many boundary $1$-strata. As a consequence, by using a general criterion for…

In this article we prove the following version of the Weak-BAB conjecture for $3$-folds in char $p>5$: Fix a DCC set $I\subset [0, 1)$ and an algebraically closed field $k$ of characteristic $p>5$. Let $\mathfrak{D}$ be a collection of klt…

In this paper we investigate non-rationality of divisors on 3-fold log Fano fibrations $(X,B)\to Z$ under mild conditions. We show that if $D$ is a component of $B$ with coefficient $\ge t>0$ which is contracted to a point on $Z$, then $D$…

We show boundedness of $3$-folds of $\epsilon$-Fano type with Mori fibration structures. The proof is based on the birational boundedness result in our previous work arXiv:1509.08722 combining with arguments in Kawamata \cite{K} and…

We establish two results on three-dimensional del Pezzo fibrations in positive characteristic. First, we give an explicit bound for torsion index of relatively torsion line bundles. Second, we show the existence of purely inseparable…

In this note, we study the normal compact K\"ahler (possibly singular) threefold $X$ admitting the action of a free abelian group $G$ of maximal rank, all the non-trivial elements of which are of positive entropy. If such $X$ is further…

We prove non-rationality and birational super-rigidity of a Q-factorial double cover X of P^3 ramified along a sextic surface with at most simple double points. We also show that the condition #|Sing(X)| < 15 implies Q-factoriality of X. In…

We give a bound on the minimal number of singularities of a nodal projective complete intersection threefold which contains a smooth complete intersection surface that is not a Cartier divisor.

We study Fano threefolds with~terminal singularities admitting a "minimal" action of a finite group. We prove that under certain additional assumptions such a variety does not contain planes. We also obtain an upper bounds of the number of…

We obtain restrictions on the topology of a closed connected manifold B that bounds a (possibly noncompact) manifold whose interior V admits a complete Riemannian metric of nonpositive sectional curvature. If G denotes the fundamental group…

We present a generic condition for Lorentzian manifolds to have a barrier that limits the reach of boundary-anchored extremal surfaces of arbitrary dimension. We show that any surface with nonpositive extrinsic curvature is a barrier, in…

Let $X$ be a normal projective threefold with mild singularities, and $L_X$ a strictly nef $\mathbb{Q}$-divisor on $X$. First, we show the ampleness of $K_X+tL_X$ with sufficiently large $t$ if either the Kodaira dimension $\kappa(X)\neq 0$…