Related papers: A note on lc-trivial fibrations

Let $X$ be an irreducible projective variety and $f$ a morphism $X \rightarrow \mathbb{P}^n$. We give a new proof of the fact that the preimage of any linear variety of dimension $k\ge n+1-\dim f(X)$ is connected. We prove that the…

Suppose $(X, \Delta)$ is a log-$\bQ$-Gorenstein pair. Recent work of M. Blickle and the first two authors gives a uniform description of the multiplier ideal $\mJ(X;\Delta)$ (in characteristic zero) and the test ideal $\tau(X;\Delta)$ (in…

It is well known that the general fibers of a fibration $f\colon X\to B$ are isomorphic if the general Kodaira-Spencer class vanishes. In this paper we consider the birational analogue when the general Kodaira-Spencer class is supported on…

Let $X$ be a general divisor in $|3M+nL|$ on the rational scroll $\mathrm{Proj}(\oplus_{i=1}^{4}\mathcal{O}_{\mathbb{P}^{1}}(d_{i}))$, where $d_{i}$ and $n$ are integers, $M$ is the tautological line bundle, $L$ is a fibre of the natural…

We prove the following theorem: Fibered Power Theorem: Let $X\rar B$ be a smooth family of positive dimensional varieties of general type, with $B$ irreducible. Then there exists an integer $n>0$, a positive dimensional variety of general…

Following ideas of Nori, Biswas, ..., we show that given an integer r>0, a noetherian scheme X, and an effective Cartier divisor D on it, the parabolic vector bundles on (X,D) with weights multiples of 1/r (in the sense of…

Let $\mathcal{X}$ be a finite-dimensional complex vector space and let k be a positive integer. An explicit formula for the k-reflexivity defect of the image of a generalized derivation on $L(\mathcal{X})$, the space of all linear…

Given a Fano type log Calabi-Yau fibration $(X,B)\to Z$ with $(X,B)$ being $\epsilon$-lc, the first author in \cite{Bi23} proved that the generalised pair $(Z,B_Z+M_Z)$ given by the canonical bundle formula is generalised $\delta$-lc where…

We prove a weak version of a conjecture of Matsushita saying that for a Lagrangian fibration on a hyper-Kaehler manifold $X$, the moduli map for the fibers is either generically of maximal rank or constant. Assuming the base is smooth and…

Let $X$ be a Cantor set, and let $A$ be a unital separable simple amenable $C$*-algebra with tracial rank zero which satisfies the Universal Coefficient Theorem, we use $C(X,A)$ to denote the set of all continuous functions from $X$ to $A$,…

Let $\Delta(n)$ denote the smallest positive integer $m$ such that $a^3+a(1\le a\le n)$ are pairwise distinct modulo $m$. The purpose of this paper is to determine $\Delta(n)$ for all positive integers $n$.

Let $X$ be a non-singular projective threefold over an algebraically closed field of any characteristic, and let $A^2(X)$ be the group of algebraically trivial codimension 2 algebraic cycles on $X$ modulo rational equivalence with…

In this paper we study a special class of fibrations on Delsarte surfaces. We call these fibrations Delsarte fibrations. We show that after a specific cyclic base change the fibration is the pull back of a fibration with three singular…

In this article we prove the following boundedness result: Fix a DCC set $I\subset [0, 1]$. Let $\mathfrak{D}$ be the set of all log pairs $(X, \Delta)$ satisfying the following properties: (i) $X$ is a projective surface defined over an…

A positive integer $n$ is said to be a Zumkeller number if the positive divisors of $n$ can be partitioned into two disjoint subsets of equal sum \cite{zumkeller}. In this paper, in the first section, we investigate differences between…

Let $G$ be an additive finite abelian group with exponent $\exp(G)=m$. For any positive integer $k$, the $k$-th generalized Erd\H{o}s-Ginzburg-Ziv constant $\mathsf s_{km}(G)$ is defined as the smallest positive integer $t$ such that every…

In this short note, we will show the following weak evidence of S. Lang conjecture over function fields. Let f : X ---> Y be a projective and surjective morphism of algebraic varieties over an algebraically closed field k of characteristic…

Let $X$ be a projective klt threefold in characteristic $p>5$ and let $L$ be a nef Cartier divisor on $X$. We show that $H^1(X, -L)=0$ for the following two cases: (1) $K_X$ is not big and $L$ is big; (2) $-K_X$ is nef and $L$ is of…

We propose a variant of the effective adjunction conjecture for lc-trivial fibrations. This variant is suitable for inductions and can be used to treat real coefficients.

In this study, we give an alternative and elementary proof to Tsuji's criterion for a Cartier divisor to be numerically trivial.