Related papers: On irregular threefolds and fourfolds with numeric…

In this paper we study birational immersions from a very general smooth plane curve to a non-rational surface with $p_g=q=0$ to treat dominant rational maps from a very general surface $X$ of degree$\geq 5$ in ${\mathbb P}^3$ to smooth…

Let $X$ be an irreducible smooth complex projective curve of genus at least two. Let $N$ be a connected component of the moduli space of semistable principal ${\rm PGL}_r({\mathbb C})$- bundles over $X$; it is a normal unirational complex…

In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle.…

We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong K\"ahler with torsion…

We show, using [14], that a smooth projective fibration f : X $\rightarrow$ Y between connected complex quasi-projective manifolds satisfies the equality $\kappa$(X) = $\kappa$(X y) + $\kappa$(Y) of Logarithmic Kodaira dimensions if its…

Let $X$ be a four-dimensional projective variety defined over the field of complex numbers with only terminal singularities. We prove that if the intersection number of the canonical divisor $K$ with every very general curve is positive…

Let f: X \to Z be a surjective morphism of smooth complex projective varieties with connected fibers. Suppose that L is a pseudo-effective divisor on X that is f-numerically trivial. We show that there is a divisor D on Z such that L is…

Let $X^{(n)}$ denote $n$-th symmetric power of a cubic surface $X$. We show that $X^{(4)}\times X$ is stably birational to $X^{(3)}\times X$, despite examples when $X^{(4)}$ is not stably birational to $X^{(3)}$.

We define a categorical birational invariant for minimal geometrically rational surfaces with a conic bundle structure over a perfect field via components of a natural semiorthogonal decomposition. Together with the similar known result on…

In this paper, we classify irregular threefolds with numerically trivial canonical divisors in positive characteristic. For such a variety, if its Albanese dimension is not maximal, then the Albanese morphism will induce a fibration which…

Given a very ample line bundle L on a projective variety X, the syzygy bundle M_L associated to L is the kernel of the evaluation map on sections of L. Our main result is that if X is a smooth projective surface defined over an…

Let B be a nef and big line bundle on a smooth complex threefold X with canonical bundle K. Let x be a point on X and suppose that BC\ge3 for any curve C passing x, B^2S\ge7 for any surface S containing x, and B^3\ge51. Then K+B is spanned…

This is the unabridged web version of the paper that will be published on the American Journal of Mathematics. In this paper, we study the birational geometry of certain examples of mildly singular quartic 3-folds. A quartic 3-fold is an…

In this paper we study pluricanonical maps of minimal projective 3-folds of general type with geometric genus $1$, $2$ and $3$. We go in the direction pioneered by Enriques and Bombieri, and other authors, pinning down, for low projective…

The goal of this paper is to present a modified version (GML) of ML invariant which should take into account rulings over a projective base and allow further stratification of smooth affine rational surfaces. We provide a non-trivial…

Let $f:X\to Y$ be a fibration from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field $k$ of characteristic $p >5$. We prove that if the generic fiber $X_{\eta}$ has big canonical divisor…

For $\epsilon$-lc Fano type varieties $X$ of dimension $d$ and a given finite set $\Gamma$, we show that there exists a positive integer $m_0$ which only depends on $\epsilon,d$ and $\Gamma$, such that both $|-mK_X-\sum_i\lceil mb_i\rceil…

We first state a condition ensuring that having a birational map onto the image is an open property for families of irreducible normal non uniruled varieties. We give then some criteria to ensure general birationality for a family of…

The following conjecture arose out of discussions between B. Harbourne, J. Ro\'e, C. Cilberto and R. Miranda: for a smooth projective surface $X$ there exists a positive constant $c_X$ such that $h^1(\mathcal O_X(C))\le c_X h^0(\mathcal…

In this article we establish the following results: Let $(X, B)$ be a dlt pair, where $X$ is a $\mathbb Q$-factorial K\"ahler $4$-fold -- (i) if $X$ is compact and $K_X+B\sim_{\mathbb Q} D\geq 0$ for some effective $\mathbb Q$-divisor, then…