Related papers: Pseudo-split fibres and arithmetic surjectivity

Let f:S ->B be a relatively minimal fibred surface. In this note we give a partial affirmative answer to a conjecture of Xiao, proving that the direct image of the relative dualizing sheaf of $f$ is ample when the slope of the fibration is…

A variety X over a field K is of Hilbert type if the set of rational points X(K) is not thin. We prove that if f: X\to S is a dominant morphism of K-varieties and both S and all fibers f^{-1}(s), s in S(K), are of Hilbert type, then so is…

Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Koll\'ar proved that, if W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces.…

Around 1960, R. Palais and J. Cerf proved a fundamental result relating spaces of diffeomorphisms and imbeddings of manifolds: If V is a submanifold of M, then the map from Diff(M) to Imb(V,M) that takes f to its restriction to V is locally…

The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov-Witten invariants of X and Gromov-Witten invariants of complete intersections Y in X is established.

We prove a uniform version of the Dynamical Mordell-Lang Conjecture for \'etale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined…

Let $X$ be a quasi projective scheme over a noetherian affine scheme $Spec(A)$, $U\subseteq X$ be an open subset, and $Z=X-U$.Assume that $Z$ is complete intersection, with $k=codim Z$. Consider the map $$ q:{\mathbb K}\left({\mathscr…

Let $k$ be a number field and $U$ a smooth integral $k$-variety. Let $X \to U$ be an abelian scheme. We consider the set $\mathcal{R}$ of rational points $m \in U(k)$ such that the Mordell-Weil rank of the fibre $U_{m}$ is strictly bigger…

Let $X$ be a normal, connected and projective variety over an algebraically closed field $k$. It is known that a vector bundle $V$ on $X$ is essentially finite if and only if it is trivialized by a proper surjective morphism $f:Y\to X$. In…

Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex…

We prove that a Noetherian ring $R$ is a splinter if and only if for every equidimensional surjective morphism $\operatorname{Spec}(S) \to \operatorname{Spec}(R)$, the map $R \to S$ is pure. This yields a large, nontrivial class of ring…

The goal of this work is to pursue the study of pseudo-effective line bundles and vector bundles. Our first result is a generalization of the Hard Lefschetz theorem for cohomology with values in a pseudo-effective line bundle. The Lefschetz…

A finite morphism $f:X\to \mathbb P^2$ of a a smooth irreducible projective surface $X$ is called an almost generic cover if for each point $p\in \mathbb P^2$ the fibre $f^{-1}(p)$ is supported at least on $deg(f)-2$ distinct points and $f$…

We assume that the existence and termination conjecture for flips holds. A complex projective manifold is said to be {\it of almost general type} if the intersection number of the canonical divisor with every very general curve is strictly…

By analogy with algebraic geometry, we define a category of non-linear sheaves (quasi-coherent homotopy-sheaves of topological spaces) on projective toric varieties and prove a splitting result for its algebraic K-theory, generalising…

Let W be a projective variety of dimension n+1, L a free line bundle on W, X in $H^0(L^d)$ a hypersurface of degree d which is generic among those given by sums of monomials from $L$, and let $f : Y \to X$ be a generically finite map from a…

In this paper we study some properties of degenerations of surfaces whose general fibre is a smooth projective surface and whose central fibre is a reduced, connected surface $X \subset IP^r$, $r \geq 3$, which is assumed to be a union of…

We derive the geodesic equation for relatively K\"ahler metrics on fibrations and prove that any two such metrics with fibrewise constant scalar curvature are joined by a unique smooth geodesic. We then show convexity of the log-norm…

In this paper, with the aim of establishing a structure theorem for a compact K\"ahler manifold $X$ with semi-positive holomorphic sectional curvature, we study a morphism $\phi: X \to Y$ to a compact K\"ahler manifold $Y$ with…

One version of the classical Lefschetz hyperplane theorem states that for $U \subset \mathbb P^n$ a smooth quasi-projective variety of dimension at least $2$, and $H \cap U$ a general hyperplane section, the resulting map on \'etale…