Related papers: Numerical properties of isotrivial fibrations

Let $X$ be a minimal projective Gorenstein 3-fold of general type. We give two applications of an inequality between $\chi (\omega_X)$ and $p_g(X)$: 1) Assume that the canonical map $\Phi_{|K_X|}$ is of fiber type. Let $F$ be a smooth model…

Let f : X -> S be any elliptic fibration. If X has dimension 3 and is not uniruled, then X has a minimal model (with terminal singularities) [Mori]. In earlier work we have shown that there exists a birationally equivalent elliptic…

Let $S$ be a smooth minimal surface of general type with a (rational) pencil of hyperelliptic curves of minimal genus $g$. We prove that if $K_S^2<4\chi(\mathcal O_S)-6,$ then $g$ is bounded. The surface $S$ is determined by the branch…

We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves $B$ in smooth projective surfaces $X$ defined over an algebraically closed field of characteristic zero. We observe that…

Let $f:X \to \mathbb{P}^1$ be a non-isotrivial family of semi-stable curves of genus $g\geq 1$ defined over an algebraically closed field $k$ with $s_{nc}$ singular fibers whose Jacobians are non-compact. We prove that $s_{nc}\geq 5$ if…

We construct examples of non-isotrivial algebraic families of smooth complex projective curves over a curve of genus 2. This solves a problem from Kirby's list of problems in low-dimensional topology. Namely, we show that 2 is the smallest…

The Noether-Horikawa surfaces are the minimal surfaces S with K^2=2p_g-4. For 8 | K^2 they belong to two families of respective type C and N (connected, resp. non connected branch locus for the canonical map). For 16 | K^2 the two types are…

In this paper, we classify the minimal surfaces of general type with $\chi=5$, $K^{2}=9$ whose canonical map is composed with an involution. We obtain 6 families, whose dimensions in the moduli space are 28, 27, 33, 32, 31 and 32…

Let $f: X \to B$ be a relatively minimal fibration of maximal Albanese dimension from a variety $X$ of dimension $n \ge 2$ to a curve $B$ defined over an algebraically closed field of characteristic zero. We prove that $K_{X/B}^n \ge 2n!…

A viable and still unproved conjecture states that, if $X$ is a smooth algebraic surface and $C$ is a smooth algebraic curve in $X$, then $C$ realizes the smallest possible genus amongst all smoothly embedded $2$-manifolds in its homology…

In this part of the series, we shall investigate Deligne-Mumford semistable reductions from the point of view of numerical invariants. As an application, we obtain two numerical criterions for a base change to be stabilizing, and for a…

We call a projective surface $X$ mixed quasi-\'etale quotient if there exists a curve $C$ of genus $g(C)\geq 2$ and a finite group $G$ that acts on $C\times C$ exchanging the factors such that $X=(C\times C)/G$ and the map $C\times C…

Let $X$ be a projective variety with log terminal singularities and vanishing augmented irregularity. In this paper we prove that if $X$ admits a relatively minimal genus one fibration then it does contain a subvariety of codimension one…

Given a positive singular hermitian metric of a pseudoeffective line bundle on a complex Kaehler manifold, a singular foliation is constructed satisfying certain analytic analogues of numerical conditions. This foliation refines Tsuji's…

As a generalization of a quasi-elliptic surface, there is a quasi-hyperelliptic surface, a nonsingular projective surface which has a fibration structure whose general fiber is a quasi-hyperelliptic curve ($=$ singular hyperelliptic curve…

We study non-isotrivial projective families of elliptic surfaces of Kodaira dimension one, over complex projective curves. If the base is an elliptic curve, we show that the family must have a singular fibre, and that over the projective…

In this paper we show that the relative canonical sheaf of a relatively minimal fibration of curves over a curve is semi-ample ; in fact, its m-tensored product is base point free for any m > 1. We use Koszul cohomology with it to prove…

For a smooth minimal surface of general type $S$ with $Albdim(S) = 2$, Severi inequality says that $K_S^2 \geq 4\chi(S)$, which was proved by Pardini. It is expected that when the equality is attained, $S$ is birational to a double cover…

In this paper we give for all $n \geq 2$, d>0, $g \geq 0$ necessary and sufficient conditions for the existence of a pair (X,C), where X is a K3 surface of degree 2n in $\matbf{P}^{n+1}$ and C is a smooth (reduced and irreducible) curve of…

A mixed quasi-\'etale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely out of a finite subset. A mixed quasi-\'etale surface is the…