Given a relatively minimal non locally trivial fibred surface f: S->B, the slope of the fibration is a numerical invariant associated to the fibration. In this paper we explore how properties of the general fibre of $f$ and global properties of S influence on the lower bound of the slope. First of all we obtain lower bounds of the slope when the general fibre is a double cover. We also obtain a lower bound depending as an increasing function on the relative irregularity of the fibration, extending previous results of Xiao. We construct several families of examples to check the assimptotical sharpness of our bounds.
Let f :S\to B be a non locally trivial fibred surface. We prove a lower bound for the slope of f depending increasingly from the relative irregularity of f and the Clifford index of the general fibres.
We prove new slope inequalities for relatively minimal fibred surfaces, showing an influence of the relative irregularity, of the unitary rank and of the Clifford index on the slope. The argument uses Xiao's method and a new Clifford-type inequality for subcanonical systems on non-hyperelliptic curves.
In this paper we present a generalisation of a theorem due to Cornalba and Harris, which is an application of Geometric Invariant Theory to the study of invariants of fibrations. In particular, our generalisation makes it possible to treat the problem of bounding the invariants of general fibred surfaces. As a first application, we give a new proof of the slope inequality and of a bound for the invariants associated to double cover fibrations.
Let $f:\,S \to B$ be a locally non-trivial relatively minimal fibration of genus $g\geq 2$ with relative irregularity $q_f$. It was conjectured by Barja and Stoppino that the slope $λ_f\geq \frac{4(g-1)}{g-q_f}$. We prove the conjecture when $q_f$ is small with respect to $g$; we also construct counterexamples when $g$ is odd and $q_f=(g+1)/2$.
We study from a geographical point of view fibrations of threefolds over smooth curves, such that the general fibre is of general type. We prove the non-negativity of certain relative invariants under general hypotheses and give lower bounds for the self-interssection of the relative canonical divisor of the fibration, depending on other relative invariants. We also study the influence of the relative irregularity on these bounds. A more detailed study of the lowest cases of the bounds is given.
We complement Catanese's results on isotrivially fibred surfaces by completely describing the components containing an isotrivial surface with monodromy group $\ZZ/2\ZZ$. We also give an example for deformation equivalent isotrivial surfaces with different monodromy group.
We generalise a method of Xiao Gang to construct 'prototypes' of fibred surfaces with maximal irregularity without being a product. This enables us, in the case of fibre genus g=3 to describe the possible singular fibres and to calculate the invariants of these surfaces. We also prove structure theorems on the moduli space for fibred surfaces with fibre genus g=2 and g=3.
Let f: X->B be a fibred surface of genus g whose general fibre is a double cover of a smooth curve of genus gamma. We show that, for g > 4gamma+1, the number 4(g-1)/(g-gamma) is a sharp lower bound for the slope of f, proving a conjecture of Barja. Moreover, we give a characterisation of the fibred surfaces that reach the bound. In the case g = 4gamma+1 we obtain the same sharp bound under the assumption that the involutions on the general fibres glue to a global involution on X.
We study threefolds fibred by Kummer surfaces associated to products of elliptic curves, that arise as resolved quotients of threefolds fibred by certain lattice polarized K3 surfaces under a fibrewise Nikulin involution. We present a general construction for such surfaces, before specializing our results to study Calabi-Yau threefolds arising as resolved quotients of threefolds fibred by mirror quartic K3 surfaces. Finally, we give some geometric properties of the Calabi-Yau threefolds that we have constructed, including expressions for Hodge numbers.
Let $f:S \fr B$ be a surface fibration with fibres of genus 5. We find a linear relation between the fundamental invariants of the surface. Namely $K_f^2=χ_f+N$ where $N$ is the number of trigonal fibres. Our proof is based on the analysis of the relative canonical algebra $\cal{R}(f)$.
We study and obtain Slope inequalities for fibred irregular varieties of non-maximal Albanese dimension. We give a comparison theorem between Clifford-Severi and Slope inequalities for this type of fibrations. We also obtain a set of Slope inequalities considering the geometry of the Albanese map and the associated eventual maps.
An isotrivially fibred surface is a smooth projective surface endowed with a morphism onto a curve such that all the smooth fibres are isomorphic to each other. The first goal of this paper is to classify the isotrivially fibred surfaces with $p_g=q=2$ completing and extending a result of Zucconi. As an important byproduct, we provide new examples of minimal surfaces of general type with $p_g=q=2$ and $K^2=4,5$ and a first example with $K^2=6$.
We study threefolds fibred by mirror quartic K3 surfaces. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic K3 surfaces. This is then used to give a complete explicit description of all Calabi-Yau threefolds fibred by mirror quartic K3 surfaces. We conclude by studying the properties of such Calabi-Yau threefolds, including their Hodge numbers and deformation theory.
We prove that the genus $g$, the relative irregularity $q_f$ and the Clifford index $c_f$ of a non-isotrivial fibration $f$ satisfy the inequality $q_f \leq g - c_f$. This gives in particular a proof of Xiao's conjecture for fibrations whose general fibres have maximal Clifford index.
This note aims to improve known numerical bounds proved earlier by Chen \cite{PAMS} and Chen-Hacon \cite{Chen-Hacon} and to present some new examples of smooth minimal 3-folds canonically fibred by surfaces (resp. curves) of geometric genus as large as 19 (resp. 13). As an interesting by-product, we present a new class of general type surfaces which are canonically fibred by curves of genus 13.
We study threefolds fibred by K3 surfaces admitting a lattice polarization by a certain class of rank 19 lattices. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the appropriate K3 moduli space, which we call the generalized functional invariant. Then we show that if the threefold total space is a smooth Calabi-Yau, there are only finitely many possibilities for the polarizing lattice and the form of the generalized functional invariant. Finally, we construct explicit examples of Calabi-Yau threefolds realizing each case…
We consider threefolds that admit a fibration by K3 surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarisation of degree two on the general fibre. Under certain assumptions on the threefold we show that its relative log canonical model exists and can be explicitly reconstructed from a small set of data determined by the original fibration. Finally we prove a converse to the above statement: under certain assumptions, any such set of data determines a threefold that arises as the relative log canonical model of a threefold admitting a fibration by K3 surface…
We consider a rational surface with a relatively minimal fibration. Picard number of a such fibred surface is bounded in terms of the genus of a general fibre. When Picard number is the maximum for any given genus, we characterize a such fibred surface whose Mordell-Weil group is trivial by singular fibres. Furthermore, we describe the defining equation explicitly.
We study holomorphic symplectic manifolds which are fibred by abelian varieties. This structure is a higher dimensional analogue of an elliptic fibration on a K3 surface. We investigate when a holomorphic symplectic manifold is fibred in this way, and are led to several natural conjectures. We then study the geometry of these fibrations. The expectation is that this point of view will prove useful in understanding holomorphic symplectic manifolds, and possibly lead to a classification.
K. Konno proved a slope equality for fibred surfaces with fibres of odd genus and general fibre of maximal gonality. More precisely he found a relation between the invariants of the fibration and certain weights of special fibres (called the Horikawa numbers). We give an alternative and more geometric proof in the case of a genus 5 fibration, under generality assumptions. In our setting we are able to prove that the fibre with positive Horikawa numbers are precisely the trigonal ones, we compute their weights explicitly and thus we exhibit explicit examples of regular surfaces with assigned in…
In "Seshadri fibrations of algebraic surfaces" [arXiv:0709.2592v1] we showed that if the multiple point Seshadri constants of an ample line bundle on a smooth projective surface in very general points satisfy certain inequality then the surface is fibred by curves computing these constants. Here we characterize the border case of polarized surfaces whose Seshadri constants in general points fulfill the equality instead of inequality and which are not fibred by Seshadri curves. It turns out that these surfaces are the projective plane and surfaces of minimal degree.
We describe the minimal number of critical points and the minimal number $s$ of singular fibres for a non isotrivial fibration of a surface $S$ over a curve $B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular fibre with $4$ nodes. Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group $Sp (2g, \mathbb{Z})$.
In this paper we investigate two stratifications of the moduli space of elliptically fibred K3 surfaces. The first comes from Shimada's classification of connected components of elliptically fibred K3 surfaces and is closely related to the root lattice of the fibration. The second is the monodromy stratification defined by Bogomolov, Petrov and Tschinkel. The main result of the paper is a classification of all positive-dimensional ambi-typical strata, that is strata which are both Shimada root strata and monodromy strata. We further discuss the connection with moduli spaces of lattice-polarise…
Motivated by a conjecture of Xiao, we study supporting divisors of fibred surfaces. On the one hand, after developing a formalism to treat one-dimensional families of varieties of any dimension, we give a structure theorem for fibred surfaces supported on relatively rigid divisors. On the other hand, we study how to produce supporting divisors by constructing a global adjoint map for a fibration over a curve (generalizing the infinitesimal constructions of Collino, Pirola, Rizzi and Zucconi).
We consider relationships between families of K3 surfaces, in the context of string theory. An important ingredient of string theory also of interest in algebraic geometry is T-duality. Donagi and Pantev have extended the original duality on genus one fibred K3 surfaces with a section to the case of any genus one fibration, via a Fourier-Mukai transform. We investigate possibilities of extending this result to the more general case of non fibred K3s, in the context of Caldararu's conjecture. We also provide an explicit construction of a quasi-universal sheaf associated to the classical Mukai c…
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 less than 4, if the general fibre of $f$ is non-hyperelliptic or the genus of the fibre or of the base curve is low.
Being inspired by Ross' construction of unstable products of certain smooth curves, we show that the product $C\times C$ of every smooth curve $C$ of genus at least 2 is not slope semistable with respect to certain polarisations. Besides, we produce examples of Kodaira-fibred surfaces of nonzero signature, which are not slope semistable with respect to some polarisations, and so they admit Kähler classes that do not contain any constant scalar curvature Kähler metrics.
We consider relatively minimal fibrations of curves of genus two on rational surfaces whose Picard numbers are not maximal. By birational morphisms, such fibred surfaces are interpreted as pencils of plane curves. We show that only four are canonical, among a variety of possible models. For each canonical pencil, we give an example with trivial Mordell-Weil group.
We get a new inequality on the Hodge number $h^{1,1}(S)$ of fibred algebraic complex surfaces $S$, which is a generalization of an inequality of Beauville. Our inequality implies the Arakelov type inequalities due to Arakelov, Faltings, Viehweg and Zuo, respectively.
We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of pola…
We study slopes of finite cyclic covering fibrations of a fibered surface. We give the best possible lower bound of the slope of these fibrations. We also give the slope equality of finite cyclic covering fibrations of a ruled surface and observe the local concentration of the global signature of these surfaces on a finite number of fiber germs. We also give an upper bound of the slope of finite cyclic covering fibrations of a ruled surface.
In this paper, the Euler characteristic formula for projective logarithmic minimal degenerations of surfaces with Kodaira dimension zero over a 1-dimensional complex disk is proved under a reasonable assumption and as its application, the singularity of logarithmic minimal degenerations are determined in the abelian or hyperelliptic case. By globalizing this local analysis of singular fibres via generalized canonical bundle formulae due to Fujino-Mori, we bound the number of singular fibres of abelian fibred Calabi-Yau threefolds from above,which was previously done by Oguiso in the potentiall…
We establish the slope equality and give an upper bound of the slope for finite cyclic covering fibrations of an elliptic surface including bielliptic fibrations of genus greater than 5. We also give an upper bound of the slope for triple cyclic covering fibrations of a ruled surface and hyperelliptic fibrations, which provides a new proof of Xiao's upper bound.
In this article we study asymptotic slopes of strongly semistable vector bundles on a smooth projective surface. A connection between asymptotic slopes and strong restriction theorem of a strongly semistable vector bundle is shown. We also give an equivalent criterion of strong semistability of a vector bundle in terms of its asymptotic slopes under some assumptions on the surface and on the bundle.
We give a sharp lower bound for the selfintersection of a nef line bundle $L$ on an irregular variety $X$ in terms of its continuous global sections and the Albanese dimension of $X$, which we call the Generalized Clifford-Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibred irregular varieties. As a byproduct we obtain a lower bound for the volume of irregular varieties; when $X$ is of maximal Albanese dimension the bound is $Vol(X) \geq 2 n! χω_X$ and it is sharp.
We study slope stability of smooth surfaces and its connection with exceptional divisors. We show that a surface containing an exceptional divisor with arithmetic genus at least two is slope unstable for some polarisation. In the converse direction we show that slope stability of surfaces can be tested with divisors, and prove that for surfaces with non-negative Kodaira dimension any destabilising divisor must have negative self-intersection and arithmetic genus at least two. We also prove that a destabilising divisor can never be nef, and as an application give an example of a surface that is…
On a Weierstraß elliptic surface $X$, we define a `limit' of Bridgeland stability conditions, denoted $Z^l$-stability, by varying the polarisation in the definition of Bridgeland stability along a curve in the ample cone of $X$. We show that a slope stable torsion-free sheaf of positive (twisted) degree or a slope stable locally free sheaf is taken by a Fourier-Mukai transform on $D^b(X)$ to a $Z^l$-stable object, while a $Z^l$-semistable object of nonzero fiber degree can be modified so that its inverse Fourier-Mukai transform is a slope semistable torsion-free sheaf.
Given a complex algebraic variety X, we define a natural number called the motivic dimension which measures the amount of transcendental (co)homology of X. It is zero precisely when all the (co)homolgy is spanned by algebraic cycles. Most of this paper is concerned with giving estimates on this number, along with examples where it is small. As an application, we check or recheck the Hodge conjectue in a number of examples: uniruled fourfolds, rationally connected fivefolds, fourfolds fibred by surfaces with p_g=0, Hilbert schemes of a small number points on surfaces with p_g=0, and generic hyp…
Previously, many people have studied a stability of vector bundles of given rank and Chern classes on algebraic varieties. Recently, we are interested in the slope stability of the rank 2 Lazarsfeld-Mukai bundle $E_{C,Z}$ on a K3 surface $X$ associated to a very ample smooth curve $C$ on $X$ and a base point free pencil $Z$ on $C$ with respect to $\mathcal{O}_X(C)$. In this paper, we will give a sufficient condition for such a Lazarsfeld-Mukai bundle $E_{C,Z}$ to be $\mathcal{O}_X(C)$-slope semistable by ACM line bundles with respect to $\mathcal{O}_X(C)$.
In this paper, we investigate the general notion of the slope for families of curves $f: X \to Y$. The main result is an answer to the above question when $\dim Y = 2$, and we prove a lower bound for this new slope in this case over fields of any characteristic. Both the notion and the slope inequality are compatible with the theory for $\dim Y = 0, 1$ in a very natural way, and this gives a strong evidence that the slope for an $n$-fold fibration of curves $f: X \to Y$ may be $K_{X/Y}^n / \mathrm{ch}_{n-1}(f_* ω_{X/Y})$. Rather than the usual stability methods, the whole proof of the slope in…
We carry out a detailed intersection theoretic analysis of the Deligne-Mumford compactification of the divisor on M_{10} consisting of curves sitting on K3 surfaces. This divisor is not of classical Brill-Noether type, and is known to give a counterexample to the Slope Conjecture. The computation relies on the fact that this divisor has four different incarnations as a geometric subvariety of the moduli space of curves, one of them as a higher rank Brill-Noether divisor consisting of curves with an exceptional rank 2 vector bundle. As an application we describe the birational nature of the mod…
Let $f:S\to B$ be a finite cyclic covering fibration of a fibered surface. We study the lower bound of slope $λ_{f}$ when the relative irregularity $q_{f}$ is positive.
In this paper, we first construct varieties of any dimension $n>2$ fibered over curves with low slopes. These examples violate the conjectural slope inequality of Barja and Stoppino [BS14b]. Led by their conjecture, we focus on finding the lowest possible slope when $n=3$. Based on a characteristic $p > 0$ method, we prove that the sharp lower bound of the slope of fibered $3$-folds over curves is $4/3$, and it occurs only when the general fiber is a $(1, 2)$-surface. Otherwise, the sharp lower bound is $2$. We also obtain a Cornalba-Harris-Xiao type slope inequality for families of surfaces o…
We prove the slope inequality for a relative minimal surface fibration in positive characteristic via Xiao's approach. We also prove a better low bound for the slope of non-hyperelliptic fibrations.
Given a relatively minimal fibration $f: S \to \Bbb P^1$ on a rational surface $S$ with general fiber $C$ of genus $g$, we investigate under what conditions the inequality $6(g-1)\le K_f^2$ occurs, where $K_f$ is the canonical relative sheaf of $f$. We give sufficient conditions for having such inequality, depending on the genus and gonality of $C$ and the number of certain exceptional curves on $S$. We illustrate how these results can be used for constructing fibrations with the desired property. For fibrations of genus $11\le g\le 49$ we prove the inequality: $$ 6(g-1) +4 -4\sqrt g \le K_f^2…
