Minimal Fibrations Research Articles

Relative Severi inequality for fibrations of maximal Albanese dimension over curves

AbstractLet$f: X \to B$be a relatively minimal fibration of maximal Albanese dimension from a varietyXof dimension$n \ge 2$to a curveBdefined over an algebraically closed field of characteristic zero. We prove that$K_{X/B}^n \ge 2n! \chi _f$. It verifies a conjectural formulation of Barja in [2]. Via the strategy outlined in [4], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and$\chi _f> 0$, we prove that the general fibreFoffhas to satisfy the Severi equality that$K_F^{n-1} = 2(n-1)! \chi (F, \omega _F)$. We also prove some sharper results of the same type under extra assumptions.

Extremal trigonal fibrations on rational surfaces

For a relatively minimal fibration $f : X \to \mathbb{P}^1$ of non-hyperelliptic curves of genus $g$, we know the Picard number $\rho(X) \leq 3g + 8$. We study the case where $\rho(X) = 3g + 8$ and the Mordell–Weil group of $f$ is trivial. Such an $f$ occurs only if $g \equiv 0$ or $1 \pmod{3}$, and we describe such $f : X \to \mathbb{P}^1$ explicitly.

Another approach to the Kan\u2013Quillen model structure

By careful analysis of the embedding of a simplicial set into its image under Kan’s mathop {mathop {mathsf {Ex}}^infty } functor we obtain a new and combinatorial proof that it is a weak homotopy equivalence. Moreover, we obtain a presentation of it as a strong anodyne extension. From this description we can quickly deduce some basic facts about mathop {mathop {mathsf {Ex}}^infty } and hence provide a new construction of the Kan–Quillen model structure on simplicial sets, one which avoids the use of topological spaces or minimal fibrations.

On the slope conjecture of Barja and Stoppino for fibred surfaces

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 $\lambda_f\geq \frac{4(g-1)}{g-q_f}$. We prove the conjecture when $q_f$ is small with respect to $g$; we also construct counterexamples when $g$ is odd and $q_f=(g+1)/2$.

Minimality in diagrams of simplicial sets

We formulate the concept of minimal fibration in the context of fibrations in the model category $${\mathbf {S}}^{\mathcal {C}}$$ of $${\mathcal {C}}$$ -diagrams of simplicial sets, for a small index category $${\mathcal {C}}$$ . When $${\mathcal {C}}$$ is an EI-category satisfying some mild finiteness restrictions, we show that every fibration of $${\mathcal {C}}$$ -diagrams admits a well-behaved minimal model. As a consequence, we establish a classification theorem for fibrations in $${\mathbf {S}}^{\mathcal {C}}$$ over a constant diagram, generalizing the classification theorem of Barratt, Gugenheim, and Moore for simplicial fibrations (Barratt et al. in Am J Math 81:639–657, 1959).

Complex length of short curves and Minimal Fibrations of hyperbolic three‐Manifolds fibering over the circle

We investigate the maximal solid tubes around short simple geodesics in hyperbolic three-manifolds and how complex length of curves relate to closed, incompressible, least area minimal surfaces. As applications, we prove, there are some closed hyperbolic three-manifolds fibering over the circle which are not foliated by closed incompressible minimal surfaces diffeomorphic to the fiber. We also show, the existence of quasi-Fuchsian manifolds containing arbitrarily many embedded closed incompressible minimal surfaces. Our strategy is to prove main theorems under natural geometric conditions on the complex length of closed curves on a fibered hyperbolic three-manifold, then we find explicit examples where these conditions are satisfied via computer programs.

On the gonality and the slope of a fibered surface

Let f:X→B be a locally non-trivial relatively minimal fibration of curves of genus g≥2. We obtain a lower bound of the slope λ(f) increasing with the gonality of the general fiber of f. In particular, we show that λ(f)≥4 provided that f is non-hyperelliptic and g≥16.

The Frobenius condition, right properness, and uniform fibrations

We develop further the theory of weak factorization systems and algebraic weak factorization systems. In particular, we give a method for constructing (algebraic) weak factorization systems whose right maps can be thought of as (uniform) fibrations and that satisfy the (functorial) Frobenius condition. As applications, we obtain a new proof that the Quillen model structure for Kan complexes is right proper, avoiding entirely the use of topological realization and minimal fibrations, and we solve an open problem in the study of Voevodsky's simplicial model of type theory, proving a constructive version of the preservation of Kan fibrations by pushforward along Kan fibrations. Our results also subsume and extend work by Coquand and others on cubical sets.

Minimal fibrations of dendroidal sets

We prove the existence of minimal models for fibrations between dendroidal sets in the model structure for infinity-operads, as well as in the covariant model structure for algebras and in the stable one for connective spectra. In an appendix, we explain how our arguments can be used to extend the results of Cisinski, giving the existence of minimal fibrations in model categories of presheaves over generalised Reedy categories of a rather common type. Besides some applications to the theory of algebras over infinity-operads, we also prove a gluing result for parametrized connective spectra (or Gamma-spaces).

On the slope of hyperelliptic fibrations with positive relative irregularity

Let $f:\, S \to B$ be a locally non-trivial relatively minimal fibration of hyperelliptic curves of genus $g\geq 2$ with relative irregularity $q_f$. We show a sharp lower bound on the slope $\lambda_f$ of $f$. As a consequence, we prove a conjecture of Barja and Stoppino on the lower bound of $\lambda_f$ as an increasing function of $q_f$ in this case, and we also prove a conjecture of Xiao on the ampleness of the direct image of the relative canonical sheaf if $\lambda_f<4$.

A Lower Bound on the Euler–Poincaré Characteristic of Certain Surfaces of General Type with a Linear Pencil of Hyperelliptic Curves

AbstractLet S be a surface of general type. In this article, when there exists a relatively minimal hyperelliptic fibration whose slope is less than or equal to four, we give a lower bound on the Euler–Poincaré characteristic of S. Furthermore, we prove that our bound is the best possible by giving required hyperelliptic fibrations.

Fibrations by curves with more than one nonsmooth point

Bertini’s theorem on variable singular pointsmay fail in positive characteristic, as was discovered by Zariski in 1944. In fact, he found fibrations by nonsmooth curves. In this work we continue to classify this phenomenon in characteristic three by constructing the first example, rising in the literature, of fibrations with more than one nonsmooth point. Our approach has been motivated by the close relation between it and the theory of regular but nonsmooth curves, or equivalently, nonconservative function fields in one variable. In analogy to the Kodaira-Neron classification of special fibers of minimal fibrations by elliptic curves, we also construct the minimal proper regular model of some fibrations by nonsmooth projective plane quartic curves and determine the structure of the bad fibers.

Minimal fibrations and the organizing theorem of simplicial homotopy theory

Quillen showed that simplicial sets form a model category (with appropriate choices of three classes of morphisms), which organized the homotopy theory of simplicial sets. His proof is very difficult and uses even the classification theory of principal bundles. Thus, Goerss–Jardine appealed to topological methods for the verification. In this paper we give a new proof of this organizing theorem of simplicial homotopy theory which is elementary in the sense that it does not use the classifying theory of principal bundles or appeal to topological methods.

Fibrations by nonsmooth genus three curves in characteristic three

Bertini’s theorem on variable singular points may fail in positive characteristic, as was discovered by Zariski in 1944. In fact, he found fibrations by nonsmooth curves. In this work we continue to classify this phenomenon in characteristic three by constructing a two-dimensional algebraic fibration by nonsmooth plane projective quartic curves, that is universal in the sense that the data about some fibrations by nonsmooth plane projective quartics are condensed in it. Our approach has been motivated by the close relation between it and the theory of regular but nonsmooth curves, or equivalently, nonconservative function fields in one variable. Actually, it also provides an understanding of the interesting effect of the relative Frobenius morphism in fibrations by nonsmooth curves. In analogy to the Kodaira–Néron classification of special fibers of minimal fibrations by elliptic curves, we also construct the minimal proper regular model of some fibrations by nonsmooth projective plane quartic curves, determine the structure of the bad fibers, and study the global geometry of the total spaces.

On the slope of relatively minimal fibrations on rational complex surfaces

Given a relatively minimal fibration \({f:S \to \mathbb{P}^1}\), defined on a rational surface S, with a general fiber C of genus g, we investigate under what conditions the inequality \({6(g-1)\le K_f^2}\) occurs, where Kf 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 ≤ g ≤ 49 we prove the inequality: $$6(g-1) +4 -4\sqrt g \le K_f^2.$$

Canonical fixed parts of fibred algebraic surfaces

It is shown that the fixed part of the canonical linear system of a fibre in a relatively minimal fibred surface supports at most exceptional sets of weakly elliptic singular- ities. Introduction. Let S be a non-singular projective surface and f : S → C a surjective morphism of S onto a non-singular projective curve C with connected fibres. We call f a relatively minimal fibration of genus g if a general fibre is a non-singular projective curve of genus g and there are no (−1)-curves contained in fibres. We assume that g ≥ 2 throughout the paper. Let F be afi bre off . Then the intersection form is negative semi-definite on Supp(F ) by Zariski's lemma. Furthermore, there exist a positive integer m and a 1-connected curve D such that F = mD .W henm is strictly greater than one, F is called a multiple fibre of multiplicity m and OD(D) is a torsion of order m. In (8), we considered the canonical linear system on the minimal resolution of a normal surface singularity and showed that the fixed part supports at most exceptional sets of rational singular points (cf. (1) and (2)). The present article is an extension of it to the semi-global case and we study the fixed part of the canonical linear system |KF | which we call the canonical fixed part in this paper. Recall that the canonical fixed part is closely related to the Horikawa index (see (3, p. 12)), an analytic invariant of a singular fibre germ. In fact, according to (6, Lemma 10 and Theorem 3), if g = 2, the canonical fixed part is a chain of (−2)-curves (of type A) and the Horikawa index is almost equivalent to the number of its irreducible components.

On Bertini's theorem for fibrations by plane projective quartic curves in characteristic five

Bertini's theorem on variable singular points may fail in positive characteristic. We construct in characteristic five a two-dimensional algebraic fibration π : T → S by plane projective quartic curves, that is pathological in the sense that all fibers are non-smooth though the total space T is smooth after restricting the base surface S to a dense open subset, and that is universal in the sense that each pathological fibration by plane projective quartics is up to birational equivalence obtained by a base extension either from the two-dimensional fibration π or from an one-dimensional pathological fibration π −1 ( C ) → C obtained by restricting the base of π to a uniquely determined curve C on S. These curves on the base surface S, which are bases of pathological fibrations, are classified in terms of invariant curves of an algebraic vector field. Among these fibrations there is just one whose general fiber admits a non-ordinary inflection point. In analogy to the Kodaira–Néron classification of special fibers of minimal fibrations by elliptic curves, we construct the minimal proper regular model of this pathological fibration, determine the structure of the bad fibers, and study the global geometry of the total space.

Relative canonical sheaves of a family of curves

In this paper we study the relative canonical sheaf of a relatively minimal fibration of curves of genus g⩾2 over a one-dimensional regular scheme. Using the configurations of (−2)-chains, we show that its m-tensored product is base point free for any m⩾2. We utilize Koszul cohomology to prove that the relative canonical algebra of the fibration is generated in degree up to five. It is a generalization of K. Konno's work on the 1-2-3 Conjecture of M. Reid.

On Bertini's theorem in characteristic ${p}$ for families of canonical curves in ${\Bbb P}^{(p-3)/2}$

Bertini's theorem on variable singular points may fail in characteristic p; there are algebraic fibrations that are pathological in the sense that all fibres are non-smooth though the total space admits only a finite number of singular points. We prove that if an algebraic fibration by plane projective quartic curves in characteristic 7 is pathological then, after an eventual cyclic base extension of degree 3, it is, up to birational equivalence, obtained by a base extension from the pencil of quartic curves cut out by the equations zx3 + xy3 + t yz3 = 0. More generally, we classify pathological fibrations by canonical curves of arithmetic genus g = p − 1 2 in the projective space of dimension p − 3 2 . We prove that the gap sequences of the smooth Weierstrass points of the fibres have the property that a positive integer ℓ is a gap if and only if 2g + 1 − ℓ is a non-gap. In analogy to the Kodaira–Néron classification of special fibres of minimal fibrations by elliptic curves, we construct minimal proper regular models of pathological fibrations, determine the structure of the bad fibres, and study the global geometry of the total spaces. 2000 Mathematics Subject Classification 14H05, 14H10, 14H20, 14H50.

Some inequalities for minimal fibrations of surfaces of general type over curves

Some inequalities for minimal fibrations of surfaces of general type over curves