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*This paper is from 2018* In this paper, we try to classify moduli spaces of arrangements of $12$ lines with sextic points. We show that moduli spaces of arrangements of $12$ lines with sextic points can consist of more than two connected components. We also present defining equations of the arrangements whose moduli spaces are not irreducible taking quotients by the complex conjugation by supply some potential Zariski pairs. Through complex conjugation we take quotients and supply some potential Zariski pairs., Comment: *A paper from 2018*. arXiv admin note: text overlap with arXiv:1206.2486

We find a new Zariski pair with non-isomorphic fundamental groups that consists of degree $ 8 $ conic-line arrangements. Each arrangement has three conics and two lines. We use the Van Kampen Theorem and some known Coxeter groups to determine the fundamental groups. Two examples of Zariski pairs that were introduced in 2014 by the last named author, are given as well. They consist of a pair of conic-line arrangements with three conics in each (and thus, each has a single line) and a pair with two conics in each (and thus, each has three lines). By our methods, we are able to show that these two Zariski pairs have non-isomorphic fundamental groups., Comment: Includes an appendix with the full computation of the fundamental groups and a similar computations for Zariski pairs of conic-line arrangements discovered by the last named author in 2014

In this paper we consider the Galois covers of algebraic surfaces of degree 6, with all associated planar degenerations. We compute the fundamental groups of those Galois covers, using their degeneration. We show that for 8 types of degenerations the fundamental group of the Galois cover is non-trivial and for 20 types it is trivial. Moreover, we compute the Chern numbers of all the surfaces with this type of degeneration and prove that the signatures of all their Galois covers are negative. We formulate a conjecture regarding the structure of the fundamental groups of the Galois covers based on our findings. With an appendix by the authors listing the detailed computations and an appendix by Guo Zhiming classifying degree 6 planar degenerations., Comment: Main paper is 21 pages and contains 7 figures. The appendix is 65 pages and contains 42 figures. arXiv admin note: substantial text overlap with arXiv:1610.09612

Modular invariants of families of curves are Arakelov invariants in arithmetic algebraic geometry. All the known uniform lower bounds of these invariants are not sharp. In this paper, we aim to give explicit lower bounds of modular invariants of families of curves, which is sharp for genus 2. According to the relation between fractional Dehn twists and modular invariants, we give the sharp lower bounds of fractional Dehn twist coefficients and classify pseudo-periodic maps with minimal coefficients for genus 2 and 3 firstly. We also obtain a rigidity property for families with minimal modular invariants, and other applications.

We prove the generic exclusion of certain Shimura varieties of unitary and orthogonal types from the Torelli locus. The proof relies on a slope inequality on surface fibration due to G. Xiao, and the main result implies that certain Shimura varieties only meet the Torelli locus in dimension zero., Comment: Dedicated to Professor Ngaiming Mok on the occasion of his sixtieth birthday

Let $f:\,X \to \mathbb{P}^1$ be a non-isotrivial semi-stable family of varieties of dimension $m$ over $\mathbb{P}^1$ with $s$ singular fibers. Assume that the smooth fibers $F$ are minimal, i.e., their canonical line bundles are semiample. Then $\kappa(X)\leq \kappa(F)+1$. If $\kappa(X)=\kappa(F)+1$, then $s>\frac{4}m+2$. If $\kappa(X)\geq 0$, then $s\geq\frac{4}m+2$. In particular, if $m=1$, $s=6$ and $\kappa(X)=0$, then the family $f$ is Teichm\"uller., Comment: any comment is warmly welcome

We determine positive-dimensional G-periodic proper subvarieties of an n-dimensional normal projective variety X under the action of an abelian group G of maximal rank n-1 and of positive entropy. The motivation of the paper is to understand the obstruction for X to be G-equivariant birational to the quotient variety of an abelian variety modulo the action of a finite group., Comment: Asian Journal of Mathematics (to appear), Special issue on the occasion of Prof N. Mok's 60th birthday

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 $k=\mathbb C$ and $g\geq 5$; we also prove that $s_{nc}\geq 4$ if ${\rm char}~k>0$ and the relative Jacobian of $f$ is non-smooth., Comment: comments are welcome

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., Comment: 11 pages, 3 figures

We establish the canonical class inequality for families of higher dimensional projective manifolds. As an application, we get a new inequality between the Chern numbers of 3-folds with smooth families of minimal surfaces of general type over a curve, $c_1^3<18c_3$., Comment: 9 pages

In a family of curves, the Chern numbers of a singular fiber are the local contributions to the Chern numbers of the total space. We will give some inequalities between the Chern numbers of a singular fiber as well as their lower and upper bounds. We introduce the dual fiber of a singular fiber, and prove a duality theorem. As an application, we will classify singular fibers with large or small Chern numbers., Comment: 23 pages

Let $f:X@>>>\Bbb P^1$ be a fibered surface with fibers of genus g>1. If f is semistable and non isotrivial we prove that X of non negative Kodaira dimension implies that the number s of singular fibers is at least 5. Information about the nature of X if s=6,5 and g<6 is given. If f is any relatively minimal fibration we bound by below K_f^2, the bound depending on g and the Kodaira dimension of X, a classification of rational surfaces with minimal K_f^2 is given. Moreover, we prove several properties of positivity for the linear system K_X+F (F a fibre of f). Examples of a K3 surfaces admitting a semistable fibration with s=6 and g=3 and of a surface of general type admitting a semistable fibration with s=7 and g=4 are provided., Comment: AMS Tex file, 12 pages. This new version is based in previous preprints by S.-L. Tan and Y.Tu and A. G. Zamora

We give optimal effective bounds for some well-known theorems on complex algebraic surfaces, which are respectively due to Serre, Zariski (1962), Castelnuovo (1897), Artin (1962, 1966), Benveniste (1984), Cutkosky and Srinivas (1993). These theorems are about Riemann-Roch problem (on the behavior of the function dim |nD| of n), vanishing theorems, base-point freeness and k-very ampleness of the linear systems |nD| and |nA+L|, where D is effective, A is nef and big and L is arbitrary. As a consequence, we obtain an effective version of Matsusaka's big theorem, and we give also examples to show that our bound is the best possible one., Comment: Latex, 17 pages. Asian Journal of Mathematics (to appear)

We express explicitly the integral closures of some ring extensions; this is done for all Bring-Jerrard extensions of any degree as well as for all general extensions of degree < 6; so far such an explicit expression is known only for degree < 4 extensions. As a geometric application we present explicitly the structure sheaf of every Bring-Jerrard covering space in terms of coefficients of the equation defining the covering; in particular, we show that a degree-3 morphism f : Y --> X is quasi-etale if and only if the first Chern class of the sheaf f_*(O_Y) is trivial (details in Theorem 5.3). We also try to get a geometric Galoisness criterion for an arbitrary degree-n finite morphism; this is successfully done when n = 3 and less satifactorily done when n = 5., Comment: Advances in Mathematics, to appear (no changes, just add this info)

Let $f:X\to C$ be a family of semistable K3 surfaces with non-empty set $S$ of singular fibres having infinite local monodromy. Then, when the so called Arakelov-Yau inequality reaches equality, we prove that $C\setminus S$ is a modular curve and the family comes essentially from a family of elliptic curves through a so called Nikulin-Kummer construction. In particular, when $C=\BBb P^1$, the family of elliptic curves must be one of Beauville's 6 examples where Arakelov inequality reaches equality., Comment: 18 pages, Latex

The purpose of this paper is to give a linear and effective height inequality for algebraic points on curves over functional fields. Our height inequality can be viewed as the logarithmic canonical class inequality of a punctured curve over a functional field (a fibered surface minus a section)., Comment: 14 pages, AmSTeX 2.1 This paper will appear in J. reine angew. Math

In this paper, we shall prove Beauville's conjecture: if $f:S \to P^1$ is a non-trivial semistable fibration of genus g>1, then $f$ admits at least 5 singular fibers. We have also constructed an example of genus 2 with 5 singular fibers. This paper will appear in the Journal of Algebraic Geometry., Comment: 6 pages, AmSTeX

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 fibration to be isotrivial. We also obtain a canonical class inequality for any fibration. Some other applications are presented. Most of the results of this paper have arithmetical analogues. This paper will appear in Math. Z., Comment: 21 pages, AmSTeX