In this paper, we study adjoint foliated structures of the form K_ℱ+D on algebraic surfaces and their minimal and canonical models. We investigate the effective behavior of the linear systems |m(K_ℱ+D)| for sufficiently divisible m>0. As an application, we obtain an effective answer to a boundedness problem for foliated surfaces of general type posed by Hacon and Langer.

The slopes of fibrations of genus g >= 2 have strict lower (resp., upper) bound, namely lambda m(g) (resp., lambda(M)(g)). In this paper, we show that if g not equal 3, then each rational number r is an element of[lambda(m)(g),lambda(M)(g)] can occur as the slope of some fibration of genus g. A similar result is also true for g=3 and r is an element of[lambda(m)(3),9].

In this paper, we describe the structure of the negative part of a Zariski decomposition of K_X+K_ℱ for a relatively minimal foliation (X,ℱ) whenever K_X+K_ℱ is pseudoeffective.

Let F be a periodic singular fiber of genus g with dual fiber F-& lowast;, and let T (resp. T-& lowast;) be the set of the components of F (resp. F-& lowast;) by removing one component with multiplicity one. We give a formula to compute the determinant | det T | of the intersect form of T. As a consequence, we prove that | det T | = | det T-& lowast; |. As an application, we compute the Mordell-Weil group of a fibration f : S -> P-1 of genus 2 with two singular fibers.

Lins Neto [Ann. Sci. École Norm. Sup. (4) 35 (2002), pp. 231–266] constructed families of foliations which are counterexamples to Poincaré’s Problem and Painlevé’s Problem. We will determine the minimal models of these families of foliations, calculate their Chern numbers, Kodaira dimension, and numerical Kodaira dimension. We prove that the slopes of Lins Neto’s foliations are at least 6, and their limits are bigger than 7 7 .

Our main purpose in this paper is to give a counterexample to some interesting conjectures on a rational fibred surface with multiple fibers proposed by R. V. Gurjar and D.-Q. Zhang.

Let f : S -> B be a surface fibration of genus g >= 2 over C. The semistable reduction theorem asserts there is a finite base change pi : B' -> B such that the fibration S x(B) B' -> B' admits a semistable model. An interesting invariant of f , denoted by N (f), is the minimum of deg (pi) for all such pi. In an early paper of Xiao, he gives a uniform multiplicative upper bound N-g for N (f) depending only on the fibre genus g. However, it is not known whether Xiao's bound is sharp or not. In this paper, we give another uniform upper bound N-g' for N (f) when f is hyperelliptic. Our N-g' is optimal in the sense that for every g >= 2 there is a hyperelliptic fibration f of genus g so that N (f) = N-g'. In particular, Xiao's upper bound N-g is optimal when N-g = N-g'. We show that this last equation N-g = N-g' holds for infinitely many g.

For a relatively minimal fibration f : X -> P-1 of non-hyperelliptic curves of genus g, we know the Picard number rho(X) <= 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 equivalent to 0 or 1 (mod 3), and we describe such f : X -> P-1 explicitly.

Let ( Y , G ) (Y, {\mathcal {G}}) be a Riccati foliation on Y Y and let π : ( X , F ) → ( Y , G ) \pi :(X,{\mathcal {F}}){\rightarrow } (Y,{\mathcal {G}}) be a double cover ramified over some normal-crossing curves. We will determine the minimal model of F {\mathcal {F}} and compute its Chern numbers c 1 2 ( F ) c_1^2({\mathcal {F}}) , c 2 ( F ) c_2({\mathcal {F}}) , and χ ( F ) = ( c 1 2 ( F ) + c 2 ( F ) ) / 12 \chi ({\mathcal {F}})=(c_1^2({\mathcal {F}})+ c_2({\mathcal {F}}))/12 . We will prove that the slope λ ( F ) = c 1 2 ( F ) / χ ( F ) \lambda ({\mathcal {F}})=c_1^2({\mathcal {F}})/\chi ({\mathcal {F}}) satisfies 4 ≤ λ ( F ) > 12 4\leq \lambda ({\mathcal {F}})>12 .

Let f : S -> C be a totally ramified triple cover fibration of type (g, gamma). We prove that the slope off has a sharp lower bound 24(g-1)/5g-6 gamma+1 given that g > 15/2 gamma + 5/2. We also characterize fibrations that achieve the bound.

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 threefolds X with smooth families of minimal surfaces of general type over a curve, -c(1)(3)(X) < -18c(3)(X).

Let f: S -> P-1 be a family of genus g >= 2 curves with two singular fibers F-1 and F-2. We show that F-1 = F-2* and F-2 = F-1* are dual to each other, S is a ruled surface, the geometric genera of the singular fibers are equal to the irregularity of the surface, and the virtual Mordell-Weil rank of f is zero. We prove also that c(1)(2)(S) <= -2 if g = 2, and c(1)(2)(S) <= -4 if g > 2. As an application, we will classify all such fibrations of genus g = 2.

We get a new inequality on the Hodge number of fibred algebraic complex surfaces , 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 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.

Let P be a normal singularity of multiplicity d = 2 or 3 of a complex surface X. It is well-known that X is locally an irreducible finite cover pi : X -> Y of degree dd over a smooth surface Y, and the singularity (X,P) can be resolved by the canonical resolution X-k -> Xk-1 -> . . . -> X-0 = X, which is the pullback of the embedded resolution of the corresponding singularity p = pi(P) of the branch locus. Let FF be the maximal ideal cycle of this resolution. We will prove that FF has a unique decomposition F = Z(1) + . . . + Z(d) with Z(1) >= Z(2) >= . . . >= Z(d) >= 0, where Z(i) is a fundamental cycle or zero. We show that w = p(a)(Z(1)) + . . . + p(a)(Z(d)) is an invariant of (X,P) that can also be computed from the multiplicity of the branch locus at pp. (X,P) is a rational singularity iff all of the singular points in the canonical resolution satisfies w <= d - 1. In order to get the minimal resolution from the canonical one, we need to blow down some exceptional curves, the number of blowing-downs is exactly that of fundamental cycles Z in the canonical resolution satisfying p(a)(Z) = 0 and Z(2) = -1.

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.

We determine the modular invariants of a family of non-hyperelliptic curves of genus 3 from its special bers. The formulas we obtained are conjectured by M. Reid.