Related papers: On the BMY inequality on surfaces

The aim of this paper is to consider a possible extension of the Bogomolov--Miyaoka--Yau inequality to differentiable orbifolds. The conjectured extension is related to the Montgomery--Yang problem about circle actions on the 5--sphere and…

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 describe the singular fibers of a parabolic fibration $f:X\to Y$ whose moduli divisor $M_Y$ is numerically trivial and discriminant divisor $B_Y$ is zero.

We show that a compact complex surface which fibers smoothly over a curve of genus >1 with fibers of genus >1 fibers holomorphically. We deduce an improvement of a result in [D Kotschick, Math. Research Letters, 5 (1998) 227-234], and a…

A surface in a Riemannian space is called of constant astigmatism if the difference between the principal radii of curvatures at each point is a constant function. In this paper we give a classification of all rotational surfaces of…

Transformations between different analytic descriptions of constant mean curvature (CMC) surfaces are established. In particular, it is demonstrated that the system \[ \begin{split} &\partial \psi_{1} = (|\psi_{1}|^{2} + |\psi_{2}|^{2})…

We find conditions under which the restriction of a divergence-free vector field $B$ to an invariant toroidal surface $S$ is linearisable. The main results are similar in conclusion to Arnold's Structure Theorems but require weaker…

Let $X$ be a smooth projective surface such that linear and numerical equivalence of divisors on $X$ coincide and let $\sigma\subseteq |D|$ be a linear pencil on $X$ with integral general fibers. A fiber of $\sigma$ will be called special…

We provide general inequalities that compare the surface area S(K) of a convex body K in ${\mathbb R}^n$ to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for…

The Minkowski inequality is a classical inequality in differential geometry, giving a bound from below, on the total mean curvature of a convex surface in Euclidean space, in terms of its area. Recently there has been interest in proving…

We generalize an inequality for convex lattice polygons -- aka toric surfaces -- to general rational surfaces.

For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of…

Suppose X is a smooth projective 3-fold of general type and |mK_X| is composed of a pencil of surfaces with m>1. This pencil naturally induces a fibration f:X->C onto a smooth curve C after the Stein-factorization, which is the main objects…

We investigate the failure of B\'ezout's Theorem for two symplectic surfaces in $\mathbb{C}\mathrm{P}^2$ (and more generally on an algebraic surface), by proving that every plane algebraic curve $C$ can be perturbed in the…

We prove two local inequalities for divisors on surfaces and study their applications.

Let f: Y -> CP^2 be a birational morphism of non-singular (rational) surfaces. We give an effective (necessary and sufficient) criterion for algebraicity of the surfaces resulting from contraction of the union of the strict transform of a…

We generalize the construction of Raynaud of smooth projective surfaces of general type in positive characteristic that violate the Kodaira vanishing theorem. This corrects an earlier paper of the same purpose. These examples are smooth…

In this paper we study Moebius applicable surfaces, i.e., conformally immersed surfaces in Moebius 3-space which admit deformations preserving the Moebius metric. We show new characterizations of Willmore surfaces, Bonnet surfaces and…

In 1944 Zariski discovered that Bertini's theorem on variable singular points is no longer true when we pass from a field of characteristic zero to a field of positive characteristic. In other words, he found fibrations by singular curves,…

In this paper, we extend our result in [3] to hypersurfaces of any smooth projective variety $Y$. Precisely we let $X_0$ be a generic hypersurface of $Y$ and $c_0:\mathbf P^1\to X_0$ be a generic birational morphism to its image, i.e.…