Related papers: On surfaces with a canonical pencil

A nodal Enriques surface can have at most 8 nodes. We give an explicit description of Enriques surfaces with 8 nodes, showing that they are quotients of products of elliptic curves by a group isomorphic to $\Z_2^2$ or to $\Z_2^3$ acting…

In this note we investigate three new pencils of symmetric surfaces in complex projective three-space. These have degree 6, 8 resp. 12 and are invariant under the action of subgroups of SO(4) containing the Heisenberg group. The pencils of…

Let $S$ be a minimal surface of general type with $p_g(S) = 0, K_S^2 = 5$ and bicanonical map of degree 4. Denote by $\Sigma$ the bicanonical image. If $\Sigma$ is smooth, then $S$ is a Burniat surface; and if $\Sigma$ is singular, then we…

In this paper, we give a classification of Codazzi hypersurfaces in a Lie group $(Nil^{4},\widetilde g)$. We also give a characterization of a class of minimal hypersurfaces in $(Nil^{4},\widetilde g)$ with an example of a minimal surface…

We establish a curvature estimate for classical minimal surfaces with total boundary curvature less than 4\pi. The main application is a bound on the genus of these surfaces depending solely on the geometry of the boundary curve. We also…

The subject of this paper is the study of various families of quartic K3 surfaces which are invariant under a certain $(\mathbb{Z}/2\mathbb{Z})^{4}$ action. In particular, we describe families whose general member contains $8,16,24$ or $32$…

We classify smooth weak del Pezzo surfaces with global vector fields over an arbitrary algebraically closed field $k$ of arbitrary characteristic $p \geq 0$. We give a complete description of the configuration of $(-1)$- and $(-2)$-curves…

Surfaces of amplitude 1 in ordinary projective space are of general type, but this need not be the case in weighted projective spaces. Indeed, there are 4 classes of quasi-smooth weighted hypersurfaces in $\mathbf{P}(1,2,a,b)$ of amplitude…

Let $S$ be a smooth projective complex algebraic surface and $f\, :\, S\, \longrightarrow\, {\mathbb C}{\mathbb P}^2$ a finite map. Consider a pencil of hyperplane sections on ${\mathbb C}{\mathbb P}^2$ and pull it back to $S$. We address…

For a minimal smooth projective surface $S$ of general type over a field of characteristic $p>0$, we prove that $K^2_S\le 32\chi(\cal{O}_S).$ Moreover, if $18\chi(\cal{O}_S)<K^2_S\le 32\chi(\cal{O}_S)$, Albanese morphism of $S$ must induces…

Smooth minimal surfaces of general type with $K^2=1$, $p_g=2$, and $q=0$ constitute a fundamental example in the geography of algebraic surfaces, and the 28-dimensional moduli space $\mathbf{M}$ of their canonical models admits a modular…

We consider a higher-order Henneberg-type minimal surfaces family using the generalized Weierstrass--Enneper representation in four-dimensional space $\mathbb{R}^4$. We derive explicit parametric equations for the surface and determine its…

We study the higher Chow groups $CH^2(X,1)$ and $CH^3(X,2)$ of smooth, projective algebraic surfaces over a field of char 0. We develop a theoretical framework to study them by using so-called higher normal functions and higher…

We study the orchard problem on cubic surfaces. We classify possibly reducible cubic surfaces $X\subseteq \mathbb{P}^3(\C)$ with smooth components on which there exist families of finite sets (of unbounded size) with quadratically many…

In the present paper, we study surfaces in the four-dimensional Euclidean space $\mathbb{R}^4$. We define special principal parameters, which we call canonical, on each surface without minimal points, and prove that the surface admits (at…

We describe a new family of triply-periodic minimal surfaces with hexagonal symmetry, related to the quartz (qtz) and its dual (the qzd net). We provide a solution to the period problem and provide a parametrisation of these surfaces, that…

We construct a surface of general type with invariants \( \chi = K^2 = 1 \) and torsion group \( \Bbb{Z}/{2} \). We use a double plane construction by finding a plane curve with certain singularities, resolving these, and taking the double…

In this note it is shown that, given a smooth minimal complex surface of general type S with p_g(S)=0, K^2_S=3, for which the bicanonical map is a morphism, then the degree of the bicanonical map of S is not equal to 3. This completes our…

Let $S$ be a minimal surface of general type with irregularity $q(S) = 1$. Well-known inequalities between characteristic numbers imply that $3 p_g(S) \le c_2(S) \le 10 p_g(S)$, where $p_g(S)$ is the geometric genus and $c_2(S)$ the…

In this paper we describe a new deformation that connects minimal disks with planar ends with minimal disks with helicoidal ends. In this way, we are able to construct a family of complete minimal surfaces with helicoidal ends that contains…