A new family of surfaces of general type with K^2=7 and p_g=0

We construct a new family of smooth minimal surfaces of general type with $K^2=7$ and $p_g=0$ . We show that a surface in this family has ample canonical divisor and birational bicanonical morphism. We also prove that these surfaces satisfy Bloch’s conjecture.     

Integrable super extensions of $$K(-2,-2)$$ equation

Theoretical and Mathematical Physics - Two coupled systems involving both bosonic and fermionic fields are proposed as super generalizations of the $$K(-2,-2)$$ equation...     

Burniat-type surfaces and a new family of surfaces with p_g = 0, K^2=3

The paper is one of a series devoted to the classification, the moduli spaces and the classification of surfaces of general type with \(p_g=0\). Here we generalize a classical construction due to P. Burniat (revised by M. Inoue). Among other results we construct a family of surfaces of general type with \(K_S^2 = 3\), \(p_g(S) = 0\) realizing a new fundamental group of order 16.     

Affine ADE bundles over surfaces with $$ p _{g}=0$$

Given any Kodaira curve C in a complex surface X, we construct a simply-laced affine Lie algebra bundle \(\mathcal {E}\) over X. When \( p _{g}(X)=0\), we construct deformations of holomorphic structures on \(\mathcal {E}\) such that the new bundle is trivial over any ADE curve \( C^{\prime }\) inside C and therefore descends to the singular surface obtained by contracting \(C^{\prime }\).     

A characterization of Burniat surfaces with K2=4 and of non nodal type

Let S be a minimal surface of general type with pg(S) = 0 and K_S~2= 4. Assume the bicanonical map ψ of S is a morphism of degree 4 such that the image of ψ is smooth. Then we prove that the surface S is a Burniat surface with K~2= 4 and of non nodal type.     

The Bicanonical Map of Surfaces with pg = 0 and K2 >= 7

A minimal surface of general type with p_g(S) = 0 satisfies 1 K² 9, and it is known that the image of the bicanonical map is a surface for , whilst for , the bicanonical map is always a morphism. In this paper it is shown that is birationalif , and that the degree of is at most 2 if or By presenting two examples of surfaces S with and 8 and bicanonical map of degree 2, it is alsoshown that this result is sharp. The example with is, to our knowledge, a new example of a surfaceof general type with p_g = 0. The degree of is also calculated for two other known surfacesof general type with p_g = 0 and . In both cases, the bicanonical map turns out to be birational.     

Characterization of a class of surfaces with p g = 0 and K 2 = 5 by their bicanonical maps

Let S be a minimal surface of general type with ${p_g(S) = 0, K_S^2 = 5}$ . We prove that S is a Burniat surface if its bicanonical map is of degree 4 and has smooth image.     

The Bicanonical Map of Surfaces with pg = 0 and K2 >= 7, II

Minimal complex surfaces of general type with p_g = 0 and K²= 7 or 8 whose bicanonical map is not birational are studied.It is shown that if S is such a surface, then the bicanonicalmap has degree 2 (see Bulletin of the London Mathematical Society33 (2001) 1–10) and there is a fibration f: S P¹ suchthat (i) the general fibre F of f is a genus 3 hyperellipticcurve; (ii) the involution induced by the bicanonical map ofS restricts to the hyperelliptic involution of F. Furthermore, if , then f isan isotrivial fibration with six double fibres, and if , then f has five double fibres andit has precisely one fibre with reducible support, consistingof two components. 2000 Mathematics Subject Classification 14J29.     

Indifferentiable hashing to ordinary elliptic $${\mathbb {F}}_{\!q}$$ F q -curves of $$j=0$$ j = 0 with the cost of one exponentiation in $${\mathbb {F}}_{\!q}$$ F q

Designs, Codes and Cryptography - Let $${\mathbb {F}}_{\!q}$$ be a finite field and $$E_b\!: y^2 = x^3 + b$$ be an ordinary (i.e., non-supersingular) elliptic curve (of j-invariant 0) such that...     

A Threefold of General Type with q 1=q 2=p g =P 2=0

One of the problems in classifying nonsingular threefolds of general type with p _g =0 lies in finding the range of the bigenus P ₂ (surfaces of general type with p _g =0 have 2P ₂10). Another problem involves finding the minimum integer m such that the m-canonical map _|mK| is birational for any threefold (m=5 in the case of surfaces). An example of a nonsingular threefold X of general type with q ₁=q ₂=0, p _g =P ₂=0,P ₃=1 is presented. In addition, the m-canonical map of X is birational if and only if m14. The threefold is obtained as a nonsingular model of a degree ten hypersurface in P ⁴ _C with the affine equation t ²=f ₁₀(x,y,z).     

On the $$D(2)$$ -Vertex Distinguishing Total Coloring of Graphs with $$\Delta=3$$

Mathematical Notes - A $$D(2)$$ -vertex-distinguishing total coloring of a graph $$G$$ is a proper total coloring such that no pair of vertices, within distance two, has the same set of colors, and...     

A characterization of $$ \cal N $$ -constrained groups

We prove that a finite group G is -constrained if and only if it contains a nilpotent subgroup I satisfying for all .Received: 22 July 2002     

Embedding some Riemann surfaces into $${\mathbb {C}^2}$$ with interpolation

We prove that several types of open Riemann surfaces, including the finitely connected planar domains, embed properly into such that the values on any given discrete sequence can be arbitrarily prescribed. Kutzschebauch supported by Schweizerische Nationalfonds grant 200021-107477/1.     

On surfaces with p g =2, q=1 and K 2=5

We consider minimal surfaces of general type with p _g =2, q=1 and K ²=5. We provide a stratification of the corresponding moduli space \(\mathcal{M}\) and we give some bounds for the number and the dimensions of its irreducible components.     

A fast $$(2 + \frac{2}{7})$$ ( 2 + 2 7 ) -approximation algorithm for capacitated cycle covering

Mathematical Programming - We consider the capacitated cycle covering problem: given an undirected, complete graph G with metric edge lengths and demands on the vertices, we want to cover the...     

$$|End \mathfrak{A}| = |Con \mathfrak{A}| = |End \mathfrak{A}| = 2^{|\mathfrak{A}|}$$ for any uncountable 1-unary algebra $$\mathfrak{A}$$

Algebra universalis -     

Gradient Estimates for the Positive Solutions of $$\mathfrak {L}u=0$$ on Self-Shrinkers

In this paper, we investigate the positive solutions of \(\mathfrak {L}u=0\) on a self-shrinker. First, we prove a global gradient estimate for the positive solutions, and obtain a strong Liouville theorem. Then by the generalized Laplacian comparison theorem for the distance function on a self-shrinker, we derive a local gradient estimate for the positive solutions. At last, we collect some applications of the local gradient estimate for the positive solutions on self-shrinkers.