Derived categories of Burniat surfaces and exceptional collections

We construct an exceptional collection $\varUpsilon $ of maximal possible length 6 on any of the Burniat surfaces with $K_X^2=6$ , a 4-dimensional family of surfaces of general type with $p_g=q=0$ . We also calculate the DG algebra of endomorphisms of this collection and show that the subcategory generated by this collection is the same for all Burniat surfaces. The semiorthogonal complement $\mathcal{A }$ of $\varUpsilon $ is an “almost phantom” category: it has trivial Hochschild homology, and $K_0(\mathcal{A })=\mathbb{Z }_2^6$ .     

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.     

A Gauss-Bonnet theorem for motivic cohomology

Partially supported by CNPq (Brasil) and NSF (USA), DMOS-8120790     

Logarithmic Kodaira dimension and the poles of the Hodge and motivic zeta functions for surfaces

To any polynomial f[x ₁,,x _n ] with f(0)=0 one associates the well-known motivic zeta function and its specialization to the level of Hodge polynomials. These zeta functions can be given very explicitly in terms of an embedded resolution of f ^–1{0} in ⁿ . In this paper, where we work with polynomials in three variables, i.e., n=3, we find a geometric condition for having a nonzero contribution to the residue at a candidate pole. More precisely, for a given embedded resolution h we fix an exceptional surface E with h(E)={0}, which induces in a canonical way a candidate pole q of the motivic zeta function. Then we prove that, when the surface E is non-rational and we are in a generic situation, the maximality of the logarithmic Kodaira dimension of E implies the non-vanishing of the contribution of E to the residue at q. Here E denotes the part of E that doesn't belong to any other irreducible component of h ^–1(f ^–1{0}). The same result is already true on the level of Hodge polynomials.Postdoctoral Fellow of the Fund for Scientific Research – Flanders (Belgium). Mathematics Subject Classificaton (2000):14B05, 14E15, 14J17 (32S45)     

A classification of projective homogeneous varieties up to motivic isomorphism

A complete classification is given for anisotropic projective homogeneous varieties of dimension less than 6 up to motivic isomorphism. Several criteria are presented for anisotropic flag varieties of type A_n to have isomorphic motives. Bibliography: 22 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 158–172.     

A bilinear form of dilogarithm and motivic regulator map

The Bloch-Wigner function D₂ is a single-valued version of a dilogarithm function and is used by Bloch to describe the Borel regulator map from K₃(C) into R explicitly (c.f. [Bloch, Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves, American Mathematical Society, Providence, RI, 2000]). We introduce a new way to formulate a single-valued dilogarithm function and use it to explicitly define a motivic regulator map for , defined in terms of the motivic complex of Goodwillie and Lichtenbaum. We also detect certain explicit nonzero elements in the motivic cohomology group. Throughout this paper, a path will be a C¹-function from the unit interval [0,1] into C-{0}.     

A study of generalized logistic distributions

Because of their flexibility, recently, much attention has been given to the study of generalized distributions. A complete study of the beta generalized logistic distribution (type IV) is proposed, introducing an approximate form for the median and deducing the mean deviation from the mean and the median. A complete parameter estimation using the method of maximum likelihood and the method of moments is presented. Some characteristic properties of the generalized logistic distribution type I are discussed. Also, a highlight to some properties of an analog distribution to the generalized logistic distribution type IV, discussed by Zografos and Balakrishnan [1], is presented.     

Soliton surfaces in the generalized symmetry approach

We investigate some features of generalized symmetries of integrable systems aiming to obtain the Fokas–Gel’fand formula for the immersion of two-dimensional soliton surfaces in Lie algebras. We show that if there exists a common symmetry of the zero-curvature representation of an integrable partial differential equation and its linear spectral problem, then the Fokas–Gel’fand immersion formula is applicable in its original form. In the general case, we show that when the symmetry of the zero-curvature representation is not a symmetry of its linear spectral problem, then the immersion function of the two-dimensional surface is determined by an extended formula involving additional terms in the expression for the tangent vectors. We illustrate these results with examples including the elliptic ordinary differential equation and the CP^N?1 sigma-model equation.     

ANOTHER TYPE OF GENERALIZED BALL CURVES AND SURFACES

In 2000,Wu presented two new types of generalized Ball curves,one of which is called an NB1 curve located between the Wang Ball curve and the Said Ball curve.In this article,the authors aim to discuss properties of NB1 curves and surfaces,including the recursive algorithms,conversion algorithms between NB1 and Bézier curves and surfaces, etc.In addition the authors compare the computation efficiency of recursive algorithms for the NB1 and above mentioned two generalized Ball curves and surfaces.     

Invariant algebraic surfaces of the generalized Lorenz system

In this paper, according to the idea of the weight of a polynomial introduced by Swinnerton-Dyer(Math Proc Camb Philos Soc 132:385–393, 2002), we successfully find all the invariant algebraic surfaces of the generalized Lorenz system x′ = a(y ? x), y′ = bx + cy ? xz, z′ = xy + dz.     

Images of two-dimensional motivic Galois representations

Let M be a two-dimensional motive which is pure of weight w over a number field K and let (: G_K Aut(H(M) )) be the system of the -adic realizations. Choose G_K-invariant -lattices T of H(M) and let (:G_K GL (T))be the corresponding system of integral representations. Then either for almost all primes (G_K) consist of all the elements of GL(T) with determinant in ( ^*)^–w or the system () is associated to algebraic Hecke characters. We also can prove an adelic version of our results.Mathematics Subject Classification (2000): 11F80     

On non-vanishing of cohomologies of generalized Raynaud polarized surfaces

We consider a family of slightly extended version of Raynaud’s surfaces X over the field of positive characteristic with Mumford-Szpiro type polarizations Z, which have Kodaira non-vanishing H¹(X,Z⁻ⁿ)≠0 for all 1≤n≤N with some N≥1. The surfaces are at least normal but smooth under a special condition. We also give a fairly large family of non-Mumford-Szpiro type polarizations Z_a,b with Kodaira non-vanishing.     

Indefinite centroaffine surfaces with vanishing generalized Pick function

In this paper, we study indefinite centroaffine surfaces with vanishing generalized Pick function. We give a classification of such surfaces.