Involutions on surfaces with pg?=?q?=?0 and K2?=?3

We study surfaces of general type S with p _g  = 0 and K ² = 3 having an involution i such that the bicanonical map of S is not composed with i. It is shown that, if S/i is not rational, then S/i is birational to an Enriques surface or it has Kodaira dimension 1 and the possibilities for the ramification divisor of the covering map S → S/i are described. We also show that these two cases do occur, providing an example. In this example S has a hyperelliptic fibration of genus 3 and the bicanonical map of S is of degree 2 onto a rational surface.     

Surfaces with pg?=?q?=?1, K2?=?8 and nonbirational bicanonical map

We prove that if the bicanonical map of a minimal surface of general type S with p _g  = q = 1 and K_S²=8{K_{S}^2=8} is nonbirational, then it is a double cover onto a rational surface. An application of this theorem is the complete classification of minimal surfaces of general type with p_g=q=1, K_S²=8{p_{g}=q=1, K_{S}^2=8} and nonbirational bicanonical map.     

On the bicanonical map of a surface section of a threefold

Let (ℳ, ℒ) be a 3-fold of log-general type polarized by a very ample line bundle ℒ. We study the pairs (ℳ, ℒ) in the case when there exists at least one smooth surface Ŝ ∈ |ℒ| such that the bicanonical map associated to |2K_Ŝ| is not birational. As one consequence of our classification we obtain the result:if a smooth projective threefold has non- negative Kodaira dimension, then given any smooth very ample divisor Ŝon the threefold, the bicanonical map associated to |2K_Ŝ|is birational.     

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.     

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.     

Projective models of Enriques surfaces in scrolls

We study Cossec's ? ‐function, which is defined for divisors with positive self‐intersection on an Enriques surface. In this paper we study the existence of pairs (C ², ? (C )) with C an irreducible curve. The ? ‐function gives in a natural way scrolls containing Enriques surfaces. We compute scroll types to some of these scrolls. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On kth-order embeddings of K3 surfaces and Enriques surfaces

We give necessary and sufficient conditions for a big and nef line bundle L of any degree on a K3 surface or on an Enriques surface S to be k-very ample and k-spanned. Furthermore, we give necessary and sufficient conditions for a spanned and big line bundle on a K3 surface S to be birationally k-very ample and birationally k-spanned (our definition), and relate these concepts to the Clifford index and gonality of smooth curves in |L| and the existence of a particular type of rank 2 bundles on S. Received: 28 March 2000 / Revised version: 20 October 2000     

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.     

Smooth 4-folds which contain a P1-bundle¶as an ample divisor

We show that if a smooth projective 4-fold M contains an ample divisor A which is P ¹-bundle π :A→S over a smooth projective surface S, π is extended to a P ²-bundle π :S→S, unless $A$ is isomorphic to P ²×P ¹. Received: 28 September 1998 / Revised version: 16 August 1999     

Halves of a real Enriques surface

The real partE _ℝ of a real Enriques surfaceE admits a natural decomposition in two halves,E _ℝ=E _ℝ ⁽¹⁾ ∪E _ℝ ⁽²⁾ , each half being a union of components ofE _ℝ. We classify the triads (E _ℝ;E _ℝ ⁽¹⁾ ,E _ℝ ⁽²⁾ ) up to homeomorphism. Most results extend to surfaces of more general nature than Enriques surfaces. We use and study in details the properties of Kalinin's filtration in the homology of the fixed point set of an involution, which is a convenient tool not widely known in topology of real algebraic varieties.     

A characterization of the Clifford torus

We provide a characterization of the Clifford torus via a Ricci type condition among minimal surfaces in S⁴. More precisely, we prove that a compact minimal surface in S⁴, with induced metric ds² and Gaussian curvature K, for which the metric is flat away from points where K = 1, is the Clifford torus, provided that m is an integer with m > 2.Received: 8 September 2004     

Generic vanishing index and the birationality of the bicanonical map of irregular varieties

We prove that any smooth complex projective variety with generic vanishing index bigger or equal than 2 has birational bicanonical map. Therefore, if X is a smooth complex projective variety φ with maximal Albanese dimension and non-birational bicanonical map, then the Albanese image of X is fibred by subvarieties of codimension at most 1 of an abelian subvariety of Alb X.     

Exceptional bundles on nodal Enriques surfaces

In this paper we prove that an Enriques surfaceX has a smooth rational curve if and only if there exists an exceptional bundleE _t of rank 2 withc ₂ (E _t )=t for any integer t onX. We describe all exceptional bundles of rank 2 on Enriques surfaces and show that they are all stable with respect to any ample divisor.     

Noncommutative unitons

By Uhlenbeck’s results, every harmonic map from the Riemann sphere S² to the unitary group U(n) decomposes into a product of so-called unitons: special maps from S² to the Grassmannians Gr _k(ℂⁿ) ⊂ U(n) satisfying certain systems of first-order differential equations. We construct a noncommutative analogue of this factorization, applicable to those solutions of the noncommutative unitary sigma model that are finite-dimensional perturbations of zero-energy solutions. In particular, we prove that the energy of each such solution is an integer multiple of 8π, give examples of solutions that are not equivalent to Grassmannian solutions, and study the realization of non-Grassmannian zero modes of the Hessian of the energy functional by directions tangent to the moduli space of solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 220–239, February, 2008.     

On automorphisms of Enriques surfaces and their entropy

Consider an arbitrary automorphism of an Enriques surface with its lift to the covering K3 surface. We prove a bound of the order of the lift acting on the anti‐invariant cohomology sublattice of the Enriques involution. We use it to obtain some mod 2 constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value. In the Appendix, we give a complete list of Salem numbers that potentially could be the minimal dynamical degree on Enriques surfaces and for which the existence of geometric automorphisms is unknown.     

Surfaces of general type with <Emphasis Type="Italic">p</Emphasis><Subscript><Emphasis Type="Italic">g</Emphasis></Subscript>=0, <Emphasis Type="Italic">K</Emphasis><Superscript>2</Superscript>=6 and non birational bicanonical map

Let S be a minimal surface of general type with p_g=0 and K²=6, such that its bicanonical map is not birational. The map is a morphism of degree 4 onto a surface. The case of deg = 4 is completely classified in [Topology, 40 (5) (2001), 977–991] and the present paper completes the characterization of these surfaces. It is proven that the degree of cannot be equal to 3, and the geometry of surfaces with deg = 2 is analysed in detail. The last section contains three examples of such surfaces, two of which appear to be new.Mathematics Subject Classification (2000): 14J29     

The equator map and the negative exponential functional

We define a negative exponential harmonic map from the ballB ⁿ of ℝ_n into the sphereS ⁿ of ℝ ⁿ⁺¹ . We prove that the equator map is a negative exponential harmonic map, but not stable for the negative exponential functional whenn≥2. Moreover, we consider maps from a ballB ⁿ into the unit sphereS ^m of ℝ_m+1 wherem≥2, and prove that no nonconstant, non surjective map can reach either the minimum or the maximum of the negative exponential functional.     

Deformations of circle-valued Morse functions on surfaces

Let M be a smooth connected orientable compact surface and let F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) be a space of all Morse functions f : M → S ¹ without critical points on ∂M such that, for any connected component V of ∂M, the restriction f : V → S ¹ is either a constant map or a covering map. The space F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) is endowed with the C ^∞-topology. We present the classification of connected components of the space F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) . This result generalizes the results obtained by Matveev, Sharko, and the author for the case of Morse functions locally constant on ∂M.     

Non-classical Godeaux surfaces

A non-classical Godeaux surface is a minimal surface of general type with χ = K ² = 1 but with h ⁰¹ ≠ 0. We prove that such surfaces fulfill h ⁰¹ = 1 and they can exist only over fields of positive characteristic at most 5. Like non-classical Enriques surfaces they fall into two classes: the singular and the supersingular ones. We give a complete classification in characteristic 5 and compute their Hodge-, Hodge–Witt- and crystalline cohomology (including torsion). Finally, we give an example of a supersingular Godeaux surface in characteristic 5.