global holomorphic 2 forms and pluricanonical systems on threefolds the change of zero locus of a global holomorphic 2 form on a threefold under birational transformations is investigated it is proved that existence of 2 forms with certain conditions on their zero loci on a threefold of nonnegative kodaira dimension limits types of terminal singularities appearing on its minimal models as a result of the restriction on the types of terminal singularities and reid s riemann roch formula a universal bound n is found such that the linear system nk defines a birational map from a threefold of general type admitting those 2 forms where k is the canonical bundle of the threefold

closed embeddings of c in c 2 part i we consider closed curves c c in the affine plane s c2 that admit a good or very good asymptote by this we mean a curve l c in s that in suitable coordinates for s is linear and tangent to c at infinity and meets c once or not at all at finite distance we classify these curves up to automorphism of s relying on the theory of open algebraic surfaces we first determine the possibilities for the singularities of c at infinity and then proceed to give explicit equations

surfaces of general type with p_g q 1 k 2 8 and bicanonical map of degree 2 we classify the minimal algebraic surfaces of general type with pg q 1 k 2 8 and bicanonical map of degree 2 it will turn out that they are isogenous to a product of curves i e if s is such a surface then there exist two smooth curves c f and a finite group g acting freely on c x f such that s c x f g we describe the c f and g that occur in particular the curve c is a hyperelliptic bielliptic curve of genus 3 and the bicanonical map of s is composed with the involution induced on s by r x id c x f c x f where is the hyperelliptic involution of c in this way we obtain three families of surfaces with pg q 1 k 2 8 which yield the first known examples of surfaces with these invariants we compute their dimension and we show that they are three generically smooth irreducible components of the moduli space m of surfaces with p g q 1 k 2 8 moreover we give an alternative description of these surfaces as double covers of the plane recovering a construction proposed by du val

locating p poly optimally in the extended low hierarchy the low hierarchy in np and the extended low hierarchy have turned out to be very useful in classifying many interesting language classes and almost all of them could be located optimally therein however until now the exact location of p poly remained open

bounded queries approximations and the boolean hierarchy this paper investigates nondeterministic bounded query classes in relation to the complexity of np hard approximation problems and the boolean hierarchy nondeterministic bounded query classes turn out be rather suitable for describing the complexity of np hard approximation problems the results in this paper take advantage of this machine based model to prove that in many cases np approximation problems have the upward collapse property that is a reduction between np approximation problems of apparently different complexity at a lower level results in a similar reduction at a higher level for example if maxclique reduces to log n approximating maxclique using many one reductions then the traveling salesman problem tsp is equivalent to maxclique under many one reductions several upward collapse theorems are presented in this paper the proofs of these theorems rely heavily on the machinery provided by the nondeterministic bounded query classes in fact these results depend on a surprising connection between the boolean hierarchy and nondeterministic bounded query classes

complexity restricted advice functions the authors consider uniform subclasses of the nonuniform complexity classes defined by karp and lipton l enseign math 28 1982 via the notion of advice functions these subclasses are obtained by restricting the complexity of computing correct advice also the effect of allowing advice functions of limited complexity to depend on the input rather than on the input s length is investigated among other results using the notions described above new characterizations of a np np cap sparse b np with a restricted access to an np oracle and c the odd levels of the boolean hierarchy are given as a consequence it is shown that every set that is nondeterministically truth table reducible to sat in the sense of rich j comput system sci 38 1989 pp 511 523 is already deterministically truth table reducible to sat furthermore it turns out that the np reduction classes of bounded versions of this reducibility coincide with the odd levels of the boolean hierarchy

bounding singular surfaces via chern numbers we prove the existence of a bound on the number of steps of the minimal model program for singular surfaces in terms of discrepancies and chern numbers as an application we prove that given r in mathbb r 0 and epsilon in 0 1 the class mathcal f r epsilon of 2 dimensional pairs x d of general type with epsilon klt singularities d with standard coefficients and 4c 2 x d c 1 2 x d le r forms a bounded family

enriques surfaces with eight nodes 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 or to acting freely in codimension 1 we use this result to show that if s is a minimal surface of general type with pg 0 such that the image of the bicanonical map is birational to an enriques surface then k2s 3 and the bicanonical map is a morphism of degree 2

periodic wave solutions and solitary wave solutions of generalized modified boussinesq equation and evolution relationship between both solutions abstract in this paper we focus on studying the exact solitary wave solutions and periodic wave solutions of the generalized modified boussinesq equation u t t u t t x x a 1 u a 2 u 2 a 3 u 3 x x 0 as well as the evolution relationship between these solutions detailed qualitative analysis is conducted on traveling wave solutions of this equation and global phase portraits in various parameter conditions are proposed various significant results about the existence of both solutions including three forms of solitary wave solutions and four exact bounded periodic wave solutions in different conditions are obtained then we further discuss the relationship between energy of hamiltonian system corresponding to this equation and the periodic wave solutions and solitary wave solutions it is concluded that the essential reason of periodic wave solutions and solitary wave solutions is the different values for the energy of hamiltonian system corresponding to this equation in addition the limited relations of periodic wave solutions and solitary wave solutions with the energy of hamiltonian system are proposed and the schematic diagram of evolution from periodic wave solutions to solitary wave solutions is drawn

some remarks on the local moduli of tangent bundles over complex surfaces using the hirzebruch s riemann roch formula for endomorphism bundles over a compact complex two fold we prove that the tangent bundle of a complex surface m of general type admits a nontrivial trace free deformation unless m is holomorphically covered by the euclidean ball it follows that the tangent bundle of the mostow siu surface which is a kahler surface with a negative definite curvature tensor does have a nontrivial trace free moduli among some other results we also point out a relationship between the kuranishi obstruction and symmetric holomorphic two tensors on a complex surface

hyperbolicity of nodal hypersurfaces we show that a nodal hypersurface x in p3 of degree d with a sufficiently large number l of nodes l 8 3 d2 5 2 d is algebraically quasi hyperbolic i e x can only have finitely many rational and elliptic curves our results use the theory of symmetric differentials and algebraic foliations and give a very striking example of the jumping of the number of symmetric differentials in families

the bogomolov miyaoka yau inequality for logarithmic surfaces in positive characteristic we generalize bogomolov s inequality for higgs sheaves and the bogomolov miyaoka yau inequality in positive characteristic to the logarithmic case we also generalize shepherd barron s results on bogomolov s inequality on surfaces of special type from rank 2 to the higher rank case we use these results to show some examples of smooth nonconnected curves on smooth rational surfaces that cannot be lifted modulo p2 these examples contradict some claims by xie

canonical surfaces with big cotangent bundle surfaces of general type with positive second segre number are known to have big cotangent bundle we give a new criterion ensuring that a surface of general type with canonical singularities has a minimal resolution with big cotangent bundle this provides many examples of surfaces with negative second segre number and big cotangent bundle

negative curves on algebraic surfaces we study curves of negative self intersection on algebraic surfaces our main result shows there exist smooth complex projective surfaces x related to hilbert modular surfaces such that x contains reduced irreducible curves c of arbitrarily negative self intersection c 2 previously the only known examples of surfaces for which c 2 was not bounded below were in positive characteristic and the general expectation was that no examples could arise over the complex numbers indeed we show that the idea underlying the examples in positive characteristic cannot produce examples over the complex number field and thus our complex examples require a different approach

algebraic montgomery yang problem the non cyclic case montgomery yang problem predicts that every pseudofree circle action on the 5 dimensional sphere has at most 3 non free orbits using a certain one to one correspondence kollar formulated the algebraic version of the montgomery yang problem every projective surface s with quotient singularities such that the second betti number b2 s 1 has at most 3 singular points if its smooth locus s0 is simply connected we prove the conjecture under the assumption that s has at least one non cyclic singularity in the course of the proof we classify projective surfaces s with quotient singularities such that i b2 s 1 ii h 1 s 0 mathbb z 0 and iii s has 4 or more singular points not all cyclic and prove that all such surfaces have pi 1 s 0 cong mathfrak a 5 the icosahedral group

the geometry of sporadic c embeddings into c2 abstract a closed algebraic embedding of c c 1 0 into c 2 is sporadic if for every curve a c 2 isomorphic to an affine line the intersection with c is at least 2 non sporadic embeddings have been classified there are very few known sporadic embeddings we establish geometric and algebraic tools to classify them based on the analysis of the minimal log resolution x d p 2 u where u is the closure of c on p 2 we show in particular that one can choose coordinates on c 2 in which the type at infinity of the c and the self intersection of its proper transform on x are sharply limited

on rational cuspidal projective plane curves in 2002 l nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a gorenstein surface singularity with rational homology sphere link with the seiberg witten invariant or one of its candidates of the link recently the last three authors found some counterexamples using superisolated singularities the theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves in the case when the corresponding curve has only one singular point one knows no counterexample in fact in this case the above seiberg witten conjecture led us to a very interesting and deep set of compatibility properties of these curves generalising the seiberg witten invariant conjecture but sitting deeply in algebraic geometry which seems to generalise some other famous conjectures and properties as well for example the noether nagata or the log bogomolov miyaoka yau inequalities namely we provide a set of compatibility conditions which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer d if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree d the conjectured compatibility properties have a weaker version too valid for any rational cuspidal curve with more than one singular point the goal of the present article is to formulate these conjectured properties and to verify them in all the situations when the logarithmic kodaira dimension of the complement of the corresponding plane curves is strictly less than 2

degree of the bicanonical map of a surface of general type let 5 be a smooth complex projective surface of general type rt 5 fpn l be the map defined by nks is called pluricanonical map for n 2 due to the works by bombieri and others in the 1970s the nature of n is determined except when n 2 and for some low values of invariants of 5 see 1 for an up to date treatment and references these maps play an important role in the classification of surfaces of general type recently the remaining case of the bicanonical map 2 for low values of invariants has aroused new research interests due to the work of francia 8 and especially the remarkable method of reider 15 see also 6 14 4 they have proved the following result among other things

the classification of double planes of general type with k2 8 and pg 0 abstract we study minimal double planes of general type with k 2 8 and p g 0 namely pairs s where s is a minimal complex algebraic surface of general type with k 2 8 and p g 0 and is an automorphism of s of order 2 such that the quotient s is a rational surface we prove that s is a free quotient f c g where c is a curve f is an hyperelliptic curve g is a finite group that acts faithfully on f and c and is induced by the automorphism id of f c being the hyperelliptic involution of f we describe all the f c and g that occur in this way we obtain 5 families of surfaces with p g 0 and k 2 8 of which we believe only one was previously known using our classification we are able to give an alternative description of these surfaces as double covers of the plane thus recovering a construction proposed by du val in addition we study the geometry of the subset of the moduli space of surfaces of general type with p g 0 and k 2 8 that admit a double plane structure

three divisible families of skew lines on a smooth projective quintic we give an example of a family of 15 skew lines on a quintic such that its class is divisible by 3 we study properties of the codes given by arrangements of disjoint lines on quintics

restricted information from nonadaptive queries to np we investigate classes of sets that can be decided by bounded truth table reductions to an np set in which evaluators donothave full access to the answers to the queries but get only restricted information such as the number of queries that are in the oracle set or even just this number modulom for somem 2 we also investigate the case in which evaluators are nondeterministic we show that when we vary the information that the evaluators get this can change the resulting power of the evaluators we locate all these classes within levels of the boolean hierarchy which allows us to compare the complexity of such classes