the einstein smoluchowski promeasure versus the boltzmann peierls equation beginning from the specific model of the boltzmann peierls equation for the distribution function f k t the time dependent theory of fluctuations is developed from einstein s inversion of boltzmann s relation the distribution function f k t that satisfies the boltzmann peierls equation is to be viewed as a random variable characterizing the macro state of a gas of quasiparticles phonons magnons rotons etc the crucial assumption entering into the present approach is simply that the randomness in the statistical distribution of the initial values of f k t can be characterized by the einstein smoluchowski promeasure it is convenient to think of as being the infinite dimensional analog of einstein s distribution law in the gaussian approximation the equilibrium promeasure is used to study the correlations in time of fluctuations in the moments of f k t some problems associated with a kinetic version of the green kubo approach to transport processes are carefully studied the exact formula for the thermal conductivity coefficient t which is derived in this paper emerges as a portion of the general formalism

domain of definition of levermore s five moment system the simplest system in levermore s moment hierarchy involving moments higher than second order is the five moment closure it is obtained by taking velocity moments of the one dimensional boltzmann equation under the assumption that the velocity distribution is a maximum entropy function the moment vectors for which a maximum entropy function exists consequently make up the domain of definition of the system the aim of this article is a complete characterization of the structure of the domain of definition and the connected maximum entropy problem the space homogeneous case of the equation and numerical aspects are also addressed

a study on the dimension of global sections of adjoint bundles for polarized manifolds abstract let x be a smooth complex projective variety of dimension n and let l be an ample line bundle on x in this paper in order to investigate the dimension of h 0 k x t l more systematically we introduce the invariant a i x l for every integer i with 0 i n furthermore we study this invariant for the case where l is ample and spanned by global sections as applications we get a lower bound resp an upper bound for the dimension of h 0 k x t l if l is ample and spanned by global sections resp very ample

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

balanced line bundles on fano varieties a conjecture of batyrev and manin relates arithmetic properties of varieties with ample anticanonical class to geometric invariants in particular counting functions defined by metrized ample line bundles and the corresponding asymptotics of rational points of bounded height are interpreted in terms of cones of effective divisors and certain thresholds with respect to these cones this framework leads to the notion of balanced line bundles whose counting functions conjecturally capture generic distributions of rational points we investigate balanced line bundles in the context of the minimal model program with special regard to the classification of fano threefolds

second chern class of fano manifolds and anti canonical geometry let x be a fano manifold of picard number one we establish a lower bound for the second chern class of x in terms of its index and degree as an application if y is a n dimensional fano manifold with k y n 3 h for some ample divisor h we prove that h 0 y h ge n 2 moreover we show that the rational map defined by vert mh vert is birational for m ge 5 and the linear system vert mh vert is basepoint free for m ge 7 as a by product the pluri anti canonical systems of singular weak fano varieties of dimension at most 4 are also investigated

nine moment phonon hydrodynamics based on the maximum entropy closure one dimensional flow the maximum entropy distribution functions applied to callaway s model for phonon gas dynamics lead to a hierarchy of closed systems of moment equations the system of equations for the energy density and the heat flux is the first member of this hierarchy of closures here emphasis is placed on analysing the next member the 9 moment maximum entropy system that involves the flux of the heat flux as an extra gas state variable after presenting a study of the one dimensional rotationally symmetric reduction of this system we explicitly calculate a single generating function of three lagrange multipliers in terms of which the reduced system of three evolution equations for these multipliers can be cast into a symmetric hyperbolic form in the context of determining the lagrange multipliers as explicit functions of the moment densities we discuss new aspects of the expansion of various non equilibrium quantities about quasi equilibrium states this expansion is fundamentally a non equilibrium expansion that includes the heat flux in a non perturbative manner i e there are no unphysical limitations on the magnitude of the nonvanishing component of the heat flux to maintain a theory results are presented both at the first order in the expansion and at the second order this enables us to verify the internal consistency of our approach and to justify the non equilibrium generalization of the method of grad

on the simultaneous generation of jets of the adjoint bundles abstract in this paper we investigate the problem of the simultaneous generation of r jets of the bundle x l m where l is an ample and globally generated line bundle on x the results work in arbitrary characteristics a similar problem is studied when x admits singularities

global generation of pluricanonical and adjoint linear series on smooth projective threefolds the purpose of this paper is to show how the cohomological techniques developed by kawamata reid shokurov and others lead to some effective and practical results of reider type on freeness of linear series on smooth complex projective threefolds in recent years there has been a great deal of interest in the geometric properties of pluricanonical and adjoint linear systems on surfaces and higherdimensional varieties among other things one wants to understand as explicitly as possible when the systems in question are base point free or very ample modem work in this area goes back to kodaira kod and bombieri bmb who studied pluricanonical maps of surfaces of general type more recently many of their results have emerged as special cases of a celebrated theorem of reider rdr reider uses vector bundle techniques to show that if b is a nef line bundle on a surface x such that b2 5 then kx b is globally generated unless there exist certain special curves c c x such that b c 1 he also obtains analogous criteria for kx b to be very ample a cohomological approach to these theorems based on miyaoka s vanishing theorem for zariski decompositions was given by sakai sak2

characterization of the 4 canonical birationality of algebraic threefolds for nonsingular projective threefolds x of general type whose geometric genus p g h 0 x k x is ge 5 the birationality of the fourth canonical map varphi 4 x phi 4k x was characterized by d q zhang and the first author in 2008 this paper aims at characterizing the birationality of varphi 4 x for those x with p g 4

a construction of horikawa surface via q gorenstein smoothings we prove that the fintushel stern construction of a horikawa surface which is obtained from elliptic surface via a rational blow down surgery in the smooth category can be performed in the complex category the main technique involved is mathbb q gorenstein smoothings

maximum of the local entropy production becomes minimal in stationary processes in this paper we propose a new principle for stationary thermodynamic processes the maximum of the local entropy production becomes minimal in stationary processes in order to show the usefulness of the principle we consider one dimensional stationary heat transfer in monatomic gases here we solve extended moment schemes that follow from the boltzmann equation such schemes require boundary conditions for all moments under consideration and these are constructed by means of the new principle moreover we show that the minimum principle for the global entropy production does not lead to good results in this case s0031 9007 98 06306 6

extended thermodynamics of fluids versus the revised enskog equation abstract the joint labours of specialists in mechanics and thermodynamics have suggested certain purely phenomenological hypotheses regarding the structure of the extended thermodynamics et of gases and fluids the microscopic counterparts to these propositions are however known for very few kinds of kinetic models of continuum systems boltzmann s equation while boltzmann s h theorem lies at the very basis of the kinetic origin of the et of dilute gases in particular in justifying the local constitutive assumption lca underlying the liu muller theory no such interrelation between macroscopic and microscopic approaches has been proved as yet for the revised enskog equation ree with the appropriate h theorem accordingly the purpose of this paper is to construct the et of dense gases and fluids within the framework of the ree and to investigate the nature of the resulting discrepancies with the existing phenomenological theories

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

irreducible tensor description i a classical gas a classical moderately rarefied simple monatomic gas is considered and it is supposed that its behavior is governed by boltzmann s equation then after a brief review of the fundamental properties of the symmetric traceless tensors the irreducible equations of transfer for the relative symmetric traceless moments of the one point distribution function f are systematically derived and related to those of johnston subsequently grad s expansion of the distribution function in terms of reducible three dimensional hermite polynomials which does not fit in together and with the equations of transfer just mentioned is rigorously transformed into its irreducible counterpart fashioned by mathematical apparatus such as one dimensional laguerre polynomials and ikenberry s tensorial harmonics finally some useful conversion formulas between the relative symmetric traceless moments and the tensorial laguerre ikenberry expansion coefficients of f are deduced and the irreducible variant of grad s moment t

on the projective normality of double coverings over a rational surface abstract we study the projective normality of a minimal surface x which is a ramified double covering over a rational surface s with dim k s 1 in particular horikawa surfaces the minimal surfaces of general type with k x 2 2 p g x 4 are of this type up to resolution of singularities let be the covering map from x to s we show that the z 2 invariant adjoint divisors k x r a are normally generated where the integer r 3 and a is an ample divisor on s

random fixed points and viable random solutions of functional differential inclusions abstract there are given conditions under which the existence of viable solutions yields the existence of viable random solutions to functional differential inclusions

a numerical criterion for very ample line bundles let x be a projective algebraic manifold of dimension n and let l be an ample line bundle over x we give a numerical criterion ensuring that the adjoint bundle kx l is very ample the sufficient conditions are expressed in terms of lower bounds for the intersection numbers l y over subvarieties y of x in the case of surfaces our criterion gives universal bounds and is only slightly weaker than i reider s criterion when dimx 3 and codimy 2 the lower bounds for l y involve a numerical constant which depends on the geometry of x by means of an iteration process it is finally shown that 2kx ml is very ample form 12n our approach is mostly analytic and based on a combination of hormander s l estimates for the operator lelong number theory and the aubin calabi yau theorem

on birational geometry of minimal threefolds with numerically trivial canonical divisors for a minimal threefold x with k x equiv 0 and a nef and big weil divisor l on x we investigate the birational geometry inspired by l we prove that ml and k x ml give birational maps for all m ge 17 the result remains true under weaker assumption that l is big and has no stable base components

k3 surfaces with involution equivariant analytic torsion and automorphic forms on the moduli space in this paper we introduce an invariant of a k3 surface with 2 action equipped with a 2 invariant kahler metric which we obtain using the equivariant analytic torsion of the trivial line bundle this invariant is shown to be independent of the choice of the kahler metric it can be viewed as a function on the moduli space of k3 surfaces with involution the main result of this paper is that this function can be identified with an automorphic form which characterizes the discriminant locus in particular we show that the ray singer analytic torsion of the trivial line bundle on an enriques surface with ricci flat kahler metric is given by the value of the norm of the borcherds function at its period point

symmetric positive semidefinite and positive definite real solutions of ax xat and ax yb abstract given the equations ax xa t and ax yb with arbitrary nonzero real matrices a and b of the same size we seek all real solutions x and y which are 1 symmetric 2 symmetric and positive semidefinite and 3 symmetric and positive definite necessary and sufficient conditions for the existence of such solutions and their general forms are derived

surfaces of general type with pg 0 k2 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 open image in new window 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

random fixed points of k set and pseudo contractive random maps abstract let be a measurable space x and y separable banach spaces and c a weakly compact subset of x let f c y and t c y be continuous random operators then the deterministic solvability of the equation f x t x 0 x c implies the stochastic solvability of it provided that f t is demiclosed at zero and t c is bounded for each as applications random fixed points of various types of pseudo contractive and k set contractive random operators are obtained

a sharp lower bound for the canonical volume of 3 folds of general type let v be a smooth projective 3 fold of general type denote by k 3 a rational number the self intersection of the canonical sheaf of any minimal model of v one defines k 3 as the canonical volume of v assume p g v 2 we show that k 3 ge frac 1 3 which is a sharp lower bound then we classify those v with small volume we also give some new examples with p g 2 which have maximal canonical stability index finally we give an application to 4 folds of general type

on the adjunction mapping of very ample vector bundles of corank one let e be a very ample vector bundle of rank n 1 over a smooth complex projective variety x of dimension n 3 the structure of x e being known when k x det e 0 we investigate the structure of the adjunction mapping when 0 k x det e n

on irregular threefolds and fourfolds with numerically trivial canonical bundle we prove that for a smooth projective irregular threefold x with k x equiv 0 and a nef and big divisor l on x ml p gives a birational map for all m ge 3 and all p in mathrm pic 0 x we also use the same method to deal with fourfolds and prove that for a smooth projective irregular fourfold x with k x equiv 0 and an ample divisor l on x ml p gives a birational map for all m ge 5 and all p in mathrm pic 0 x these results are optimal

response to z banach s comments on the modified moment method and irreversible thermodynamics abstract in this paper the author responds to the comments on the modified moment method and irreversible thermodynamics made by z banach physica a 145 1987 105 in this paper banach suggests a variational method in which the lagrange multipliers are determined from the constraints alone by disregarding the entropy balance equation it is shown that since this method does not yield an extended gibbs relation consistent with the entropy balance equation or the h theorem there is no irreversible thermodynamics formalism afforded by the method consequently his criticism cannot be supported from the viewpoint of irreversible thermodynamics it is also pointed out that neither is there a mathematical and physical support for the criticism he makes on the cumulant expansion for the boltzmann collision integral

on positivity of line bundles on enriques surfaces we study linear systems on enriques surfaces we prove rationality of seshadri constants of ample line bundles on enriques surfaces and provide lower bounds on these numbers we show the nonexistence of k very ample line bundles on enriques surfaces of degree 4k 4 for k 1 thus answering an old question of ballico and sommese introduction in this note we answer a question posed by ballico and sommese 1 see also terakawa 14 concerning the existence of k very ample line bundles of degree 4k 4 on enriques surfaces we show that such line bundles exist if and only if k 0 and that in fact the degree of a k very ample line bundle on an enriques surface is subject to a much more restrictive bound theorem 2 4 this has strong consequences for the local positivity of line bundles on enriques surfaces we address this questions in section 3 in particular we show that seshadri constants of ample line bundles on enriques surfaces are always rational 1 auxiliary material and notation the basic reference for this paper is the monograph 6 by cossec and dolgachev here we briefly recall properties of enriques surfaces needed for our considerations first of all an enriques surface is a surface y of kodaira dimension zero with no differential forms in particular ho ky 0 and the canonical bundle ky is a 2torsion element in pico y i e 2ky oy to alleviate notation it is convenient to use the following convention in the sequel we make frequent use of vanishing theorems and statements on adjoint line bundles since the canonical divisor ky is numerically trivial the adjoint line bundle ky l has the same numerical properties as l in particular if l is nef and big ample then so is ky l by the nakai moishezon criterion so in order to get vanishing statements for l we apply vanishing theorems to ky l and use the fact that 2ky is trivial in the sequel we do so without comment and hope to cause no confusion this way received by the editors june 1 2000 2000 mathematics subject classification primary 14j28 secondary 14c20 14e25

nine moment phonon hydrodynamics based on the modified grad type approach hyperbolicity of the one dimensional flow after expanding the distribution function about an anisotropic planck function the new moment closure method of banach and larecki applied to the boltzmann peierls equation for the phonon gas dynamics leads to a whole hierarchy of closed systems of moment equations the system of equations for the energy density and the heat flux is the first non perturbative member of this hierarchy of closures in our previous paper 2004 j phys a math gen 37 9805 emphasis was placed on deriving the next member the 9 moment anisotropic closure that involves the flux of the heat flux as an extra gas state variable here as a first step in effectively analysing this system we present a study of the one dimensional rotationally symmetric reduction of these equations under the assumption of callaway s model a systematic procedure is derived which shows that the obtained system of three evolution equations for three nonvanishing gas state variables can be cast into a symmetric hyperbolic form for the sake of completeness we describe explicitly the region of symmetric hyperbolicity in parameter space the space defined by the gas state variables the evolution system is symmetric hyperbolic for significant ranges of physical conditions i e there are effectively no unphysical limitations on the magnitude of the energy density and the heat flux this paper also deals with the eigenvalue problem and calculates approximately the characteristic speeds

irreducibility brill noether loci and vojta s inequality this paper deals with generalizations of hilbert s irreducibility theorem the classical hilbert irreducibility theorem states that for any cover f of the projective line defined over a number field k there exist infinitely many k rational points on the projective line such that the fiber of f over p is irreducible over k in this paper we consider similar statements about algebraic points of higher degree on curves of any genus we prove that hilbert s irreducibility theorem admits a natural generalization to rational points on an elliptic curve and thus via a theorem of abramovich and harris to points of degree 3 or less on any curve we also present examples that show that this generalization does not hold for points of degree 4 or more these examples come from an earlier geometric construction of debarre and fahlaoui some additional necessary facts about this construction can be found in the appendix provided by debarre we exhibit a connection between these irreducibility questions and the sharpness of vojta s inequality for algebraic points on curves in particular we show that vojta s inequality is not sharp for the algebraic points arising in our examples

positivity of line bundles on general blow ups of p2 abstract let x be the blow up of p 2 at r general points p 1 p r p 2 we study line bundles on x given by plane curves of degree d passing through p i with multiplicity at least m i motivated by results in 39 we establish conditions for ampleness very ampleness and global generation of such line bundles

on syzygies of abelian varieties in this paper we prove the following result let x be a co mplex torus and m a normally generated line bundle on x then for every p 0 the line bundle mp 1 satisfies property np of green lazarsfeld

the gonality of complete intersection curves abstract the purpose of this paper is to show that for a complete intersection curve c in projective space other than a few exceptions stated below any morphism f c p r satisfying deg f o p r 1 deg c is obtained by projection from a linear space in particular we obtain bounds on the gonality of such curves and compute the gonality of general complete intersection curves we also prove a special case of one of the well known cayley bacharach conjectures posed by eisenbud green and harris

fundamental groups of galois closures of generic projections for the galois closure x gal of a generic projection from a surface x it is believed that 1 x gal gives rise to new invariants of x however in all examples this group is surprisingly simple in this article we offer an explanation for this phenomenon we compute a quotient of 1 x gal that depends on 1 x and data from the generic projection only in all known examples this quotient is in fact isomorphic to 1 x gal as a byproduct we simplify the computations of moishezon teicher and others

tilt stability vanishing theorems and bogomolov gieseker type inequalities abstract we investigate the tilt stability of stable sheaves on projective varieties with respect to certain tilt stability conditions depends on two parameters constructed by bridgeland 11 see also 1 6 5 for a stable sheaf we give effective bounds of these parameters such that the stable sheaf is tilt stable these allow us to prove new vanishing theorems for stable sheaves and an effective serre vanishing theorem for torsion free sheaves using these results we also prove bogomolov gieseker type inequalities for the third chern character of a stable sheaf on p 3

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

on the second sectional h arithmetic genus of polarized manifolds let x l be a polarized variety defined over the complex number field with dim x n in this paper we introduce the notion of the i th sectional h arithmetic genus hi x l for every integer i with 0 i n we expect that this invariant has a property similar to the euler poincare characteristic of the structure sheaf of i dimensional varieties in this paper we consider the case where x is smooth and i 2 and we study a polarized version of some results in the theory of surfaces

irreducible tensor description iii thermodynamics of a low temperature phonon gas let one assume that the interacting phonon gas whose behavior is governed by the boltzmann peierls equation inhabits an insulating crystal at sufficiently low temperature then within the framework of a single acoustic phonon branch and of an isotropic long wavelength approximation to the dispersion relation the simplest acceptable version of extended irreversible thermodynamics based upon the nine moment system of differential equations for the slow and fast gas state variables is carefully investigated it is clearly demonstrated that in virtue of the structure simplifications just mentioned the conceptually different macroscopic kinetic and variational procedures which are discussed in this paper appear complementary to each other finally with the help of a suitable contraction of the nine moment system of field equations for callaway s relaxation model a slightly generalized nonlinear variant of ordinary low temperature phonon hydrodynamics is explicitly derived

on the foundations of extended irreversible thermodynamics a theory of macroscopic systems which takes as independent variables the slow conserved ones plus the fast dissipative fluxes is carefully analyzed at three levels of description macroscopic thermodynamic microscopic projection operators and mesoscopic fluctuation theory such a description is compared with the memory function approach based only on the conserved variables we find that the first theory is richer and wider than the second one and some misunderstandings in this connection are clarified and discussed

examples of smooth non general type surfaces in p4 smooth projective varieties with small invariants have received renewed interest in recent years primarily due to the fine study of the adjunction mapping now through the effort of several mathematicians a complete classification of smooth surfaces in bbb p 4 has been worked out up to degree 10 and a partial one is available in degree 11 on the other side recently ellingsrud and peskine have proved hartshorne s conjecture that there are only finitely many families of smooth surfaces in bbb p 4 not of general type it is believed that the degree of the smooth non general type surfaces in bbb p 4 should be less than or equal to 15 the aim of this paper is to provide a series of examples of smooth surfaces in bbb p 4 not of general type in degrees varying from 12 up to 14 and to describe their geometry by using mainly syzygies and liaison techniques we construct the following families of surfaces begin enumerate item minimal proper elliptic surfaces of degree 12 and sectional genus pi 13 item two types of non minimal proper elliptic surfaces of degree 12 and sectional genus pi 14 item non minimal k3 surfaces of degree 13 and sectional genus 16 and item non minimal k3 surfaces of degree 14 and sectional genus 19 1991 mathematics subject classification 14m07 14j25 14j26 14j28 14c05

on vector bundles on 3 folds with sectional genus 1 here we give a classification in characteristic zero of pairs v e with v being a smooth connected complete 3 fold and e a rank 2 spanned ample vector bundle on v with sectional genus 1 the proof uses the partial classification of fano 3 folds and mori theory let x be an integral complete manifold dim x n and e a rank n 1 vector bundle on x we define the sectional genus g e using the following formula 2g e 2 kx c 1 e cn e it is easy to check see remark 4 1 that g e is an integer if e has a section with zero locus c of codimension n 1 then g e pa c in this paper we give a reasonable classification in characteristic zero see theorem 0 of the pairs v e with v being a smooth complete 3 fold and e a rank 2 ample spanned vector bundle on e with g e 1 note that if g e 0 then kv cl e is not nef under this assumption but a far weaker assumption on v e wisniewski in w gave a classification using mori theory see also 4 2 for a similar result the case g e 1 seems to be of a different order of difficulty but we will use very much of wisniewski s work w the partial classification of fano 3folds and classifications e g io based on mori theory in an interesting paper fl fujita gave two definitions of sectional genus for higher rank vector bundles the second one is the sectional genus of the tautological line bundle on p e the first one and more interesting is called the cl sectional genus of e it is the sectional genus of the line bundle det e i e g is defined by 2g 2 kx n 1 c1 e c1 e n 1 for instance if e is the direct sum of n 1 line bundles all isomorphic to a line bundle a with our definition 2g e 2 kx n 1 a an i while the cl sectional genus g of e is defined by 2g 2 n 1 n kx n 1 2a an i which is much bigger if a is ample for instance the cl sectional genus of the bundle in b in the statement of theorem 0 is 33 the first three sections of this paper are devoted to the proof of the following result over an algebraically closed base field with characteristic zero received by the editors february 20 1989 1980 mathematics subject classification 1985 revision primary 1 4j30 secondary 14f05

on the spannedness of adjunction of vector bundles let x be a smooth projective variety of dimension n defined over the field of complex numbers and e be an ample vector bundle on x the linear system kx c e is called the adjoint bundle for vector bundle e the nefness and the ampleness of the adjoint bundle kx cl e have been investigated by several authors abw f p1 yz in this paper we shall investigate the base point freeness spannedness of the adjoint bundle kx c e i e we ask the following

on the geometry of principal homogeneous spaces let b be a curve defined over an algebraically closed field k and let pi colon x to b be an elliptic surface with base curve b we investigate the geometry of everywhere locally trivial principal homogeneous spaces for x i e elements of the tate shafarevich group of the generic fiber of pi if y is such a principal homogeneous space of order n we find strong restrictions on the bbb p n 1 bundle over b into which y embeds examples for small values of n show that at least in some cases these restrictions are sharp finally we determine these bundles in case k has characteristic zero b bbb p 1 and x is generic in a suitable sense

effective bounds for very ample line bundles let l be an ample line bundle on a non singular projective n fold x it is first shown that 2k x ml is very ample for m ge 2 3n 1 choose n the proof develops an original idea of y t siu and is based on a combination of the riemann roch theorem together with an improved noetherian induction technique for the nadel multiplier ideal sheaves in the second part an effective version of the big matsusaka theorem is obtained refining an earlier version of y t siu there is an explicit polynomial bound m 0 m 0 l n l n 1 cdot k x of degree le n3 n in the arguments such that ml is very ample for m ge m 0 the refinement is obtained through a new sharp upper bound for the dualizing sheaves of algebraic varieties embedded in projective space

maximisation of the entropy in non equilibrium this author proves that the maximisation of the entropy in non equilibrium is equivalent to the exploitation of the entropy inequality in extended thermodynamics

on tensor products of ample line bundles on abelian varieties in the present note we consider the question of global generation and very ampleness of tensor products of ample line bundles on abelian varieties it is a classical result of lefschetz 3 theorem 4 5 1 that the third tensor power of an ample line bundle on an abelian variety is very ample it is also well known that the second power is globally generated we generalize these results to the product of three resp two arbitrary ample line bundles further we give criteria for a tensor product of two ample line bundles to be very ample the essential tool in the proof of the results for the powers of an ample line bundle is the theorem of the square 3 theorem 2 3 3 our method is to replace this theorem by arguments involving the isogeny associated to an ample line bundle throughout this note the base field is c

explicit equations of a fake projective plane fake projective planes are smooth complex surfaces of general type with betti numbers equal to those of the usual projective plane they come in complex conjugate pairs and have been classified as quotients of the 2 dimensional ball by explicitly written arithmetic subgroups in the following we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order 7 automorphism

families of nodal curves on projective threefolds and their regularity via postulation of nodes the main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds precisely given x a smooth projective threefold e a rank two vector bundle on x l a very ample line bundle on x and k 0 0 integers and denoted by v e l k the subscheme of p h 0 e l k parametrizing global sections of e l k whose zero loci are irreducible and nodal curves on x we present a new cohomological description of the tangent space t s v e l k at a point s 2 v e l k this description enable us to determine effective and uniform upper bounds for which are linear polynomials in k such that the family v e lk is smooth and of the expected dimension regular for short the almost sharpness of our bounds is shown by some interesting examples furthermore when x is assumed to be a fano or a calaby yau threefold we study in detail the regularity property of a point s 2 v e lk related to the postulation of the nodes of its zero locus c v s x roughly speaking when the nodes of c are assumed to be in general position either on x or on an irreducible divisor of x having at worst log terminal singularities or to lie on a l c i and subcanonical curve in x we find upper bounds onwhich are respectively cubic quadratic and linear polynomials in k ensuring the regularity of v e l k at s finally when x p 3 we also discuss some interesting geometric properties of the curves given by sections parametrized by v e ox k