slope inequality for families of curves over surfaces in this paper we investigate the general notion of the slope for families of curves f x rightarrow y the main result is an answer to the above question when dim y 2 and we prove a lower bound for this new slope in this case over fields of any characteristic both the notion and the slope inequality are compatible with the theory for dim y 0 1 in a very natural way and this gives a strong evidence that the slope for an n fold fibration of curves f x rightarrow y may be k x y n mathrm ch n 1 f omega x y rather than the usual stability methods the whole proof of the slope inequality here is based on a completely new method using characteristic p 0 geometry a simpler version of this method yields a new proof of the slope inequality when dim y 1

optimal routing under capacity and distance restrictions this paper describes an integer linear programming algorithm for vehicle routing problems involving capacity and distance constraints the method uses constraint relaxation and a new class of subtour elimination constraints two versions of the algorithm are presented depending upon the nature of the distance matrix exact solutions are obtained for problems involving up to sixty cities

the traveling salesman problem an overview of exact and approximate algorithms in this paper some of the main known algorithms for the traveling salesman problem are surveyed the paper is organized as follows 1 definition 2 applications 3 complexity analysis 4 exact algorithms 5 heuristic algorithms 6 conclusion

semistable principal g bundles in positive characteristic let x be a normal projective variety defined over an algebraically closed field k of positive characteristic let g be a connected reductive group defined over k we prove that some frobenius pullback of a principal g bundle admits the canonical reduction ep such that its extension by p p rup is strongly semistable see theorem 5 1 then we show that there is only a small difference between semistability of a principal g bundle and semistability of its frobenius pullback see theorem 6 3 this and the boundedness of the family of semistable torsion free sheaves imply the boundedness of semistable rational principal g bundles

transformation and reduction formulae for double q clausen series of type φ1 1 μ1 2 λ abstract the sears transformations are employed to establish several general series transformations for double q clausen hypergeometric series of type 1 1 1 2 these transformations yield further a number of reduction and summation formulae on the double basic hypergeometric series

some formulae for double clausenian functions abstract for certain f q s s p r r x y functions with p r 3 and q s 2 reduction and transformation formulae under various conditions upon variables and parameters are derived from an euler integral representation that requires one relation between parameters

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

reduction and transformation formulae for bivariate basic hypergeometric series the main object of this paper is to establish several bivariate basic hypergeometric series identities by means of elementary series manipulation some of them can be applied to yield transformation and reduction formulae for q kampe de feriet functions

transformation and summation formulae for kampe de feriet series abstract euler s two transformations are employed to establish several general transformation formulae for double hypergeometric series these transformations yield further a number of reduction formulae for kampe de feriet series of types f 1 0 1 1 f 1 0 0 2 and f 1 0 2 0 by specializing certain parameters a list of new summation formulae for these kampe de feriet series are obtained

partial regularity and amplitude the author continues the study of frobenius amplitude introduced in an earlier paper and compares this to various other positivity notions these are applied to deduce vanishing theorems a refinement of castelnuovo mumford regularity is also given

frobenius amplitude and strong vanishing theorems for vector bundles the primary goal of this paper is to systematically exploit the method of deligne illusie to obtain kodaira type vanishing theorems for vector bundles and more generally coherent sheaves on algebraic varieties the key idea is to introduce a number which provides a cohomological measure of the positivity of a coherent sheaf called the frobenius or f amplitude the f amplitude enters into the statement of the basic vanishing theorem and this leads to the problem of calculating or at least estimating this number most of the work in this paper is devoted to doing this various situations

fujita s conjecture and frobenius amplitude we prove a version of fujita s conjecture in arbitrary characteristic generalizing results of k e smith our methods use the frobenius morphism but avoid tight closure theory we also obtain versions of fujita s conjecture for coherent sheaves with certain ampleness properties

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

exact algorithms for the vehicle routing problem based on spanning tree and shortest path relaxations we consider the problem of routing vehicles stationed at a central facility depot to supply customers with known demands in such a way as to minimize the total distance travelled the problem is referred to as the vehicle routing problem vrp and is a generalization of the multiple travelling salesman problem that has many practical applications we present tree search algorithms for the exact solution of the vrp incorporating lower bounds computed from i shortest spanning k degree centre tree k dct and ii q routes the final algorithms also include problem reduction and dominance tests computational results are presented for a number of problems derived from the literature the results show that the bounds derived from the q routes are superior to those from k dct and that vrps of up to about 25 customers can be solved exactly

kodaira dimension and zeros of holomorphic one forms we show that every holomorphic one form on a smooth complex projective variety of general type must vanish at some point the proof uses generic vanishing theory for hodge modules on abelian varieties

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

bogomolov s inequality for higgs sheaves in positive characteristic we prove bogomolov s inequality for higgs sheaves on varieties in positive characteristic p that can be lifted modulo p 2 this implies the miyaoka yau inequality on surfaces of non negative kodaira dimension liftable modulo p 2 this result is a strong version of shepherd barron s conjecture our inequality also gives the first algebraic proof of bogomolov s inequality for higgs sheaves in characteristic zero solving the problem posed by narasimhan

ordinary and basic bivariate hypergeometric transformations associated with the appell and kampe de feriet functions by means of formal power series method and series rearrangement technique the authors investigate and establish several ordinary as well as basic or q bivariate hypergeometric transformation and reduction formulas these latter formulas may be considered as basic or q extensions of some recent results considered by lee et al internat j comput math 78 2001 303 321 and karlsson et al internat j math statist sci 9 2000 211 226 the bivariate hypergeometric series which are considered in this paper are closely related to the familiar appell and kampe de feriet functions

vanishing via lifting to second witt vectors and a proof of an isotriviality result abstract a proof based on reduction to finite fields of esnault and viehweg s stronger version of the sommese vanishing theorem for k ample line bundles is given this result is used to give different proofs of isotriviality results of parshin and migliorini

a new subtour elimination constraint for the vehicle routing problem abstract vehicle routing problems vrp are concerned with the delivery of a single commodity from a centralized depot to a number of specified customer locations with known demands in this paper we consider the vrp characterized by fixed or variable number of vehicles common vehicle capacity distance restrictions and minimization of total distance travelled by all vehicles as the objective we develop an exact algorithm based on a new subtour elimination constraint the algorithm is implemented using the cplex package for solving the relaxed subproblems computational results on 1590 simulated problems and 10 literature problems without distance restrictions are reported and a comparative analysis is carried out

new transformation and reduction formulae for double q clausen series abstract the sears transformations for non terminating 3 2 series are applied to derive several general series transformations for another two double q clausen series of type 2 0 1 2 and 2 0 0 3 further new transformations and reduction formulae are established for the basic clausen hypergeometric series in particular several bibasic reduction formulae are obtained

tight closure and plus closure in dimension two we prove that the tight closure and the graded plus closure of a homogeneous ideal coincide for a two dimensional n graded domain of finite type over the algebraic closure of a finite field this answers in this case a tantalizing question of hochster

noncommutative ampleness from finite endomorphisms abstract let x be a projective integral scheme with endomorphism where is finite but not an automorphism we examine noncommutative ampleness of bimodules defined by in contrast to the automorphism case one sided ampleness is possible we also find that rings and bimodule algebras associated with are not noetherian

ample filters of invertible sheaves abstract let x be a scheme proper over a commutative noetherian ring a we introduce the concept of an ample filter of invertible sheaves on x and generalize the most important equivalent criteria for ampleness of an invertible sheaf we also prove the theorem of the base for x and generalize serre s vanishing theorem we then generalize results for twisted homogeneous coordinate rings which were previously known only when x was projective over an algebraically closed field specifically we show that the concepts of left and right ampleness are equivalent and that the associated twisted homogeneous coordinate ring must be noetherian

polarized endomorphisms of normal projective threefolds in arbitrary characteristic let x be a projective variety over an algebraically closed field k of arbitrary characteristic p ge 0 a surjective endomorphism f of x is q polarized if f h sim qh for some ample cartier divisor h and integer q 1 suppose f is separable and x is mathbb q gorenstein and normal we show that the anti canonical divisor k x is numerically equivalent to an effective mathbb q cartier divisor strengthening slightly the conclusion of boucksom de fernex and favre duke math j 161 8 1455 1520 2012 theorem c and also covering singular varieties over an algebraically closed field of arbitrary characteristic suppose f is separable and x is normal we show that the albanese morphism of x is an algebraic fibre space and f induces polarized endomorphisms on the albanese and also the picard variety of x and k x being pseudo effective and mathbb q cartier means being a torsion mathbb q divisor let f text gal overline x rightarrow x be the galois closure of f we show that if p 5 and co prime to deg f text gal then one can run the minimal model program mmp f equivariantly after replacing f by a positive power for a mildly singular threefold x and reach a variety y with torsion canonical divisor and also with y being a quasi etale quotient of an abelian variety when dim y le 2 along the way we show that a power of f acts as a scalar multiplication on the neron severi group of x modulo torsion when x is a smooth and rationally chain connected projective variety of dimension at most three

integer linear programming formulation for a vehicle routing problem abstract a mixed integer linear programming formulation is given for the vehicle routing problem constrained by vehicle capacity and distance travelled this corrects an error in the formulation recently proposed by kulkarni and bhave

ample filters and frobenius amplitude abstract let x be a projective scheme over a field we show that the vanishing cohomology of any sequence of coherent sheaves is closely related to vanishing under pullbacks by the frobenius morphism we also compare various definitions of ample locally free sheaf and show that the vanishing given by the frobenius morphism is in a certain sense the strongest possible our work can be viewed as various generalizations of the serre vanishing theorem

naive noncommutative blowing up let b x mathscr l be the twisted homogeneous coordinate ring of an irreducible variety x over an algebraically closed field k with dim x 2 assume that c in x and in aut x are in sufficiently general position we show that if one follows the commutative prescription for blowing up x at c but in this noncommutative setting one obtains a noncommutative ring r r x c mathscr l with surprising properties 1 r is always noetherian but never strongly noetherian 2 if r is generated in degree one then the images of the r point modules in qgr r are naturally in one to one correspondence with the closed points of x however in both qgr r and gr r the r point modules are not parametrized by a projective scheme 3 while qgr r has finite cohomological dimension dim k h 1 mathscr o

generic positivity and foliations in positive characteristic abstract we prove generic semipositivity of the tangent bundle of a non uniruled calabi yau variety in positive characteristic we also construct an example of a nef line bundle in characteristic zero whose each reduction to positive characteristic is not nef

canonical maps to twisted rings if a is a strongly noetherian graded algebra generated in degree one then there is a canonically constructed graded ring homomorphism from a to a twisted homogeneous coordinate ring open image in new window which is surjective in large degree this result is a key step in the study of projectively simple rings the proof relies on some results concerning the growth of graded rings which are of independent interest