inseparable local uniformization abstract it is known since the works of zariski in the early 40s that desingularization of varieties along valuations called local uniformization of valuations can be considered as the local part of the desingularization problem it is still an open problem if local uniformization exists in positive characteristic and dimension larger than three in this paper we prove that zariski local uniformization of algebraic varieties is always possible after a purely inseparable extension of the field of rational functions and therefore any valuation can be uniformized by a purely inseparable alteration

resolution of singularities of threefolds in positive characteristic ii abstract together with vincent cossart olivier piltant resolution of singularities of threefolds in positive characteristic i reduction to local uniformization on artin schreier and purely inseparable coverings j algebra 320 3 2008 1051 1082 this article gives a complete proof of desingularization of quasiprojective varieties of dimensional 3 on fields which are differentially finite over perfect fields

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

symmetries as integrability criteria for differential difference equations in this paper we review the results obtained by the generalized symmetry method in the case of differential difference equations during the last 20 years together with general theory of the method classification results are discussed for classes of equations which include the volterra toda and relativistic toda lattice equations

big line bundles over arithmetic varieties 5 equidistribution theory 26 5 1 a generic equidistribution theorem 27 5 2 equidistribution at infinite places 30 5 3 equidistribution at finite places 32 5 4 equidistribution of small subvarieties 35 5 5 equidistribution over algebraic dynamical systems 37 5 6 equidistribution over multiplicative groups 39 5 7 equidistribution theory 41

the bogomolov conjecture for totally degenerate abelian varieties we prove the bogomolov conjecture for a totally degenerate abelian variety a over a function field we adapt zhang s proof of the number field case replacing the complex analytic tools by tropical analytic geometry a key step is the tropical equidistribution theorem for a at the totally degenerate place v as a corollary we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v

intersecting subvarieties of g_m n with algebraic subgroups let x subset g m n be an irreducible closed subvariety defined over bar q we bound the height of algebraic points on x that are in a certain sense close to the union of all algebraic subgroup of g m n of dimension m n dim x the bound we obtain is effective and will be expressed as a function of the height of x the degree of x and n we then apply this bound to derive certain finiteness results if m is also strictly less than n dim x

ramification of valuations abstract in this paper we develop a very explicit theory of ramification of general valuations in algebraic function fields in characteristic zero and arbitrary dimension we obtain the strongest possible generalization of the classical ramification theory of local dedekind domains we further develop a ramification theory of algebraic functions fields of dimension two in positive characteristic we prove that local monomialization and simultaneous resolution hold under very mild assumptions and give pathological examples

counterexamples to local monomialization in positive characteristic a local monomialization of an algebraic map between varieties along a valuation is a sequence of blowups of nonsingular subvarieties along the valuation in the domain and target of the map such that the resulting map can be expressed by monomial inclusion in a suitable sense in this paper we give examples showing that a local monomialization does not always exist over fields of positive characteristic it was shown in earlier work of the author that a local monomialization always exists over fields of characteristic zero

singularities in positive characteristic stratification and simplification of the singular locus abstract we introduce an upper semi continuous function that stratifies the highest multiplicity locus of a hypersurface in arbitrary characteristic the blow up along the maximum stratum defined by this function leads to a form of simplification of the singularities known as the reduction to the monomial case

the number of maximal torsion cosets in subvarieties of tori we present sharp bounds on the number of maximal torsion cosets in a subvariety of the complex algebraic torus gm n our first main result gives a bound in terms of the degree of the defining polynomials a second result gives a bound in terms of the toric degree of the subvariety as a consequence we prove the conjectures of ruppert and of aliev and smyth on the number of isolated torsion points of a hypersurface these conjectures bound this number in terms of the multidegree and the volume of the newton polytope of a polynomial defining the hypersurface respectively

the τ invariant and elimination abstract in this paper we present some results showing the good behavior of the invariant of a rees algebra with integral closure and elimination of variables

hasse principles for higher dimensional fields for rather general excellent schemes x k kato deflned complexes of gerstenbloch ogus type involving the galois cohomology groups of all residue flelds of x for arithmetically interesting schemes i e schemes over global or local flelds or over their rings of integers he developed a web of conjectures on some of these complexes generalizing the fundamental exact sequence of brauer groups for a global fleld he proved these conjectures for low dimensions we prove kato s conjecture with inflnite coe cients over number flelds in particular it gives a hasse principle for function flelds f over a number fleld k involving the corresponding function flelds fv over the completions kv of k we get a conditional result over global flelds k of positive characteristic assuming resolution of singularities this is unconditional for dim x 3 due to recent results on resolution there are also applications to other cases considered by kato

generalized symmetry classification of discrete equations of a class depending on twelve parameters we carry out the generalized symmetry classification of polylinear autonomous discrete equations defined on the square which belong to a twelve parametric class the direct result of this classification is a list of equations containing no new examples however as an indirect result of this work we find a number of integrable examples pretending to be new one of them has a nonstandard symmetry structure the others are analogues of the liouville equation in the sense that they are darboux integrable we also enumerate all equations of the class which are linearizable via a two point first integral and specify the nature of integrability of some known equations

severi inequality for varieties of maximal albanese dimension in this paper we prove the general severi inequality for varieties of maximal albanese dimension suppose that x is an n dimensional projective normal minimal and mathbb q gorenstein variety of general type in characteristic zero if x is of maximal albanese dimension then k n x ge 2 n chi omega x

places of algebraic function fields in arbitrary characteristic we consider the zariski space of all places of an algebraic function field f k of arbitrary characteristic and investigate its structure by means of its patch topology we show that certain sets of places with nice properties e g prime divisors places of maximal rank zero dimensional discrete places lie dense in this topology further we give several equivalent characterizations of fields that are large in the sense of f pop s annals paper embedding problems over large fields we also study the question whether a field k is existentially closed in an extension field l if l admits a k rational place in the appendix we prove the fact that the zariski space with the zariski topology is quasi compact and that it is a spectral space

positive line bundles on arithmetic varieties p 5 u integers is defined to count the volume of the lattice i l n of integral sections in the space j7 lon of real sections with supremum norm we want to prove that the leading term of this hilbert function as n f oo is given in terms of the height of x in as 1 this is known as a theorem of gillet and soule gs2 if x has a regular generic fiber beside this known result our proof uses hironaka s theorem on resolutions of singularities and minkowski s theorem on successive minima by hironaka s theorem we may construct

examples of smash nilpotent cycles on rationally connected varieties abstract voevodsky has conjectured that numerical and smash equivalence coincide on a smooth projective variety we prove this conjecture holds for uniruled 3 folds 4 folds whose mrcc quotient has dimension 2 and for smooth complete intersections with very small degree

local monomialization of analytic maps abstract in this paper local monomialization theorems are proven for morphisms of complex and real analytic spaces this gives the generalization of the local monomialization theorem for morphisms of algebraic varieties over a field of characteristic zero proven in 16 and 18 to analytic spaces

rost nilpotence and higher unramified cohomology abstract we develop an approach to proving the rost nilpotence principle involving higher unramified cohomology along with a decomposition of the diagonal technique we use this to prove the principle for certain threefolds over a perfect field

binormal motion of curves of constant curvature and torsion generation of soliton surfaces the purely binormal motion of curves of constant curvature or torsion respectively is shown to lead to integrable extensions of the dym and classical sine gordon equations in the case of the extended dym equation a reciprocal invariance is used to establish the existence of novel dual soliton surfaces associated with a given soliton surface a cc ideal formulation is adduced to obtain a matrix darboux invariance for the extended dym and reciprocally linked m 2 kdv equations a backlund transformation with a classical constant length property is thereby constructed which allows the generation of associated soliton surfaces analogues of both backlund9s and bianchi9s classical transformations are derived for the extended sine gordon system

subadditivity of kodaira dimensions for fibrations of three folds in positive characteristics abstract in this paper we will prove subadditivity of kodaira dimensions for a fibration with possibly singular geometric generic fiber under certain nefness and relative semi ampleness conditions as an application for a fibration f x y of a smooth projective threefold over an algebraically closed field of characteristic p 5 under the assumption that y is of general type and non uniruled we prove subadditivity of kodaira dimensions when general fibers are smooth or when k x y is relatively big over y

critical curves for a santalo problem in 3 space forms abstract l santalo 1956 studied in 15 the minimization of frenet first curvature dependent energies for spaces of curves constrained to lie in a surface of the euclidean 3 space in this paper we analyze an extension of that problem by considering energy functionals determined by lagrangians which depend not only on the curvature but also on the torsion acting on spaces of surface contained clamped curves of real 3 space forms

characteristic polyhedra of singularities without completion in hironaka j math kyoto univ 7 3 251 293 1967 defines characteristic polyhedra of a singularity embedded in spec r where r is a regular local ring unfortunately to build these polyhedra he must pick up coordinates in the completion widehat r in this article we show that in the hypersurface case whenever r is a g ring by using a different algorithm than hironaka s it is not necessary to extend r to compute the polyhedra

murthy s conjecture on 0 cycles we show that the levine weibel chow group of 0 cycles mathrm ch d a of a reduced affine algebra a of dimension d ge 2 over an algebraically closed field is torsion free among several applications it implies an affirmative solution to an old conjecture of murthy in classical k theory

a quantitative version of the absolute subspace theorem the celebrated subspace theorem of w m schmidt 12 says the following subspace theorem let l1 ln be linearly independent linear forms in n variables with real or complex algebraic coefficients suppose 0 consider the inequality 1 1 l1 x ln x 0 put 1 5 q c x r li x qi 1 i n then the parametric subspace theorem can be stated as follows parametric subspace theorem let c be a fixed tuple satisfying 1 3 let 0 then there are finitely many proper linear subspaces t1 tb of q n such that for any q which is sufficiently large there exists a subspace ti t1 tb with 1 6 q q c z ti

resolution of singularities and modular galois theory i shall sketch a brief history of the desingularization problem from riemann thru zariski to hironaka including the part i played in it and the work on galois theory which this led me to and how that caused me to search out many group theory gurus i shall also formulate several conjectures and suggest numerous thesis problems

extremal curves for weighted elastic energy in surfaces of 3 space forms a variational problem closely related to the bending energy of curves contained is surfaces of real 3 space forms is considered we seek curves in a surface which are critical for the elastic energy when this is weighted by the total squared normal curvature energy under two different sets of constraints clamped curves and one free end curves of constant length we start by deriving the first variation formula and the corresponding euler lagrange equations and natural boundary conditions of these energies and characterize critical geodesics we show how surfaces locally foliated by critical geodesics can be found by using the fundamental theorem of submanifolds in order to find explicit solutions we classify complete rotation surfaces in a real space form for which every parallel is critical

rotation hypersurfaces in spaces of constant curvature rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed a local characterization of such hypersurfaces with dimensions greater than two is given in terms of principal curvatures some special cases of rotation hypersurfaces with constant mean curvature in hyperbolic space are studied in particular it is shown that the well known conjugation between the helicoid and the catenoid in euclidean three space extends naturally to hyperbolic three space h3 in the latter case catenoids are of three different types and the explicit correspondence is given it is also shown that there exists a family of simply connected complete embedded nontotally geodesic stable minimal surfaces in h3

non autonomous svinolupov jordan kdv systems non autonomous svinolupov jordan kdv systems are considered the integrability criteria of such systems are associated with the existence of recursion operators a new non autonomous kdv system and its recursion operator is obtained for all n the examples for n 2 and 3 are studied in detail some possible transformations which map some systems to autonomous ones are also discussed pacs numbers 0230i 0220 0230j

hypersurfaces with defect abstract a projective hypersurface x p n has defect if h i x h i p n for some i n 2 n 2 in a suitable cohomology theory this occurs for example when x p 4 is not q factorial we show that hypersurfaces with defect tend to be very singular in characteristic 0 we present a lower bound on the tjurina number where x is allowed to have arbitrary isolated singularities for x with mild singularities we prove a similar result in positive characteristic as an application we obtain an estimate on the asymptotic density of hypersurfaces without defect over a finite field

equimultiplicity algebraic elimination and blowing up abstract given a variety x over a perfect field we study the partition defined on x by the multiplicity into equimultiple points and the effect of blowing up at smooth equimultiple centers over fields of characteristic zero we prove resolution of singularities by using the multiplicity as an invariant instead of the hilbert samuel function

existence of mori fibre spaces for 3 folds in char p abstract we prove the following results for projective klt pairs of dimension 3 over an algebraically closed field of characteristic p 5 the cone theorem the base point free theorem the contraction theorem finiteness of minimal models termination with scaling existence of mori fibre spaces etc

standard conjectures for the arithmetic grassmannian g 2 n and racah polynomials we prove the arithmetic hodge index and hard lefschetz conjectures for the grassmannian g g 2 n parametrizing lines in projective space for the natural arithmetic lefschetz operator defined via the plucker embedding of g in projective space the analysis of the hodge index inequality involves estimates on values of certain racah polynomials

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

techniques for the study of singularities with applications to resolution of 2 dimensional schemes we give an overview of invariants of algebraic singularities over perfect fields we then show how they lead to a synthetic proof of embedded resolution of singularities of 2 dimensional schemes

on the classification of darboux integrable chains we study a differential difference equation of the form tx n 1 f t n t n 1 tx n with unknown t t n x depending on x and n the equation is called a darboux integrable if there exist functions f called an x integral and i called an n integral both of a finite number of variables x t n t n 1 t n 2 tx n txx n such that dxf 0 and di i where dx is the operator of total differentiation with respect to x and d is the shift operator dp n p n 1 the darboux integrability property is reformulated in terms of characteristic lie algebras that give an effective tool for classification of integrable equations the complete list of equations of the form above admitting nontrivial x integrals is given in the case when the function f is of the special form f x y z z d x y

three dimensional canonical singularities in codimension two in positive characteristic abstract we investigate local structure of a three dimensional variety x defined over an algebraically closed field k of characteristic p 0 with at most canonical singularities under the assumption that p 3 and a general hyperplane cut of x has at most rational singularities we show that local structure of x in codimension two is well understood in the level of local equations consequently we find that i any singularity of such a variety x in codimension two is compound du val ii it has a crepant resolution iii it is analytically a product of a rational double point and a nonsingular curve when p 3 with two exceptions in p 3 and iv r 1 o x r 1 k x 0 holds outside some finite points of x for any resolution of singularities x x

arithmetic height functions over finitely generated fields in this paper we propose a new height function for a variety defined over a finitely generated field over for this height function we prove northcott s theorem and bogomolov s conjecture so that we can recover the original raynaud s theorem manin mumford s conjecture

explicit uniform estimation of rational points i estimation of heights by using the slope method in arakelov geometry we study the complexity of the singular locus of an arithmetic projective variety and explicit estimations of the arithmetic hilbert samuel function

resolution of singularities of arithmetical threefolds ii abstract we prove grothendieck s conjecture on resolution of singularities for quasi excellent schemes x of dimension three and of arbitrary characteristic this applies in particular to x spec a a a reduced complete noetherian local ring of dimension three and to algebraic or arithmetical varieties of dimension three similarly if f is a number field a complete discretely valued field or more generally the quotient field of any excellent dedekind domain o any regular projective surface x f has a proper and flat model x over o which is everywhere regular

on darboux integrable semi discrete chains a differential difference equation with unknown t n x depending on the continuous and discrete variables x and n is studied we call an equation of such kind darboux integrable if there exist two functions called integrals f and i of a finite number of dynamical variables such that dxf 0 and di i where dx is the operator of total differentiation with respect to x and d is the shift operator dp n p n 1 it is proved that the integrals can be brought to some canonical form a method of construction of an explicit formula for a general solution to darboux integrable chains is discussed and such solutions are found for a class of chains

on rational connectedness of globally f regular threefolds in this paper we show that projective globally f regular threefolds defined over an algebraically closed field of characteristic p 11 are rationally chain connected

on the abundance problem for 3 folds in characteristic p 5 in this article we prove two cases of the abundance conjecture for 3 folds in characteristic p 5 i x delta is klt and kappa x k x delta 1 and ii x delta is klt k x delta equiv 0 and x is not uniruled

a note on divisible points of curves let c be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3 we partially answer a question posed by levin on points on c for which a non trivial power lies again on c our results have connections to zilber s conjecture on intersections with tori and yield to methods arising in transcendence theory and the theory of o minimal structures

computations of bott chern classes on ℙ e we compute the bott chern classes of the metric eulersequence describing the relative tangent bundle of the variety p e of hyperplans of a holomorphic hermitian vector bundle e h on a complex manifold we give applications to the construction of the arithmetic characteristic classes of an arithmetic vector bundle and to the computation of the height of p e with respect to the tautological quotient bundle o e 1

every place admits local uniformization in a finite extension of the function field we prove that every place p of an algebraic function field f k of arbitrary characteristic admits local uniformization in a finite extension f of f we show that f f can be chosen to be galois after a finite purely inseparable extension of the ground field k instead of being galois the extension can also be chosen such that the induced extension fp fp of the residue fields is purely inseparable and the value group of f only gets divided by the residue characteristic if f lies in the completion of an abhyankar place then no extension of f is needed our proofs are based solely on valuation theoretical theorems which are of particular importance in positive characteristic they are also applicable when working over a subring r k and yield similar results if r is regular and of dimension smaller than 3

key polynomials and minimal pairs abstract in this paper we establish the relation between key polynomials as defined in 12 and minimal pairs of definition of a valuation we also discuss truncations of valuations on a polynomial ring k x we prove that a valuation is equal to its truncation on some polynomial if and only if is valuation transcendental another important result of this paper is that if is any extension of to k x and is a complete sequence of key polynomials for without last element then for each q there exists a suitable root a q k of q such that a q q is a pseudo convergent sequence defining

effective bounds of linear series on algebraic varieties and arithmetic varieties in this paper we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes they are effective versions of the hilbert samuel formula and the arithmetic hilbert samuel formula the treatments are high dimensional generalizations of duke math j 162 2013 1723 1770 and relative noether inequality on fibered surfaces preprint 2013 similar results are obtained independently by huayi chen majorations explicites de fonctions de hilbert samuel geometrique et arithmetique preprint 2014 with less explicit error terms

darboux integrable two component nonlinear hyperbolic systems of equations an approach based on a study of the structure of characteristic rings is used to solve the classification problem of darboux integrable nonlinear hyperbolic systems of equations we classify two component systems of equations with two integrals of the first order and two integrals of the second order for the resulting systems of equations x and y integrals are constructed and their general solutions are found

resolution of singularities for 3 folds in positive characteristic a concise complete proof of resolution of singularities of 3 folds in positive characteristic 5 is given abhyankar first proved this theorem in 1966

a further improvement of the quantitative subspace theorem in 2002 evertse and schlickewei 11 obtained a quantitative version of the so called absolute parametric subspace theorem this result deals with a parametrized class of twisted heights one of the consequences of this result is a quantitative version of the absolute subspace theorem giving an explicit upper bound for the number of subspaces containing the solutions of the diophantine inequality under consideration in the present paper we further improve evertse s and schlickewei s quantitative version of the absolute parametric subspace theorem and deduce an improved quantitative version of the absolute subspace theorem we combine ideas from the proof of evertse and schlickewei which is basically a substantial refinement of schmidt s proof of his subspace theorem from 1972 22 with ideas from faltings and wustholz proof of the subspace theorem 14 a new feature is an interval result which gives more precise information on the distribution of the heights of the solutions of the system of inequalities considered in the subspace theorem

resolution of singularities of threefolds in positive characteristic i reduction to local uniformization on artin schreier and purely inseparable coverings abstract in dimension 3 the desingularization problem is reduced to local uniformization on artin schreier and purely inseparable coverings of regular space

method for searching higher symmetries for quad graph equations a generalized symmetry integrability test for discrete equations on the square lattice is studied integrability conditions are discussed a method for searching higher symmetries including non autonomous ones for quad graph equations is suggested based on characteristic vector fields