characterization of log del pezzo pairs via anticanonical models we give a characterization of log del pezzo pairs in terms of their anticanonical models as a first application we show that every non rational del pezzo surface with log canonical singularities is of picard number one and contains a unique singularity which is simple elliptic as a second application we verify a conjecture of smith and schwede in dimension two which says that every surface of globally f regular type is of fano type

relative canonical sheaves of a family of curves abstract in this paper we study the relative canonical sheaf of a relatively minimal fibration of curves of genus g 2 over a one dimensional regular scheme using the configurations of 2 chains we show that its m tensored product is base point free for any m 2 we utilize koszul cohomology to prove that the relative canonical algebra of the fibration is generated in degree up to five it is a generalization of k konno s work on the 1 2 3 conjecture of m reid

variational analysis and optimal control of dynamic unilateral contact models with friction abstract the target of the present work is to consider the optimal control for a class of evolutionary variational hemivariational inequalities which modeling the dynamic viscoelastic unilateral contact problems with normal damped response and friction after studying the weak solvability of the unilateral contact model we consider its optimal control by three aspects the optimal control via external forces and initial conditions the time optimal control problem and the maximum stay control problem finally the conditions which guarantee the existence of optimal solutions to the corresponding control problems are delivered

okounkov bodies and seshadri constants abstract okounkov bodies which are closed convex sets defined for big line bundles have rich information on the line bundles on the other hand seshadri constants are invariants which measure the positivity of line bundles in this paper we prove that okounkov bodies give lower bounds of seshadri constants

stability of test ideals of divisors with small multiplicity let x vardelta be a log pair in characteristic p 0 and p be a not necessarily closed point of x we show that there exists a constant delta 0 such that the test ideal tau x vardelta a characteristic p analogue of a multiplier ideal does not change at p under the perturbation of vardelta by any mathbb r divisor with multiplicity less than delta as an application we prove that if d is an mathbb r cartier mathbb r divisor on a strongly f regular projective variety then the non nef locus of d coincides with the restricted base locus of d this is a generalization of a result of musta to the singular case and can be viewed as a characteristic p analogue of a result of cacciola di biagio

the cone of type a level 1 conformal blocks divisors abstract we prove that the type a level 1 conformal blocks divisors on m 0 n span a finitely generated full dimensional subcone of the nef cone each such divisor induces a morphism from m 0 n and we identify its image as a git quotient parameterizing configurations of points supported on a flat limit of veronese curves we show how scaling git linearizations gives geometric meaning to certain identities among conformal blocks divisor classes this also gives modular interpretations in the form of git constructions to the images of the hyperelliptic and cyclic trigonal loci in m g under an extended torelli map

okounkov bodies and restricted volumes along very general curves abstract given a big divisor d on a normal complex projective variety x we show that the restricted volume of d along a very general complete intersection curve c x can be read off from the okounkov body of d with respect to an admissible flag containing c from this we deduce that if two big divisors d 1 and d 2 on x have the same okounkov body with respect to every admissible flag then d 1 and d 2 are numerically equivalent

a stable theorem of the alternative an extension of the gordan theorem abstract a theorem with a number of equivalent alternatives is proposed as an extension of the classical gordan theorem of the alternative the theorem can handle nonzero unrestricted variables which cannot be directly treated by ordinary theorems of the alternative like the gordan theorem the extended theorem has the stability feature that small perturbations in the data will not invalidate an alternative that is in force the theorem has useful applications in establishing the boundedness and uniqueness of feasible points of polyhedral sets and of solutions to linear programming problems

analysis of an adhesive contact problem for viscoelastic materials with long memory abstract the goal of this paper is to study a mathematical model which describes the adhesive contact between a deformable body and a foundation the body consists of a viscoelastic material with long memory and the process is assumed to be quasistatic the adhesive contact condition on the normal plane is modeled by a version of normal compliance condition with unilateral constraint in which adhesion is taken into account the adhesive contact condition on the tangential plane is described by an adhesive clarke subdifferential condition and the evolution of the bonding field is described by an ordinary differential equation we derive the variational formulation of this problem which is a system of a variational hemivariational inequality for the displacement field and an ordinary differential equation for the bonding field then we obtain existence and uniqueness results on abstract inclusions and abstract variational hemivariational inequalities finally we apply the abstract results to prove the existence of a unique weak solution to the contact problem

m 0 n is not a mori dream space building on the work of goto nishida and watanabe on symbolic rees algebras of monomial primes we prove that the moduli space of stable rational curves with n punctures is not a mori dream space for n 133 this answers a question posed by hu and keel

congruence for rational points over finite fields and coniveau over local fields if the l adic cohomology of a projective smooth variety defined over a local field k with finite residue field k is supported in codimension 1 then every model over the ring of integers of k has a k rational point for k a p adic field this is proved in esnault 2007 theorem 1 1 if the model x is regular one has a congruence x k 1 modulo k for the number of k rational points esnault 2006 theorem 1 1 the congruence is violated if one drops the regularity assumption

an implementation of karmarkar s algorithm for linear programming this paper describes the implementation of power series dual affine scaling variants of karmarkar s algorithm for linear programming based on a continuous version of karmarkar s algorithm two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested linear programs are expressed in an inequality form which allows for the inexact computation of the algorithm s direction of improvement resulting in a significant computational advantage implementation issues particular to this family of algorithms such as treatment of dense columns are discussed the code is tested on several standard linear programming problems and compares favorably with the simplex codeminos 4 0

on the effective cone of m g n abstract for every g 2 and n g 1 we exhibit infinitely many extremal effective divisors in m g n coming from the strata of abelian differentials

on the dual of the mobile cone we prove that the mobile cone and the cone of curves birationally movable in codimension 1 are dual to each other in the k b negative part for a klt pair x b this duality of the cones gives a partial answer to the problem posed by sam payne we also prove the cone theorem and the contraction theorem for the expanded cone of curves birationally movable in codimension 1

augmented base loci and restricted volumes on normal varieties we extend to normal projective varieties defined over an arbitrary algebraically closed field a result of ein lazarsfeld musta nakamaye and popa characterizing the augmented base locus aka non ample locus of a line bundle on a smooth projective complex variety as the union of subvarieties on which the restricted volume vanishes we also give a proof of the folklore fact that the complement of the augmented base locus is the largest open subset on which the kodaira map defined by large and divisible multiples of the line bundle is an isomorphism

birational superrigidity and k stability of fano complete intersections of index 1 we prove that every smooth fano complete intersection of index 1 and codimension r in pn r is birationally superrigid and k stable if n 10r we also propose a generalization of tian s criterion of k stability and as an application prove the k stability of the complete intersection of a quadric and a cubic in p5 in the appendix written jointly with c stibitz we prove the conditional birational superrigidity of fano complete intersections of higher index in large dimension

basepoint freeness for nef and big line bundles in positive characteristic a necessary and sufficient condition is given for semi ampleness of a numerically effective nef and big line bundle in positive characteristic one application is to the geometry of the universal stable curve over mg specifically the semi ampleness of the relative dualizing sheaf in positive characteristic an example is given which shows this and the semi ampleness criterion fail in characteristic zero a second application is to mori s program for minimal models of 3 folds in positive characteristic namely to the existence of birational extremal contractions

towards the ample cone of mg n in this paper we study the ample cone of the moduli space m g n of stable n pointed curves of genus g our motivating conjecture is that a divisor on m g n is ample iff it has positive intersection with all 1 dimensional strata the components of the locus of curves with at least 3g n 2 nodes this translates into a simple conjectural description of the cone by linear inequalities and as all the 1 strata are rational includes the conjecture that the mori cone is polyhedral and generated by rational curves our main result is that the conjecture holds iff it holds for g 0 more precisely there is a natural finite map r m0 2g n m g n whose image is the locus rg n of curves with all components rational any 1 strata either lies in rg n or is numerically equivalent to a family e of elliptic tails and we show that a divisor d is nef iff d e 0 and r d is nef we also give results on contractions i e morphisms with connected fibers to projective varieties of m g n for g 1 showing that any fibration factors through a tautological one given by forgetting points and that the exceptional locus of any birational contraction is contained in the boundary

extremal effective divisors on overline mathcal m _ 1 n for every n ge 3 we exhibit infinitely many extremal effective divisors on overline mathcal m 1 n the deligne mumford moduli space parameterizing stable genus one curves with n ordered marked points

successive minima of line bundles abstract we introduce and study the successive minima of line bundles on proper algebraic varieties the first resp last minima are the width resp seshadri constant of the line bundle at very general points the volume of the line bundle is equivalent to the product of the successive minima for line bundles on toric varieties the successive minima are equivalent to the reciprocal of usual successive minima of the difference of the moment polytope

effective cones of quotients of moduli spaces of stable n pointed curves of genus zero let x n m 0 n the moduli space of n pointed stable genus zero curves and let x n m be the quotient of x n by the action of s n m on the last n m marked points the cones of effective divisors ne 1 x n m m 0 1 2 are calculated using this upper bounds for the cones mov x n m generated by divisors with moving linear systems are calculated m 0 1 along with the induced bounds on the cones of ample divisors of m g and m g 1 as an application the cone ne 1 m 2 1 is analyzed in detail

asymptotic base loci via okounkov bodies abstract an okounkov body is a convex subset of euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag in this paper we recover the asymptotic base loci from the okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving seshadri constants consequently we obtain the nefness and ampleness criteria of divisors in terms of the okounkov bodies furthermore we compute the divisorial zariski decomposition by the okounkov bodies and find upper and lower bounds for moving seshadri constants given by the size of simplexes contained in the okounkov bodies

vanishing sequences and okounkov bodies we define and study the vanishing sequence along a real valuation of sections of a line bundle on a normal projective variety building on previous work of the first author with huayi chen we prove an equidistribution result for vanishing sequences of large powers of a big line bundle and study the limit measure in particular the latter is described in terms of restricted volumes for divisorial valuations we also show on an example that the associated concave function on the okounkov body can be discontinuous at boundary points

naive blowups and canonical birationally commutative factors in 2008 rogalski and zhang math z 259 2 433 455 2008 showed that if r is a strongly noetherian connected graded algebra over an algebraically closed field bbbk then r has a canonical birationally commutative factor this factor is up to finite dimension a twisted homogeneous coordinate ring b x mathcal l sigma here x is the projective parameter scheme for point modules over r as well as tails of points in mathrm qgr r as usual sigma is an automorphism of x and mathcal l is a sigma ample invertible sheaf on x we extend this result to a large class of noetherian but not strongly noetherian algebras specifically let r be a noetherian connected graded bbbk algebra where bbbk is an uncountable algebraically closed field let y infty denote the parameter space or stack or proscheme parameterizing r point modules and suppose there is a projective variety x that corepresents tails of points there is a canonical map p y infty rightarrow x if the indeterminacy locus of p 1 is 0 dimensional and x satisfies a mild technical assumption we show that there is a homomorphism g r rightarrow b x mathcal l sigma and that g r is up to finite dimension a naive blowup on x in the sense of keeler et al duke math j 126 3 491 546 2005 rogalski and stafford j algebra 318 2 794 833 2007 and satisfies a universal property we further show that the point space y infty is noetherian

interior dual proximal point algorithm for linear programs abstract a new algorithm for solving a linear program based on an interior point method applied to the dual of a proximal point formulation of the linear program is presented this dual formulation contains only the nonnegativity constraint on some of the variables this simple constraint allows us to start the algorithm without a phase 1 method required by many other variants of the interior point method numerical results from a large set of test problems show that the proposed algorithm can be very competitive with other interior point methods and with minos 5 3 a state of the art linear programming package based on the simplex method global convergence of the algorithm is also established

on the projectivity of the moduli spaces of curves 1 there are basically three algebraic proofs of the projectivity of the moduli spaces of stable curves the original one by knudsen 8 9 the proof by mumford 12 and gieseker 5 which makes heavy use of the machinery of geometric invariant theory and the more recent one by viehweg 15 and kollar 10 which relies instead on the semipositivity of the direct images of powers of the relative dualizing sheaf and at least in kollar s version does not use geometric invariant theory at all in this note combining ideas from the above papers and from 4 we shall outline a proof of projectivity which we believe is simpler and more direct than any of the existing ones however while the proofs by mumford gieseker viehweg and kollar are applicable at least in principle to a wide variety of moduli problems ours uses in an essential way the peculiarities of the problem at hand and it is hard to see how it could be extended to other situations mumford s idea is to use the machinery of geometric invariant theory to directly construct the moduli space of stable curves as a projective quotient of the chow or hilbert scheme of pluricanonically embedded stable curves by the action of the projective linear group to be able to do so one needs to show that the k canonical images of stable curves are stable in the invariant theoretic sense for high k this is quite well understood although not really easy for smooth curves one notices that k canonically embedded stable curves are linearly stable and then proves that for smooth curves linear stability implies invariant theoretic stability in the singular case by contrast both mumford and gieseker have to rely on indirect arguments which are quite long and involved our main point is that one can avoid proving the invariant theoretic stability of singular stable curves provided one is willing to grant that moduli space exists as a complete algebraic space that this is the case incidentally is relatively easy to prove as in 10 we prove projectivity by applying to a suitable invertible sheaf one of the standard numerical ampleness criteria seshadri s criterion in our case the necessary numerical estimates are obtained using the techniques of 4 more precisely the invariant theoretic stability of linearly stable smooth curves is used to prove an inequality for families of stable curves which suitably interpreted says that the hypothesis of seshadri s criterion is satisfied for curves in moduli which are not contained in the boundary the case of curves lying in the boundary is then reduced to the previous one by standard elementary techniques in order to properly carry out this procedure which can be viewed as a sort of induction on the genus it is convenient to deal simultaneously with the moduli spaces of stable n pointed curves as well in a sense then our approach is to use invariant theoretic stability as a means of obtaining numerical inequalities rather than as a step in the construction of quotients by group actions

restricted volumes and base loci of linear series we introduce and study the restricted volume of a divisor along a subvariety our main result is a description of the irreducible components of the augmented base locus by the vanishing of the restricted volume

restricted volumes and divisorial zariski decompositions we give a relation between the existence of a zariski decomposition and the behavior of the restricted volume of a big divisor on a smooth complex projective variety moreover we give an analytic description of the restricted volume in the line of boucksom s work it enables us to define the restricted volume of a transcendental class on a compact k ahler manifold in a natural way the relation can be extended to a transcendental class

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

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

iitaka s fibrations via multiplier ideals we give a new characterization of iitaka s fibration of algebraic varieties associated to line bundles introducing an intersection number of line bundles and curves by using the notion of multiplier ideal sheaves iitaka s fibration can be regarded as a numerically trivial fibration in terms of this intersection theory

connecting interpolation and multiplicity estimates in commutative algebraic groups let g be a commutative algebraic group embedded in projective space and gamma a finitely generated subgroup of g from these data we construct a chain of algebraic subgroups of g which is intimately related to obstructions to multiplicity or interpolation estimates used in transcendental number theory and algebraic independence let gamma 1 ldots gamma l denote a family of generators of gamma and for any s 1 let gamma s be the set of elements n 1 gamma 1 cdots n l gamma l with integers n j such that n j s then this chain of subgroups controls for large values of s the distribution of gamma s with respect to algebraic subgroups of g as an application we essentially determine up to multiplicative constants the locus of common zeros of all p in h 0 overline g mathcal o d which vanish to at least some given order at all points of gamma s when d is very small this result reduces to a multiplicity estimate when d is very large it is a kind of interpolation estimate

on the volume of graded linear series and monge ampere mass we give an analytic description of the volume of a graded linear series as the monge ampere mass of a certain equilibrium metric associated to any smooth hermitian metric on the line bundle we also show the continuity of this equilibrium metric on some zariski open subset under a geometric assumption

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

moving codimension one subvarieties over finite fields we give the first examples of nef line bundles on smooth projective varieties over finite fields which are not semi ample more concretely we find smooth curves on smooth projective surfaces over finite fields such that the normal bundle has degree zero but no positive multiple of the curve moves in a family of disjoint curves this answers questions by keel and mumford the proof uses an obstruction theory in the spirit of homotopy theory which links the infinitely many obstructions to moving higher and higher multiples of a given codimension one subvariety on 3 folds over a finite field we find nef and big line bundles which are not semi ample finally we reprove some of the known positive results about semi ampleness over finite fields in topology the normal bundle of a submanifold determines a neighborhood of the submanifold up to isomorphism in particular the normal bundle of a codimension one submanifold is trivial if and only if the submanifold can be moved in a family of disjoint submanifolds in algebraic geometry however there are higher order obstructions to moving a given subvariety in this paper we develop an obstruction theory in the spirit of homotopy theory which gives some control over when a codimension one subvariety moves in a family of disjoint subvarieties even if a subvariety does not move in a family some positive multiple of it may we find a pattern linking the infinitely many obstructions to moving higher and higher multiples of a given subvariety as an application we find the first examples of line bundles l on smooth projective varieties over finite fields which are nef l has nonnegative degree on every curve but not semi ample no positive power of l is spanned by its global sections this answers questions by keel and mumford determining which line bundles are spanned by their global sections or more generally are semi ample is a fundamental issue in algebraic geometry if a line bundle l is semi ample then the powers of l determine a morphism from the given variety onto some projective variety one of the main problems of the minimal model program the abundance conjecture predicts that a variety with nef canonical bundle should have semi ample canonical bundle 15 conjecture

bergman kernels and the pseudoeffectivity of relative canonical bundles a criterion for the pseudoeffectivity of twisted relative canonical bundles of surjective projective maps

newton okounkov bodies on projective bundles over curves in this article we study newton okounkov bodies on projective vector bundles over curves inspired by wolfe s estimates used to compute the volume function on these varieties we compute all newton okounkov bodies with respect to linear flags moreover we characterize semi stable vector bundles over curves via newton okounkov bodies

projectivity of the moduli space of stable log varieties and subadditivity of log kodaira dimension reference epfl article 231100doi 10 1090 jams 871 url http dx doi org 10 1090 jams 871 record created on 2017 09 19 modified on 2017 10 02

surfaces of globally f regular and f split type we prove that normal projective surfaces of dense globally f split type respectively globally f regular type are of calabi yau type respectively fano type

asymptotic base loci on singular varieties we prove that the non nef locus and the restricted base locus of a pseudoeffective divisor coincide on klt pairs we also extend to klt pairs f russo s characterization of nef and abundant divisors by means of asymptotic multiplier ideals

intersection theory of moduli space of stable n pointed curves of genus zero we give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the chow ring

evolutionary variational hemivariational inequalities existence and comparison results we consider an evolutionary quasilinear hemivariational inequality under constraints represented by some closed and convex subset our main goal is to systematically develop the method of sub supersolution on the basis of which we then prove existence comparison compactness and extremality results the obtained results are applied to a general obstacle problem we improve the corresponding results in the recent monograph s carl v k le d motreanu nonsmooth variational problems and their inequalities comparison principles and applications springer monogr math springer new york 2007

projectivity of the witt vector affine grassmannian we prove that the witt vector affine grassmannian which parametrizes w k lattices in w k frac 1 p n for a perfect field k of characteristic p is representable by an ind perfect scheme over k this improves on previous results of zhu by constructing a natural ample line bundle along the way we establish various foundational results on perfect schemes notably h descent results for vector bundles

on base point free theorem and mori dream spaces for log canonical threefolds over the algebraic closure of a finite field the authors and d martinelli proved in algebra number theory 9 3 725 747 2015 the base point free theorem for big line bundles on a three dimensional log canonical projective pair defined over the algebraic closure of a finite field in this paper we drop the bigness condition when the characteristic is larger than five additionally we discuss mori dream spaces defined over the algebraic closure of a finite field

cone theorem via deligne mumford stacks we prove the cone theorem for varieties with lciq singularities using deformation theory of stable maps into deligne mumford stacks we also obtain a sharper bound on kx d degree of kx d negative extremal rays for projective mathbb q factorial log terminal threefold pairs x d

boundedness of pluricanonical maps of varieties of general type using the techniques of 20 and 10 we prove that certain log forms may be lifted from a divisor to the ambient variety as a consequence of this result following 22 we show that for any positive integer n there exists an integer rn such that if x is a smooth projective variety of general type and dimension n then phi rk x colon x dasharrow mathbb p h 0 mathcal o x rk x is birational for all r rn

base loci of linear series are numerically determined we introduce a numerical invariant called a moving seshadri constant which measures the local positivity of a big line bundle at a point we then show how moving seshadri constants determine the stable base locus of a big line bundle

the adventures of a simple algorithm abstract this paper presents analyzes and tests a row action method for solving the regularized linear programming problem minimize 1 2 x 2 2 c t x subject to a x b where is a given positive constant the proposed method is based on the observation that the dual of this problem has the form maximize b t y 1 2 a t y c 2 2 subject to y 0 and if y solves the dual then the point x a t y c provides the unique solution of the primal maximizing the dual objective function by changing one variable at a time results in an effective row action method which is suitable for solving large sparse problems one aim of this paper is to clarify the convergence properties of the proposed scheme let y k denote the estimated dual solution at the end of the k th iteration and let x k a t y k c denote the corresponding primal estimate it is proved that the sequence x k converges to x while the sequence y k converges to a point y that solves the dual the only assumption which is needed in order to establish these claims is that the feasible region is not empty yet perhaps the more striking features of the algorithm are related to temporary situations in which it attempts to solve an inconsistent system in such cases the sequence y k obeys the rule y k u k k v where u k is a fastly converging sequence and v is a fixed vector that satisfies a t v 0 and b t v 0 so the sequence x k is almost unchanged for many iterations the paper ends with numerical experiments that illustrate the effects of this phenomenon and the close links with kaczmarz s sor method

the augmented base locus of real divisors over arbitrary fields we show that the augmented base locus coincides with the exceptional locus i e null locus for any nef mathbb r cartier divisor on any scheme projective over a field of any characteristic next we prove a semi ampleness criterion in terms of the exceptional locus generalizing a result of keel we also discuss some problems related to augmented base loci of log divisors

computational experience with a primal dual interior point method for linear programming abstract a new comprehensive implementation of a primal dual algorithm for linear programming is described it allows for easy handling of simple bounds on the primal variables and incorporates free variables which have not previously been included in a primal dual implementation we discuss in detail a variety of computational issues concerning the primal dual implementation and barrier methods for linear programming in general we show that in a certain way lustig s method for obtaining feasibility is equivalent to newton s method this demonstrates that the method is in some sense the natural way to reduce infeasibility the role of the barrier parameter in computational practice is studied in detail numerical results are given for the entire expanded netlib test set for the basic algorithm and its variants as well as version 5 3 of minos

a special cayley octad first published in transactions of the american mathematical society 370 pp 5359 5379 published by the american mathematical society 2017 by a prendergast smith

factoring 3 fold flips and divisorial contractions to curves flips flops and divisorial contractions are the elementary birational maps of the minimal model program divisorial contractions are the higher dimensional analog of the inverse of blowing up a smooth point on a surface flips are instead a new operation which consists of a surgery in codimension 2 which replaces certain kx negative curves by some kx positive curves flops are also surgeries in codimension 2 but they replace certain kx trivial curves by some new kx trivial curves not surprisingly the geometry of divisorial contractions is better understood than that of flips and flops in fact the proof of the existence of flips and flops cf 1 7 is very abstract and gives little insight to their geometry in dimension 3 these operations are reasonably well understood kollar has shown 23 that if x 99k x is a flop of terminal 3 folds then x and x have analytically isomorphic singularities in 34 mori gives a precise understanding of the geometry of all flips with irreducible flipping curve we would like to remark that 3 fold divisorial contractions may be subdivided into two classes those that map the divisor to a curve and those that map the divisor to a point divisorial contractions to a point are classified in 4 14 18 19 20 21 and 31 there are also several partial results on divisorial contractions to a curve cf 5 37 38 39 40 and 41 the purpose of this talk is try to understand what happened to the singularities under these elementary birational map in dimension three and how can we decompose these maps into simplers maps that explicit description of singularities is possible in fact we can factor flips and divisorial contractions to curves via the simpler operations