modules of highest weight type for the algebra of derivations over a rational quantum torus for the lie algebra of derivations over a rational quantum torus we construct a class of irreducible weight modules with all finite dimensional weight spaces which are called the modules of highest weight type generating the notion for the case of commutative torus for the lie algebra of derivations over a rational quantum torus we construct a class of irreducible weight modules with all finite dimensional weight spaces which are called the modules of highest weight type generating the notion for the case of commutative torus

multiplicity one representations of divergence free lie algebras abstract divergence free lie algebras also known as the special lie algebras of cartan type are lie algebras of volume preserving transformation groups they are simple in generic case dokovic and zhao found a certain graded generalization of them in this paper we classify all the irreducible and indecomposable multiplicity one modules of the simple generalized divergence free lie algebras

three dimensional divisorial extremal neighborhoods in this paper we study divisorial extremal neighborhoods open image in new window such that 0 x is a can type threefold terminal singularity and f e is a smooth curve where e is the f exceptional divisor we view a divisorial extremal neighborhood as a one parameter smoothing of certain surface singularities and based on this we give a classification of such neighborhoods

divisorial contractions in dimension three which contract divisors to smooth points we deal with a divisorial contraction in dimension three which contracts its exceptional divisor to a smooth point we prove that any such contraction can be obtained by a suitable weighted blow up

elementary contractions of gorenstein threefolds in this paper we show that a generalization of the classification theorem of mori in 113 can be easily obtained from the contraction theorem of 5 a fine analysis is made of the contractions of extremal rays on projective threefolds with normal gorenstein q factorial terminal singularities so that x is in fact factorial a solution of the minimal model problem for threefolds has recently been completed by mori in ll we will first establish notation and give some preliminary results all of the definitions used in this paper can be found in 7 let x be a normal gorenstein projective threefold with q factorial terminal singularities suppose that x is defined over an algebraically closed field k of characteristic zero let r be an extremal ray of x it follows from arguments in 5 or 7 that there is a morphism f x y onto a normal variety y such that f is given by the linear system induced by a cartier divisor h and such that 1 f tp x 9 r 2 for an arbitrary curve c on x d imf c 0 holds if and only if c er 3 the pair f y is uniquely determined by conditions 1 and 2 up to isomorphism 4 s 0 pic y c pic x i sexact for i c i e r

families of d minimal models and applications to 3 fold divisorial contractions let x t be a one parameter family of canonical 3 folds and let d be a weil divisor on it flat over t we study the problem of when the d t minimal models of x t form a family and we obtain conditions for this to happen as an application of this we classify terminal divisorial contractions e subset y leftarrow c subset x contracting an irreducible surface e onto the smooth curve c in the case when the general section of x through c is a d 5 duval singularity

general elephants of three fold divisorial contractions the theory of minimal models has enriched the study of higher dimensional algebraic geometry see 10 and 12 for a variety with mild singularities this theory produces another variety which possesses good properties after finite elementary transformations called divisorial contractions and flips since mori completed this program in dimension three in 17 it has become desirable to study three folds explicitly this paper aims to complete the explicit study of three fold divisorial contractions whose exceptional divisors contract to gorenstein points after the papers 7 and 8 reid pointed out that general elements in the anti canonical systems of threefolds have at worst du val singularities in appropriate situations involving contractions of extremal faces this has become known as the general elephant conjecture the papers 19 21 and 22 support it with affirmative answers for fano threefolds with singularities and this approach has settled the problem of the existence of three fold flips in 11 and 17 our main theorem is that this conjecture holds for our divisorial contractions

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

errata to classification of three dimensional flips remark 1 the statement km92 2 2 3 is false as it stands 2 2 3 comes out of two sources one from 2 13 4 and the other from 2 13 10 the latter 2 13 10 is correct and proves that ra 5 and the index two point is of type ca 2 however the former 2 13 4 proves only a weaker assertion that ra 3 not ra 5 the index two point is of type ca 2 cax 2 or cd 2 not just of type c l 2 and with axial multiplicity k 2 because the index one cover is not smooth since f x d c y 3 q is divisorial in the case 2 13 4 km92 prop 9 3 the following revision 2 2 3 rev should replace 2 2 3 2 2 3 rev case for exceptional ia i a the two ia points are an ordinary point of odd index m ra 5 if is isolated that is a flipping contraction ra 3 if divisorial and an index two point of type ca 2 if is isolated of type ca 2 cax 2 or c d 2 if divisorial and with axial multiplicity k with 2k ra 7 and we have kx c l 2m ey q is d2k m sing x is am i d2k am i a1 ax if k 1 and a ex d c is

representations of some quantum tori lie subalgebras in this paper we define the q analog virasoro like lie subalgebras in x a b c d the embedding formulas into x are introduced irreducible highest weight representations of aq bq and cq series of the q analog virasoro like lie algebras in terms of vertex operators are constructed we also construct the polynomial representations of the aq bq cq and dq series of the q analog virasoro like lie algebras

quantum tori and the structure of elliptic quasi simple lie algebras abstract we study and classify those tame irreducible elliptic quasi simple lie algebras which are simply laced and of rankl 3 the first step is to identify the core of such an algebra up to central isogeny by identifying the coordinates when the type isdorethe coordinates are laurent polynomials in variables while for typeathe coordinates can be any quantum torus in variables the next step is to study the universal central extension as well as the derivation algebra of the core these are related to the first connes cyclic homology group of the coordinates the final step is to use this information to give constructions of lie algebras which we then prove yield representatives of all isomorphism classes of the above types of algebras

representations of the lie algebra of derivations for quantum torus abstract let der c q be the derivation lie algebra of the quantum torus c q we construct a functor from gld modules to der c q modules which generalizes the constructions obtained by shen scientia sinica a 29 1986 570 larsson internat j modern phys a 7 1992 6493 and rao j algebra 182 1996 401 we also give a complete description of the structure of the der c q modules when all the entries of the quantum torus matrix q qij d d are roots of unity

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

extensions of inhomogeneous polynomial representations for sl m 1 n we extend an inhomogeneous polynomial representation of special linear lie superalgebras sl m 1 n to an inhomogeneous representation on the tensor space of any irreducible tensor representation of the general linear lie superalgebras gl m n with this polynomial space via shen s mixed product we find a sufficient and necessary condition for irreducibility of these inhomogeneous representations and get the jordan holder series of those which are not irreducible furthermore we point out that all these irreducible modules are lowest weight modules their lowest weights and character formulas are obtained

divisorial extremal contractions of threefolds divisor to point extremal contractions which contract divisors to points in projective threefolds with inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i factorial terminal singularities are studied and divided into two categories index increasing contractions and index strictly decreasing contractions a complete classification of those in the first category is given examples of contractions in the second category are constructed to demonstrate that they are much more difficult to deal with an extremal contraction which contracts a divisor to a curve is always index decreasing an example of such a contraction to a curve with a non gorenstein terminal singularity is given based on a method of kollar and mori the classification result is then used to find a bound n depending on the picard number of a smooth projective threefold x of general type such that the linear system nk x defines a birational map

influence of gaussian noise on the correlation exponent self similarity of fractal point sets is broken if the points are contaminated by measurement noise different approaches have been taken to calculate the influence of gaussian noise on the scaling of interpoint distances we bring the results into a form which makes a comparison possible although the approaches use different norms and ways to quantify the scaling behavior we find that the influence of noise leads to similar deviations from the pure scaling behavior

q gorenstein deformations of nonnormal surfaces let delta subset h be the germ of a nonnormal surface along a proper curve with smooth components such that the high index points of h are semi log terminal and the gorenstein singular points are semi log canonical of embedding dimension at most 4 we describe the sheaf t 1 qg h of bbb q gorenstein deformations of h and we obtain criteria for delta subset h to have bbb q gorenstein terminal smoothings

vertex representations for n toroidal lie algebras and a generalization of the virasoro algebra vertex representations are obtained for toroidal lie algebras for any number of variables these representations afford representations of certainn variable generalizations of the virasoro algebra that are abelian extensions of the lie algebra of vector fields on a torus

irreducible representations of the lie algebra of the diffeomorphisms of ad dimensional torus in this paper we investigate the irreducibility of certain modules for the lie algebra of diffeomorphisms of torus t the lie algebra of diffeomorw 1 1 x phisms can be described as the derivations of a s c t t see 1 d w x w x rss we denote this lie algebra by der a larsson l1 constructed a functor from gl modules to der a modules in this paper we prove that d the image of a finite dimensional irreducible gl module is most often d irreducible the only exceptions are fundamental modules and the onedimensional modules in these cases we describe the sub modules and the quotients surprisingly the same class of modules image of finite dimen w x sional gl module are also constructed in e without any reference to d gl modules they are motivated by the vertex operator constructions in d w x em more details near the end of this introduction we will now describe our results in more detail let gl be the d lie algebras of d d matrices over complex numbers c let gl s sl d d c id where sl is the finite dimensional simple lie algebra of trace zero d matrices and id the identity matrix which is central it is well known that finite dimensional irreducible modules and dominant integral weights are in one one correspondence for a finite dimensional simple lie algebra so let v c be finite dimensional irreducible sl module corresponding to a d dominant integral weight c let id the central element act by a complex number b and denote the corresponding gl module by v c b d a d d larsson defined see 1 6 a functor f for a belonging to c rz from w x gl modules to der a modules in l1 these modules are weight modules d

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

irreducible modules over the derivation algebras of rational quantum tori abstract let g be the derivation lie algebra of a rational quantum torus c q where q q i j d d and all q i j are roots of unity in lin and tan 2004 lt a class of uniformly bounded irreducible weight modules over g were constructed which generalized the construction given by shen 1986 sh larsson 1992 l and eswara rao 1996 e these modules look mysterious because of their very unclear weight sets in this note we use very simple methods with a very short proof to re establish all even stronger results in lin and tan 2004 lt and make these mysterious modules crystal clear a necessary and sufficient condition for two such modules to be isomorphic is also given