folding of fuzzy graphs and fuzzy spheres abstract in this paper we will give a new definition for the folding of fuzzy graphs it introduces the concept of fuzzy spheres also we will deduce the relation between the folding of fuzzy graphs and fuzzy spheres theorems governing this type of folding are deduced

bipolar fuzzy graphs in this paper we introduce the notion of bipolar fuzzy graphs describe various methods of their construction discuss the concept of isomorphisms of these graphs and investigate some of their important properties we then introduce the notion of strong bipolar fuzzy graphs and study some of their properties we also discuss some propositions of self complementary and self weak complementary strong bipolar fuzzy graphs

bases axioms and circuits axioms for fuzzifying matroids abstract this paper shows that weighted graphs are examples of fuzzifying matroids axioms of bases and that of circuits are established for fuzzifying matroids

types of arcs in a fuzzy graph the concept of connectivity plays an important role in both theory and applications of fuzzy graphs depending on the strength of an arc this paper classifies arcs of a fuzzy graph into three types namely a strong b strong and d arcs the advantage of this type of classification is that it helps in understanding the basic structure of a fuzzy graph completely we analyze the relation between strong paths and strongest paths in a fuzzy graph and obtain characterizations for fuzzy bridges fuzzy trees and fuzzy cycles using the concept of a strong b strong and d arcs an arc of a fuzzy tree is a strong if and only if it is an arc of its unique maximum spanning tree also we identify different types of arcs in complete fuzzy graphs

on the shape of a tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of v i v i v d i v d i coincide we denote this common dimension by i in this paper we prove that i 0 d i for 0 i d it is already known that 0 1 if k is algebraically closed

the split decomposition of a tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i iv below i each of a a is diagonalizable ii there exists an ordering v0 v1 vd of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 vd 1 0 iii there exists an ordering v 0 v 1 v of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 v 1 0 iv there is no subspace w of v such that both aw w a w w other than w 0 and w v we call such a pair a tridiagonal pair on v in this note we obtain two results first we show that each of a a is determined up to affine transformation by the vi and v i secondly we characterize the case in which the vi and v i all have dimension one we prove both results using a certain decomposition of v called the split decomposition

stable maps in higher dimensions we formulate a notion of stability for maps between polarised varieties which generalises kontsevich s definition when the domain is a curve and tian donaldson s definition of k stability when the target is a point we give some examples such as kodaira embeddings and fibrations we prove the existence of a projective moduli space of canonically polarised stable maps generalising the kontsevich alexeev moduli space of stable maps in dimensions one and two we also state an analogue of the yau tian donaldson conjecture in this setting relating stability of maps to the existence of certain canonical kahler metrics

on computing the edge connectivity of an uncertain graph uncertain graphs are employed to describe graph models with imprecise expert data in uncertain graphs due to the existence of uncertain edges edge connectivity is essentially an uncertain variable different from that in a deterministic graph it is more suitable to investigate the possibility uncertain measure that an uncertain graph is k edge connected which is the main aim of this paper we first deduce a theoretical formula to calculate the uncertain measure and on this basis we then propose an algorithm which is derived from maximum flow algorithms to numerically calculate the uncertain measure the proposed algorithm is also proved to be a polynomial time algorithm and its effectiveness and efficiency are illustrated by numerical examples

approaches to collective decision making with fuzzy preference relations abstract recent experimental studies show that the predictive accuracy of many of the solution concepts derived from the collective decision making theory leaves much to be desired in a previous paper the author attempted to explain some of the inaccuracies in terms of the fuzzy indifference regions of the individuals participating in the voting game this paper gives straightforward generalizations of the solutions concepts in terms of the fuzzy social or individual preference relations it turns out that some of these new solution concepts cotain their nonfuzzy counterparts as subsets others in turn are subsets of their nonfuzzy counterparts we also discuss a method of aggregating individual nonfuzzy preferences so as to get a fuzzy social preference relation and furthermore a nonfuzzy social choice set

the generalized terwilliger algebra and its finite dimensional modules when d 2 terwilliger j algebraic combin 1 1992 363 388 considered the c algebra generated by a given bose mesner algebra m and the associated dual bose mesner algebra m this algebra is now known as the terwilliger algebra and is usually denoted by t terwilliger showed that each vanishing intersection number and krein parameter of m gives rise to a relation on certain generators of t these relations are often called the triple product relations they determine much of the structure of t though not all of it in general to illuminate the role these relations play the current author introduced j algebra233 2000 213 252 a generalization t of t to go from t to t we replace m and m with a pair of dual character algebras c and c the dimensions of c and c are equal let d 1 denote this common dimension intuitively t is the associative c algebra with identity generated by c and c subject to the analogues of terwilliger s triple product relations t is infinite dimensional and noncommutative in general in this paper we study t and its finite dimensional modules when d 2 and t has no extra vanishing intersection numbers or dual intersection numbers in this case we show t is c algebra isomorphic to m3 c a where m3 c denotes the c algebra consisting of all 3 by 3 matrices with entries in c and a denotes the associative c algebra with identity generated by the symbols e and f subject to the relations e2 e and f2 f we find a basis for a and we determine the center of a we classify the finite dimensional indecomposable a modules up to isomorphism there are four such a modules in every odd dimension and in every even dimension these modules are parameterized by a single complex number we also classify the finite dimensional irreducible a modules up to isomorphism using our results concerning a we find a basis for t we describe the center of t and we classify both the finite dimensional indecomposable and the finite dimensional irreducible t modules up to isomorphism

fuzzy intersection graphs abstract this paper extends the use of graphs to the modelling of systems under unceertainty when random assumptions do not adequately hold the paper includes the theoretical and computational aspects which are derived from the matrix representation of fuzzy intersection graphs their ability for applications is also discussed

generalized path finding algorithms on semirings and the fuzzy shortest path problem a new dioid structure k e e is proposed to solve a path finding problem in a fuzzy graph this algebraic structure is adapted precisely to solve the problem of the k best fuzzy shortest paths we demonstrate that the generalized gauss seidel s algorithm always converges for the solving of the k best fuzzy shortest paths problem on a valued fuzzy graph without cycles of negative weight this work starts a safe extension of the path algebra paradigm to valued fuzzy graphs for the shortest path finding problem

curve counting via stable pairs in the derived category for a nonsingular projective 3 fold x we define integer invariants virtually enumerating pairs c d where c x is an embedded curve and d c is a divisor a virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of x the resulting invariants are conjecturally equivalent after universal transformations to both the gromov witten and dt theories of x for calabi yau 3 folds the latter equivalence should be viewed as a wall crossing formula in the derived category

a refinement of the split decomposition of a tridiagonal pair let v denote a nonzero finite dimensional vector space over a field k and let a a denote a tridiagonal pair on v of diameter d let v u0 ud denote the split decomposition and let i denote the dimension of ui in this paper at first we show there exists a unique integer h 0 h d 2 such that i 1 i for d h i d we call h the height of the tridiagonal pair for 0 r h we define subspaces ui r r i d r by ui r ri r ur kerrd 2r 1 where r denotes the rasing map we show v is decomposed as a direct sum v r 0h i rd rui r this gives a refinement of the split decomposition define u r i rd rui r and observe v r 0hu r we show lu r u r 1 u r u r 1 for 0 r h where we set u 1 u h 1 0 let f r v u r denote the projection we show the lowering map l is decomposed as l l l 0 l where l r 1hf r 1 lf r l 0 r 0hf r lf r and l r 0h 1f r 1 lf r these maps satisfy l u r u r 1 l 0 u r u r and l u r u r 1 for 0 r h the main results of this paper are the following i for 0 r h 1 and r 2 i d r 1 rl l r holds on ui r for some scalar ii for 1 r h and r i d r 1 rl l r holds on ui r for some scalar iii for 0 r h and r 1 i d r 1 rl 0 l 0 r i holds on ui r for some scalars moreover we give explicit expressions of

deformations of canonical singularities m reid defined the concept of canonical singularities as those which appear on canonical models of algebraic varieties whose canonical rings are finitely generated from the view point of the classification theory of algebraic varieties it is expected that small deformations of a variety which has only canonical singularities have only canonical singularities the main result of this paper states that this is the case

fuzzy planar graphs fuzzy planar graph is a very important subclass of fuzzy graph in this paper two types of edges are mentioned for fuzzy graphs effective edges and considerable edges in addition a comparative study between kuratowski s graphs and fuzzy planar graph is made a new concept of a strong fuzzy planar graph is introduced some related results are established these results have certain applications in subway tunnels routes oil gas pipelines representation etc it is also shown that an image can be represented by a fuzzy planar graph and contraction of such an image can be made with the help of a fuzzy planar graph

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

folding of some types of fuzzy manifolds and their retractions abstract this paper is a continuation of el ghoul fuzzy sets and systems 58 1993 335 363 80 1995 389 396 it introduces a new type of some fuzzy manifolds the connection between fuzzification and a physical phenomena are deduced the ideal case of this type of phenomenalism is considered at 1 the relation between the retractions and the folding of these types of fuzzy manifolds are discussed theorems governing these relations are obtained

moduli of weierstrass fibrations with marked section abstract we study the moduli space of stable pairs x s s a i f i consisting of a weierstrass fibration x its section s and some fibers f i we find a compactification which is a dm stack and we describe the objects on the boundary we show that the fibration in the definition of weierstrass fibration extends to the boundary and it is equidimensional when s 1 we prove that there are wall crossing morphisms when the weights s and a i change when s 1 this recovers the work of la nave 36 and a special case of the work of ascher bejleri 11

unified approach fuzzy graph problems we present a taxonomy of fuzzy graphs that treats fuzziness in vertex existence edge existence edge connectivity and edge weight within that framework we formulate some standard graph theoretic problems shortest paths and minimum cut for fuzzy graphs using a unified approach distinguished by its uniform application of guiding principles such as the construction of membership grades via the ranking of fuzzy numbers the preservation of membership grade normalization and the collapsing of fuzzy sets of graphs into fuzzy graphs finally we provide algorithmic solutions to these problems with examples

on the terwilliger algebra of bipartite distance regular graphs with δ2 0 and c2 1 abstract let denote a bipartite distance regular graph with diameter d 4 and valency k 3 let x denote the vertex set of and let a denote the adjacency matrix of for x x and for 0 i d let i x denote the set of vertices in x that are distance i from vertex x define a parameter 2 in terms of the intersection numbers by 2 k 2 c 3 1 c 2 1 p 22 2 we first show that 2 0 implies that d 5 or c 2 1 2 for x x let t t x denote the subalgebra of mat x c generated by a e 0 e 1 e d where for 0 i d e i represents the projection onto the ith subconstituent of with respect to x we refer to t as the terwilliger algebra of with respect to x by the endpoint of an irreducible t module w we mean min i e i w 0 in this paper we assume has the property that for 2 i d 1 there exist complex scalars i i such that for all x y z x with x y 2 x z i y z i we have i i 1 x 1 y i 1 z i 1 x i 1 y 1 z we additionally assume that 2 0 with c 2 1 under the above assumptions we study the algebra t we show that if is not almost 2 homogeneous then up to isomorphism there exists exactly one irreducible t module with endpoint 2 we give an orthogonal basis for this t module and we give the action of a on this basis

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

a generalization of the terwilliger algebra abstract p m terwilliger 1992 j algebraic combin 1 363 388 considered the c algebra generated by a given bose mesner algebra m and the associated dual bose mesner algebra m this algebra is now known as the terwilliger algebra and is usually denoted by t terwilliger showed that each vanishing intersection number and krein parameter of m gives rise to a relation on certain generators of t these relations determine much of the structure of t thought not all of it in general to illuminate the role these relations play we consider a certain generalization t of t to go from t to t we replace m and m with a pair of dual character algebras c and c we define t by generators and relations intuitively t is the associative c algebra with identity generated by c and c subject to the analogues of terwilliger s relations t is infinite dimensional and noncommutative in general we construct an irreducible t module which we call the primary module the dimension of this module is the same as that of c and c we find two bases of the primary module one diagonalizes c and the other diagonalizes c we compute the action of the generators of t on these bases we show t is a direct sum of two sided ideals t 0 and t 1 with t 0 isomorphic to a full matrix algebra we show that the irreducible module associated with t 0 is isomorphic to the primary module we compute the central primitive idempotent of t associated with t 0 in terms of the generators of t

paths of least resistance in possibilistic production systems abstract we introduce the idea of possibilistic production systems we provide a general algorithm for finding a path from an initial node to a goal node we show that the use of a heuristic function can help in the efficient implementation of the search process we see that a significant role in determining potential optimal paths is played by concentrating on the most difficult activity to accomplish

characterizations of fuzzy interval graphs abstract we use the min operator to define the fuzzy intersection graph of a family of fuzzy sets and explore relationships between properties of this fuzzy graph and its family of cut level graphs we show that each fuzzy graph without loops is the intersection graph of some family of fuzzy sets we show that the characterization of interval graphs by gilmore and hoffman naturally extends to fuzzy interval graphs while that of fulkerson and gross does not

fuzzy graphs in the evaluation and optimization of networks abstract referring to some predecessing models the paper introduces the concept of fuzzy vertex graphs a way is shown for modeling various quantitatively featurable functional capabilities of computer communication and similar networks it is shown that functional capability of distributed hierarchical multicomponent systems networks can be described by the directed rooted tree model according to the above fuzzy graph idea the problem of optimizing the overall functional capability with given component vertex set is introduced the fuzzy formulae based on the algebraic connectives which are to be optimized are constructed in the main part of the paper a series of results are presented concerning some special however important cases also the hypotheses founded on some computer experiments are formulated finally the application of these results and hypotheses are shown by the example of a distributed communication switching system family being in development in hungary

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

analyzing employee behavior related questionnaires by combined fuzzy signature model abstract the evaluation of data obtained from responses given to questionnaires in humanities and social sciences such as management linguistics etc is a complex task with the necessity of dealing with the inherent subjectivity and vagueness in such data in this paper a method based on fuzzy signatures fsigs suitable for analyzing questionnaires with hierarchically connected partially vague responses is proposed and its applicability will be demonstrated by a real life problem the partial analysis of an ongoing research examining employee behavior in various companies the linkage of the factors hidden in the data bases obtained from the answers to the questionnaires containing various factors interconnected in a more or less tight way are represented by a hierarchical fsig system allowing further evaluation and the discovery of emerging connections and deeper patterns among the responses thus extending the idea of the original fsig model towards a more general fuzzy fuzzy signature approach the method proposed here is a combination of some statistical elements with the fuzzy signature model and it also uses kohonen maps in order to discover deeper structural components in the data pool as fsigs are suitable to express hierarchically structured connections among vague and imprecise features of the individual data the statistical analysis helps reveal the degrees of redundancies and the closeness of connectedness of the individual elements within the responses and thus enable the construction of a relevant fsig tree graph for the data on hand while further expert domain knowledge helps with determining the proper fuzzy aggregations in the intermediate nodes of the fsigs the case study presented is based on data obtained from north lithuanian companies the results of the case study focusing on the analysis of the connection between ocb and cwb and other factors disclose some interesting and partly unexpected results they indicate a strong and unambiguous relationship between career satisfaction and ocb which is not very surprising however it is found that there is no relationship with gender age and actual position in the company which are generally supposed to be determining factors these results may be further validated by expert knowledge and thus the new combined method for evaluating structured multicomponent data and internal dependencies is adequate

fibered stable varieties reference epfl article 231103doi 10 1090 tran 6386 url http dx doi org 10 1090 tran 6386 record created on 2017 09 19 modified on 2017 10 02

the terwilliger algebra of a distance regular graph of negative type abstract let denote a distance regular graph with diameter d 3 assume has classical parameters d b with b 1 let x denote the vertex set of and let a mat x c denote the adjacency matrix of fix x x and let a mat x c denote the corresponding dual adjacency matrix let t denote the subalgebra of mat x c generated by a a we call t the terwilliger algebra of with respect to x we show that up to isomorphism there exist exactly two irreducible t modules with endpoint 1 their dimensions are d and 2 d 2 for these t modules we display a basis consisting of eigenvectors for a and for each basis we give the action of a

on fuzzy tree definition abstract the connectedness and acyclicity levels are presented for a fuzzy graph several fuzzy tree definitions are given too and their relations with previous studies on the matter are analyzed

compactifications of smooth families and of moduli spaces of polarized manifolds let m h be the moduli scheme of canonically polarized manifolds with hilbert polynomial h we construct for v 2 with h v 0 a projective compactification m h of the reduced moduli scheme m h red such that the ample invertible sheaf corresponding to det f v x0 y0 on the moduli stack has a natural extension pic m h a similar result is shown for moduli of polarized minimal models of kodaira dimension zero in both cases natural means that the pullback of to a curve c mh induced by a family f 0 x 0 c o 1 m h is isomorphic to det f v x c whenever fo extends to a semistable model f x c besides of the weak semistable reduction of abramovich karu and the extension theorem of gabber there are new tools hopefully of interest by themselves in particular we will need a theorem on the flattening of multiplier sheaves in families on their compatibility with pullbacks and on base change for their direct images twisted by certain semiample sheaves

arakelov parshin rigidity of towers of curve fibrations arakelov parshin rigidity is concerned with varieties mapping rigidly to the moduli stack mathfrak m h of canonically polarized manifolds affirmative answer for any class of maps implies finiteness of the given class this article studies arakelov parshin rigidity on an open subspace of mathfrak m h on the locus mathfrak k mathfrak f h of iterated kodaira fibrations first we prove rigidity for all complete curves mapping finitely onto mathfrak k mathfrak f h then for generic affine curves mapping into mathfrak k mathfrak f h rigidity is shown when deg h 2 the method used in the latter part is showing that the iterated kodaira spencer map is injective

counting curves on surfaces in calabi yau 3 folds motivated by s duality modularity conjectures in string theory we define new invariants counting a restricted class of two dimensional torsion sheaves enumerating pairs z subset h in a calabi yau threefold x here h is a member of a sufficiently positive linear system and z is a one dimensional subscheme of it the associated sheaf is the ideal sheaf of z subset h pushed forward to x and considered as a certain joyce song pair in the derived category of x we express these invariants in terms of the mnop invariants of x

the uniformization of the moduli space of principally polarized abelian 6 folds author s alexeev valery donagi ron farkas gavril izadi elham ortega angela abstract abstractstarting from a beautiful idea of kanev we construct a uniformization of the moduli space mathcal a 6 of principally polarized abelian 6 folds in terms of curves and monodromy data we show that the general principally polarized abelian variety of dimension 6 is a prym tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the weyl group of the e 6 lattice along the way we establish numerous facts concerning the geometry of the hurwitz space of such e 6 covers including 1 a proof that the canonical class of the hurwitz space is big 2 a concrete geometric description of the hodge hurwitz eigenbundles with respect to the kanev correspondence and 3 a description of the ramification divisor of the prym tyurin map from the hurwitz space to mathcal a 6 in the terms of syzygies of the abel prym tyurin curve

folding of fuzzy torus and fuzzy graphs abstract this paper is a continuation of el ghoul 1993 it introduces a new definition for the folding of fuzzy torus it gives the concept of the connection between the folding of fuzzy graphs and the folding of fuzzy torus theorems governing this connection are established here

on the brody hyperbolicity of moduli spaces for canonically polarized manifolds we show that the moduli stack mh of canonically polarized complex manifolds with hilbert polynomial h is brody hyperbolic hence if mh denotes the corresponding coarse moduli scheme and if u mh is a quasi finite morphism induced by a family then there are no nonconstant holomorphic maps c u

fuzzy retraction and folding of fuzzy orientable compact manifold this paper is a continuation of a previous one el ghoul 1993 it introduces a new meaning for the retraction of fuzzy orientable compact manifold it also gives the concept of the connection between the retraction of fuzzy graphs and a fuzzy orientable compact manifold and its folding the theorems governing this connection are deduced

hypersurfaces in projective schemes and a moving lemma let x s be a quasi projective morphism over an affine base we develop in this article a technique for proving the existence of closed subschemes h s of x s with various favorable properties we offer several applications of this technique including the existence of finite quasi sections in certain projective morphisms and the existence of hypersurfaces in x s containing a given closed subscheme c and intersecting properly a closed set f assume now that the base s is the spectrum of a ring r such that for any finite morphism z s pic z is a torsion group this condition is satisfied if r is the ring of integers of a number field or the ring of functions of a smooth affine curve over a finite field we prove in this context a moving lemma pertaining to horizontal 1 cycles on a regular scheme x quasi projective and flat over s we also show the existence of a finite surjective s morphism to psd for any scheme x projective over s when x s has all its fibers of a fixed dimension d

general negations in fuzzy set theory abstract all possible negation operations which are optimal in some precise sense are fully described they turn out to be lowen s fuzzy complements yager s intuitionistic negation and a dual to the latter

nested hilbert schemes on surfaces virtual fundamental class abstract we construct virtual fundamental classes on nested hilbert schemes of points and curves in complex nonsingular projective surfaces these classes recover the virtual classes of seiberg witten theory as well as the reduced stable theory and play a crucial role in the reduced donaldson thomas theory of local surface threefolds that we study in 15 we show that certain integrals against the virtual fundamental classes of punctual nested hilbert schemes are expressed as integrals over the products of the hilbert scheme of points we are able to find explicit formulas for some of these integrals by relating them to carlsson okounkov s vertex operator formulas

positivity of relative canonical bundles and applications given a family f mathcal x to s of canonically polarized manifolds the unique kahler einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle mathcal k mathcal x s we use a global elliptic equation to show that this metric is strictly positive on mathcal x unless the family is infinitesimally trivial for degenerating families we show that the curvature form on the total space can be extended as a semi positive closed current by fiber integration it follows that the generalized weil petersson form on the base possesses an extension as a positive current we prove an extension theorem for hermitian line bundles whose curvature forms have this property this theorem can be applied to a determinant line bundle associated to the relative canonical bundle on the total space as an application the quasi projectivity of the moduli space mathcal m text can of canonically polarized varieties follows the direct images r n p f omega p mathcal x s mathcal k mathcal x s otimes m m 0 carry natural hermitian metrics we prove an explicit formula for the curvature tensor of these direct images we apply it to the morphisms s p mathcal t s to r p f lambda p mathcal t mathcal x s that are induced by the kodaira spencer map and obtain a differential geometric proof for hyperbolicity properties of mathcal m text can

modular compactifications of the space of marked trigonal curves abstract we construct a sequence of modular compactifications of the space of marked trigonal curves by allowing the branch points to coincide to a given extent beginning with the standard admissible cover compactification the sequence first proceeds through contractions of the boundary divisors and then through flips of the so called maroni strata culminating in a fano model for even genera and a fano fibration for odd genera while the sequence of divisorial contractions arises from a more general construction the sequence of flips uses the particular geometry of triple covers we explicitly describe the mori chamber decomposition given by this sequence of flips

erratum for boundedness of families of canonically polarized manifolds a higher dimensional analogue of shafarevich s conjecture we show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is nite and that this number is uniformly bounded in any nite type family of base varieties as a corollary we show that a direct generalization of the geometric version of shafarevich s original conjecture holds for innitesimally rigid families of canonically polarized varieties

optimal bounds for t singularities in stable surfaces abstract we explicitly bound t singularities on normal projective surfaces w with one singularity and k w ample this bound depends only on k w 2 and it is optimal when w is not rational we classify and realize surfaces attaining the bound for each nonnegative kodaira dimension of the minimal resolution of w this answers effectiveness of bounds see 1 2 17 for those surfaces