weakly lefschetz symplectic manifolds for a symplectic manifold the harmonic cohomology of symplectic divisors introduced by donaldson 1996 and of the more general symplectic zero loci introduced by auroux 1997 are compared with that of its ambient space we also study symplectic manifolds satisfying a weakly lefschetz property that is the s lefschetz property in particular we consider the symplectic blow ups cp m of the complex projective space cp m along weakly lefschetz symplectic submanifolds m cp m as an application we construct for each even integer s 2 compact symplectic manifolds which are s lefschetz but not s 1 lefschetz

markov decision evolutionary games we present a class of evolutionary games involving large populations that have many pairwise interactions between randomly selected players the fitness of a player depends not only on the actions chosen in the interaction but also on the individual state of the players players have a finite life time during which they participate in several local interactions and take actions the actions taken by a player determine not only the immediate fitness but also the transition probabilities to its next individual state we define and characterize the evolutionary stable strategies for these games and propose a method to compute them we illustrate the model and results through a networking problem

the projective geometry of freudenthal s magic square abstract we connect the algebraic geometry and representation theory associated to freudenthal s magic square we give unified geometric descriptions of several classes of orbit closures describing their hyperplane sections and desingularizations and interpreting them in terms of composition algebras in particular we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial

koszul divisors on moduli spaces of curves given a moduli space how can one construct the best in the sense of higher dimensional algebraic geometry effective divisor on it we show that at least in the case of the moduli space of curves the answer is provided by the koszul divisor defined in terms of the syzygies of the parameterized objects in this paper we find a formula for the slopes of all koszul divisors on overline cal m g in particular we obtain the first infinite series of counterexamples to the harris morrison slope conjecture and we prove the maximal rank conjecture in the case when the brill noether number of the corresponding linear series equals 0 we also find shorter proofs for the formulas of the class of the brill noether and gieseker petri divisors finally we improve most of logan s results on the kodaira dimension of the moduli spaces overline cal m g n of pointed stable curves

geometry of curves with exceptional secant planes linear series along the general curve we study linear series on a general curve of genus g whose images are exceptional with regard to their secant planes working in the framework of an extension of brill noether theory to pairs of linear series we prove that a general curve has no linear series with exceptional secant planes in a very precise sense whenever the total number of series is finite next we partially solve the problem of computing the number of linear series with exceptional secant planes in a one parameter family in terms of tautological classes associated with the family by evaluating our hypothetical formula along judiciously chosen test families as an application we compute the number of linear series with exceptional secant planes on a general curve equipped with a one dimensional family of linear series we pay special attention to the extremal case of d secant d 2 planes to 2d 1 dimensional series which appears in the study of hilbert schemes of points on surfaces in that case our formula may be rewritten in terms of hypergeometric series which allows us both to prove that it is nonzero and to deduce its asymptotics in d

a matrix game solution of the single controller stochastic game we consider a stochastic game on finitely many states with limiting average payoff further we assume that the law of motion depends on the actions of one player only say the minimizer we show that these games have the following properties a the value of the stochastic game for each state s is the same as the value of the matrix game whose rows and columns are the pure stationary strategies of the players and whose entries are the corresponding payoffs b these matrix games have a common optimal strategy for the maximizer which in turn yields his optimal stationary strategy in the stochastic game c if the value of a stochastic subgame obtained by deleting a column or a row in a particular state coincides with the value of the original game at that state then it coincides at all states d if all actions of the minimizer at each state are essential for optimal play then in the transient states under optimal play the minimizer has only one action in such a case the original game can be solve

local rigidity of quasi regular varieties for a g variety x with an open orbit we define its boundary x as the complement of the open orbit the action sheaf s x is the subsheaf of the tangent sheaf made of vector fields tangent to x we prove for a large family of smooth spherical varieties the vanishing of the cohomology groups h i x s x for i 0 extending results of bien and brion compos math 104 1 26 1996 we apply these results to study the local rigidity of the smooth projective varieties with picard number one classified in pasquier math ann in press

on fano complete intersections in rational homogeneous varieties complete intersections inside rational homogeneous varieties provide interesting examples of fano manifolds for example if x cap i 1 r d i subset g p is a smooth complete intersection of r ample divisors such that k g p otimes mathcal o g p sum i d i is ample then x is fano we first classify these fano complete intersections which are locally rigid it turns out that most of them are hyperplane sections we then classify general hyperplane sections which are quasi homogeneous

the birational type of the moduli space of even spin curves abstract we determine the kodaira dimension of the moduli space s g of even spin curves for all g precisely we show that s g is of general type for g 8 and has negative kodaira dimension for g 8

on the construction of contact submanifolds with prescribed topology we prove the existence of contact submanifolds realizing the poincare dual of the top chern class of a complex vector bundle over a closed contact manifold this result is analogue in the contact categoryto donaldson s construction of symplectic submanifolds the main tool in the construction is to show the existence of sequences of sections which are asymptotically holomorphic in an appropiate sense and that satisfya transversalitywith es timates propertydirectlyin the contact category the description of the ob tained contact submanifolds allows us to prove an extension of the lefschetz hyperplane theorem which completes their topological characterization

spin curves and scorza quartics in the paper takagi and zucconi in on blow ups of the quintic del pezzo 3 fold and varieties of power sums of quartic hypersurfaces pp 1 40 preprint submitted 2008 we construct new subvarieties in the varieties of power sums for certain quartic hypersurfaces in this paper we show that these quartics coincide with the scorza quartics of general pairs of trigonal curves and ineffective theta characteristics among other applications we give an affirmative answer to the conjecture of dolgachev and kanev on the existence of the scorza quartics for any general pairs of curves and ineffective theta characteristics we also give descriptions of the moduli spaces of trigonal even spin curves

almost holomorphic embeddings in grassmannians with applications to singular symplectic submanifolds in this paper we use donaldson s approximately holomorphic techniques to build embeddings of a closed symplectic manifold with symplectic form of integer class in the grassmannians gr r n we assure that these embeddings are asymptotically holomorphic in a precise sense we study first the particular case of cpn obtaining control on n and we improve in a sense a classical result about symplectic embeddings the main reason of our study is the construction of singular determinantal submanifolds as the intersection of the embedding with certain generalized schubert cycles defined on a product of grassmannians it is shown that the symplectic type of these submanifolds is quite more general that the ones obtained by donaldson and auroux as zeroes of very ample vector bundles

limit linear series basic theory in this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves curves of compact type the technically much simpler special case of 1 dimensional series was developed by beauville 2 knudsen 21 23 harris and mumford 17 in the guise of admissible covers it has proved very useful for studying the moduli space of curves the above papers and harris 16 and the simplest sorts of weierstrass points diaz 4 with our extended tools we are able to prove for example that in this paper we present the basic theory of limit linear series necessary for proving these results the results themselves will be taken up in our forthcoming papers 8 12 simpler applications not requiring the tools developed in this paper but perhaps clarified by them have already been given in our papers 5 7

on some smooth projective two orbit varieties with picard number 1 we classify all smooth projective horospherical varieties with picard number 1 we prove that the automorphism group of any such variety x acts with at most two orbits and that this group still acts with only two orbits on x blown up at the closed orbit we characterize all smooth projective two orbit varieties with picard number 1 that satisfy this latter property

chern schwartz macpherson classes and the euler characteristic of degeneracy loci and special divisors this concept overlaps a large family of interesting varieties for example the varieties of special divisors studied in section 3 several authors have worked out explicit formulas for the euler characteristic of dr 9 in terms of different cohomological and numerical invariants under the assumption that x is nonsingular and v is appropriately general for instance if 0 is a section of a vector bundle then the formulas for the euler characteristic x do fp were given by hirzebruch h and navarro aznar n if dr is a curve or a surface in a nonsingular x some explicit formulas for x dr v were given by harris and tu h t in terms of the chern classes of e f and x but under the extra assumption dr 1 v 0 which implies that dr q is nonsingular in loc cit the authors also posed the problem of finding a general formula for x dr o if such exists under the assumption dr 1 0 or even stronger without this assumption the first problem was solved positively by the second named author prl proposition 5 7 by the use of polynomials universally supported on degeneracy

the geometry of the moduli space of odd spin curves the spin moduli spacesg is the parameter space of theta characteristics spin structures on stable curves of genus g it has two connected components s g ands g depending on the parity of the spin structure we establish a complete birational classication by kodaira dimension of the odd components g of the spin moduli space we show thats g is uniruled for g 12 and even unirational for g 8 in this range introducing the concept of cluster for the mukai variety whose one dimensional linear sections are general canonical curves of genus g we construct new birational models ofs g these we then use to explicitly describe the birational structure of s g for instance s 8 is birational to a locally trivial p 7 bundle over the moduli space of elliptic curves with seven pairs of marked points for g 12 we prove thats g is a variety of general type in genus 12 this requires the construction of a counterexample to the slope conjecture on eective divisors on the moduli space of stable curves of genus 12

game interactions and dynamics on networked populations a new mathematical formulation of evolutionary game dynamics on networked populations is proposed the model extends the standard replicator equation to a finite set of players organized on an arbitrary network of connections graph classical results of multipopulation evolutionary game theory are used in combination with graph theory to obtain the mathematical model specifically the players located at the vertices of the graph are interpreted as subpopulations of a multipopulation dynamical game the members of each subpopulation are replicators engaged at each time instant into 2 player games with the members of other connected subpopulations this idea allows us to write an extended equation describing the game dynamics of a finite set of players connected by a graph the obtained equation does not require any assumption on the game payoff matrices nor graph topology stability of steady states nash equilibria and the relationship of the proposed model to the standard replicator equation are discussed the dynamical behavior of the model over different graphs is also investigated by means of extended simulations

hodge structures on cohomology algebras and geometry we study restrictions on cohomology algebras of compact kahler manifolds imposed by the presence of a polarized hodge structure on cohomology groups compatible with the cup product but not depending on the hp q numbers or the symplectic structure to illustrate the effectiveness of these restrictions we give a number of examples of compact symplectic manifolds satisfying the formality condition the lefschetz property and having commutative or trivial 1 but not having the cohomology algebra of a compact kaehler manifold we also prove a stability theorem for these restrictions if a compact kahler manifold is homeomorphic to a product x y with one summand satisfying b1 0 then the cohomology algebra of each summand carries a polarized hodge structure

symplectic 4 manifolds as branched coverings of ℂℙ2 we show that every compact symplectic 4 manifold x can be topologically realized as a covering of 2 branched along a smooth symplectic curve in x which projects as an immersed curve with cusps in 2 furthermore the covering map can be chosen to be approximately pseudo holomorphic with respect to any given almost complex structure on x

convergence of linear threshold decision making dynamics in finite heterogeneous populations abstract linear threshold models have been studied extensively in structured populations however less attention is paid to perception differences among the individuals of the population to focus on this effect we exclude structure and consider a well mixed population of heterogeneous agents each associated with a threshold in the form of a fixed ratio within zero and one that can be unique to this agent the agents are initialized with a choice of strategy a or b and at each time step one agent becomes active to update if the ratio of agents playing a is higher resp lower than her threshold she updates to a resp b we show that for any given initial condition after a finite number of time steps the population reaches an equilibrium where no agent s threshold is violated however the equilibrium is not necessarily uniquely determined by the initial condition but depends on the agents activation sequence we find all those possible equilibria that the dynamics may reach from a given initial condition and show that in contrast to the case of homogeneous populations heterogeneity in the agents thresholds gives rise to several equilibria where both a playing and b playing agents coexist perception heterogeneity can hence preserve decision diversity even in the absence of population structure we also investigate the asymptotic stability of the equilibria and show how to calculate the contagion probability for a population with two thresholds

curves and symmetric spaces ii let sym 3 c p k sym 3 k sym 3 k k p 13 a 1 a a det a be the veronese embedding of the space of symmetric matrices of degree 3 where a is the cofactor matrix of a the closure spg 3 6 of this image is a 6 dimensional homogeneous variety of the symplectic group sp 3 a canonical curve c 16 c p 8 of genus 9 over a perfect field k is isomorphic to a complete linear section of this projective variety spg 3 6 c p 13 unless c k k k being the algebraic closure is a covering of degree at most 5 of the projective line we prove this by means of linear systems of higher rank

moduli of theta characteristics via nikulin surfaces we study moduli spaces of k3 surfaces endowed with a nikulin involution and their image in the moduli space r g of prym curves of genus g we observe a striking analogy with mukai s well known work on ordinary k3 surfaces many of mukai s results have a very precise prym nikulin analogue for instance a general prym curve from r g is a section of a nikulin surface if and only if g 7 and g 6 furthermore r 7 has the structure of a fibre space over the corresponding moduli space of polarized nikulin surfaces we then use these results to study the geometry of the moduli space s g of even spin curves with special emphasis on the transition case of s 8 which is a 21 dimensional calabi yau variety

prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of picard number 1 under kahler deformation

classification of non degenerate projective varieties with non zero prolongation and application to target rigidity the prolongation mathfrak g k of a linear lie algebra mathfrak g subset mathfrak gl v plays an important role in the study of symmetries of g structures cartan and kobayashi nagano have given a complete classification of irreducible linear lie algebras mathfrak g subset mathfrak gl v with non zero prolongations

moduli of roots of line bundles on curves we treat the problem of completing the moduli space for roots of line bundles on curves special attention is devoted to higher spin curves within the universal picard scheme two new different constructions both using line bundles on nodal curves as boundary points are carried out and compared with pre existing ones

chern numbers of kernel and cokernel bundles appendix to on the kodaira dimension of the moduli space of curves ii the even genus case by j harris

2 adic valuations of certain ratios of products of factorials and applications abstract we prove the conjecture of falikman friedland loewy on the parity of the degrees of projective varieties of n n complex symmetric matrices of rank at most k we also characterize the parity of the degrees of projective varieties of n n complex skew symmetric matrices of rank at most 2 p we give recursive relations which determine the parity of the degrees of projective varieties of m n complex matrices of rank at most k in the case the degrees of these varieties are odd we characterize the minimal dimensions of subspaces of n n skew symmetric real matrices and of m n real matrices containing a nonzero matrix of rank at most k the parity questions studied here are also of combinatorial interest since they concern the parity of the number of plane partitions contained in a given box on the one hand and the parity of the number of symplectic tableaux of rectangular shape on the other hand

on universal severi varieties of low genus k3 surfaces we prove the irreducibility of universal severi varieties parametrizing irreducible reduced nodal hyperplane sections of primitive k3 surfaces of genus g with 3 g 11 g 10

chow rings of moduli spaces of curves ii some results on the chow ring of overline mathscr m _4 ring of the moduli space of stable curves of genus 4 these results are not complete we find generators for the chow ring of 4 and for the chow groups in codimension 1 and 2 of w4 for a2 g 4 we find fourteen generators using test surfaces we prove that the dimension of a 2 4 4 is at least 13 and explicitly determine the single relation between the fourteen generators which still can exist finally we have two proofs that this relation does indeed hold so that the dimension of a2 4 4 equals 13 this enables us to determine the chow ring of 4 4 our original proof is based on a rather delicate argument the second proof uses a result of ran see r and is much simpler