lagrangian subvarieties of the moduli space of stable vector bundles on a regular algebraic surface with pg 0 mukai m1 showed that there is a nondegenerate symplectic structures on the moduli space of stable vector bundles on a k3 surface later tyurin t2 studied generalized symplectic structures on the moduli space of stable vector bundles on any smooth regular surface x with p9 0 in the work of tyurin a symplectic structure means a nonzero regular two form on the moduli spaces in particular it may degenerate in this paper we define a lagrangian subvariety of the moduli space to be a subvariety on the zariski tangent space at any point of which the given symplectic two form is identically zero note that we do not impose any restriction on the dimension of a lagrangian subvariety this is because symplectic structures considered here may degenerate the purpose of this paper is to use bril l noether theory for curves to construct explicit ly a family of lagrangian subvarieties of the moduli space of stable vector bundles on a regular surface with pg 0 le t p be a generically smooth and irreducible component of the moduli space of gieseker stable bundles of rank r 1 with respect to a fixed polarization d on a regular algebraic surface x with p 0 by the boundedness of jf see ma after possibly twisted by the same negative line bundle on x we can assume that for any point e c j i e is generated by global sections ii hl e hz e hi e o iii h l de t e hz det e 0 for any point e c z choose a r 1 dimensional subspace v c h and consider an evaluation map e v v p x e for a general v we can make e v degenerate exactly along a smooth curve c c x and coker e v is a line bundle

tilting objects in the stable category of vector bundles on a weighted projective line of type national natural science foundation of china 11201386 10931006 11071040 11201388 natural science foundation of fujian province of china 2012j05009

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

a curve algebraically but not rationally uniformized by radicals zariski proved the general complex projective curve of genus g 6 is not rationally uniformized by radicals that is admits no map to p1 whose galois group is solvable we give an example of a genus seven complex projective curve z that is not rationally uniformized by radicals but such that there is a finite covering z z with z rationally uniformized by radicals the curve providing the example appears in a paper by debarre and fahlaoui where a construction is given to show the brill noether loci wd c in the jacobian of a curve c may contain translates of abelian subvarieties not arising from maps from c to other curves

exceptional collections of line bundles on the beauville surface abstract we construct quasi phantom admissible subcategories in the derived category of coherent sheaves on the beauville surface s these quasi phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on s we prove that there are exactly 6 exceptional collections consisting of line bundles up to a twist and these collections are spires of two helices

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

exact simulation of stochastic volatility and other affine jump diffusion processes the stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated discretization methods can be used for simulating security prices under these models however discretization introduces bias into the simulation results and a large number of time steps may be needed to reduce the discretization bias to an acceptable level this paper suggests a method for the exact simulation of the stock price and variance under hestons stochastic volatility model and other affine jump diffusion processes the sample stock price and variance from the exact distribution can then be used to generate an unbiased estimator of the price of a derivative security we compare our method with the more conventional euler discretization method and demonstrate the faster convergence rate of the error in our method specifically our method achieves an o s 1 2 convergence rate where s is the total computational budget the convergence rate for the euler discretization method is o s 1 3 or slower depending on the model coefficients and option payoff function

brill noether theory of curves on enriques surfaces i the positive cone and gonality we study the existence of linear series on curves lying on an enriques surface and general in their complete linear system using a method that works also below the bogomolov reider range we compute in all cases the gonality of such curves we also give a new result about the positive cone of line bundles on an enriques surface and we show how this relates to the gonality

exceptional collections on dolgachev surfaces associated with degenerations abstract dolgachev surfaces are simply connected minimal elliptic surfaces with p g q 0 and of kodaira dimension 1 these surfaces are constructed by logarithmic transformations of rational elliptic surfaces in this paper we explain the construction of dolgachev surfaces via q gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities this construction is based on the paper 25 also some exceptional bundles on dolgachev surfaces associated with q gorenstein smoothing have been constructed based on the idea of hacking 12 in the case if dolgachev surfaces were of type 2 3 we describe the picard group and present an exceptional collection of maximal length finally we prove that the presented exceptional collection is not full hence there exists a nontrivial phantom category in the derived category

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

volatility swaps valuation under stochastic volatility with jumps and stochastic intensity abstract in this paper a pricing formula for volatility swaps is delivered when the underlying asset follows the stochastic volatility model with jumps and stochastic intensity by using the feynman kac theorem a partial integral differential equation is obtained to derive the joint moment generating function of the previous model moreover discrete and continuous sampled volatility swap pricing formulas are given by employing the transform technique and the relationship between two pricing formulas is discussed under mild conditions finally some numerical simulations are reported to support the results presented in this paper

the generic green lazarsfeld secant conjecture using lattice theory on special k3 surfaces calculations on moduli stacks of pointed curves and voisin s proof of green s conjecture on syzygies of canonical curves we prove the prym green conjecture on the naturality of the resolution of a general prym canonical curve of odd genus as well as many cases of the green lazarsfeld secant conjecture on syzygies of non special line bundles on general curves

the space of stability conditions on the local projective plane we study the space of stability conditions on the total space of the canonical bundle over the projective plane we explicitly describe a chamber of geometric stability conditions and show that its translates via autoequivalences cover a whole connected component we prove that this connected component is simply connected we determine the group of autoequivalences preserving this connected component which turns out to be closely related to 1 3 finally we show that there is a submanifold isomorphic to the univer sal covering of a moduli space of elliptic curves with 1 3 level struc ture the morphism is 1 3 equivariant and is given by solutions of picard fuchs equations this result is motivated by the notion of stability and by mirror symmetry

on the projective normality of double coverings over a rational surface abstract we study the projective normality of a minimal surface x which is a ramified double covering over a rational surface s with dim k s 1 in particular horikawa surfaces the minimal surfaces of general type with k x 2 2 p g x 4 are of this type up to resolution of singularities let be the covering map from x to s we show that the z 2 invariant adjoint divisors k x r a are normally generated where the integer r 3 and a is an ample divisor on s

asymptotic error distributions for the euler method for stochastic differential equations we are interested in the rate of convergence of the euler scheme approximation of the solution to a stochastic differential equation driven by a general possibly discontinuous semimartingale and by the asymptotic behavior of the associated normalized error it is well known that for ito s equations the rate is 1 sqrt n we provide a necessary and sufficient condition for this rate to be 1 sqrt n when the driving semimartingale is a continuous martingale or a continuous semimartingale under a mild additional assumption we also prove that in these cases the normalized error processes converge in law the rate can also differ from 1 sqrt n this is the case for instance if the driving process is deterministic or if it is a levy process without a brownian component it is again 1 sqrt n when the driving process is levy with a nonvanishing brownian component but then the normalized error processes converge in law in the finite dimensional sense only while the discretized normalized error processes converge in law in the skorohod sense and the limit is given an explicit form

espaces abstraits de morphismes et mutations i try to extend here some results of the paper moduli spaces of decomposable morphisms of sheaves and quotients by non reductive groups let e f be decomposable sheaves on a smooth projective variety i find more linearizations of the natural action of the group mathop rm aut nolimits e times mathop rm aut nolimits f on mathop rm hom nolimits e f such that a good quotient of the open set of semi stable morphisms exists to obtain this one associates to mathop rm hom nolimits e f another space of morphisms mathop rm hom nolimits e f in such a way that there is a natural bijection between the set of orbits of an open subset of mathop rm hom nolimits e f and the set of orbits of an open subset of mathop rm hom nolimits e f we can in this case deduce the existence of good quotients of some open subsets of mathop rm hom nolimits e f from the existence of good quotients of the corresponding open subsets of mathop rm hom nolimits e f

exponential distribution of financial returns at mesoscopic time lags a new stylized fact we study the probability distribution of stock returns at mesoscopic time lags return horizons ranging from about an hour to about a month while at shorter microscopic time lags the distribution has power law tails for mesoscopic times the bulk of the distribution more than 99 of the probability follows an exponential law the slope of the exponential function is determined by the variance of returns which increases proportionally to the time lag at longer times the exponential law continuously evolves into gaussian distribution the exponential to gaussian crossover is well described by the analytical solution of the heston model with stochastic volatility r 2004 elsevier b v all rights reserved

on the valuation of variance swaps with stochastic volatility abstract this paper is an extension to a recent paper by zhu and lian 2011 1 in which a closed form exact solution was presented for the price of variance swaps with a particular definition of the realized variance here we further demonstrate that our approach is quite versatile and can be used for other definitions of the realized variance as well in particular we present a closed form formula for the price of a variance swap with the realized variance in the payoff function being defined as a logarithmic return of the underlying asset at some pre specified discretely sampling points the simple formula presented here is a result of successfully finding an exact solution of the partial differential equation pde system based on the heston 1993 s 2 two factor stochastic volatility model a distinguishable feature of this new solution is that the computational time involved in pricing variance swaps with discretely sampling time has been substantially improved

on the nef cone of symmetric products of a generic curve the nef cone of the symmetric product c of a curve c with general moduli is easily determined when k is at least equal to the gonality gon c of the curve in this paper we describe the nef cone for k gon c 1 when the genus of c is even in this case the boundary of the nef cone has rational slope and is determined using curves on c associated to the g s of c 1 introduction let c be a smooth irreducible curve of genus g 1 for any k 2 denote by c its mi symmetric product that parametrizes effective degree k divisors z x x2 on c let ni c k resp nl c k be the set of r linear combinations of 1 cycles resp divisors on k modulo numerical equivalence nx k is called the n ron severi group of c and denoted ns c k n k and nl k are finite dimensional vector spaces and the bilinear form given by the intersection pairing is nondegenerate if c has general moduli n c and nl c have rank two an interesting problem is to try to determine inside n c the convex cone of classes of effective 1 cycles denoted by ne c and called the cone of curves or mori cone dually one can study inside nl c the nef cone nef k i e the closure of the convex cone of classes of ample divisors by kleiman s criterion of ampleness

generalized lazarsfeld mukai bundles and a conjecture of donagi and morrison abstract let s be a k3 surface and assume for simplicity that it does not contain any 2 curve using coherent systems we express every non simple lazarsfeld mukai bundle on s as an extension of two sheaves of some special type that we refer to as generalized lazarsfeld mukai bundles this has interesting consequences concerning the brill noether theory of curves c lying on s from now on let g denote the genus of c and a be a complete linear series of type g d r on c such that d g 1 and the corresponding brill noether number is negative first we focus on the cases where a computes the clifford index if r 1 and with only some completely classified exceptions we show that a coincides with the restriction to c of a line bundle on s this is a refinement of green and lazarsfeld s result on the constancy of the clifford index of curves moving in the same linear system then we study a conjecture of donagi and morrison predicting that under no hypothesis on its clifford index a is contained in a g e s which is cut out from a line bundle on s and satisfies e g 1 we provide counterexamples to the last inequality already for r 2 a slight modification of the conjecture which holds for r 1 2 is proved under some hypotheses on the pair c a and its deformations we show that the result is optimal in the sense that our hypotheses cannot be avoided by exhibiting in appendix a some counterexamples obtained jointly with andreas leopold knutsen

triangle singularities ade chains and weighted projective lines abstract we study the singularity category of the triangle singularity f x a y b z c or s k x y z f attached to the weighted projective line x given by the weight triple a b c by investigating the stable category vect x of vector bundles on x obtained from the category of vector bundles by factoring out all the line bundles we show that vect x is fractional calabi yau whose cy dimension is a function of the euler characteristic of x and establish the existence of a tilting object having the shape of an a 1 b 1 c 1 cuboid the weight types 2 a b yield an explanation of happel seidel symmetry for a class of important nakayama algebras further the weight sequence 2 3 p corresponds to an ade chain the e n chain extrapolating the exceptional dynkin cases e 6 e 7 and e 8 to a whole sequence of triangulated categories

a sharp lower bound for the canonical volume of 3 folds of general type let v be a smooth projective 3 fold of general type denote by k 3 a rational number the self intersection of the canonical sheaf of any minimal model of v one defines k 3 as the canonical volume of v assume p g v 2 we show that k 3 ge frac 1 3 which is a sharp lower bound then we classify those v with small volume we also give some new examples with p g 2 which have maximal canonical stability index finally we give an application to 4 folds of general type

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

pricing forward start variance swaps with stochastic volatility abstract in this paper a general approach is presented to price forward start variance swaps with discrete sampling times based on the heston 1993 s two factor stochastic volatility model using this approach we work out two analytical closed form formulae for the price of forward start variance swap with the realized variance being defined by the actual return realized variance and the log return realized variance respectively the main features of this new approach and the developed formulae as a result include 1 treating the pricing problem of variance swaps with different definitions of discretely sampled realized variance in a highly unified way 2 easily obtaining analytical closed form solutions for forward start variance swaps with two popularly used definitions of discretely sampled realized variance 3 enabling the investigation of some important properties of the forward start variance swaps utilizing the elegant and simple form of the obtained solutions with these advantages we believe that the approach can be applied to price variance swaps based on other stochastic volatility models as well

on the adjunction mapping of very ample vector bundles of corank one let e be a very ample vector bundle of rank n 1 over a smooth complex projective variety x of dimension n 3 the structure of x e being known when k x det e 0 we investigate the structure of the adjunction mapping when 0 k x det e n

on irregular threefolds and fourfolds with numerically trivial canonical bundle we prove that for a smooth projective irregular threefold x with k x equiv 0 and a nef and big divisor l on x ml p gives a birational map for all m ge 3 and all p in mathrm pic 0 x we also use the same method to deal with fourfolds and prove that for a smooth projective irregular fourfold x with k x equiv 0 and an ample divisor l on x ml p gives a birational map for all m ge 5 and all p in mathrm pic 0 x these results are optimal

exceptional collections of line bundles on projective homogeneous varieties abstract we construct new examples of exceptional collections of line bundles on the variety of borel subgroups of a split semisimple linear algebraic group g of rank 2 over a field we exhibit exceptional collections of the expected length for types a 2 and b 2 c 2 and prove that no such collection exists for type g 2 this settles the question of the existence of full exceptional collections of line bundles on projective homogeneous g varieties for split linear algebraic groups g of rank at most 2

a transfer matrix study of conductivity and permeability exponents in continuum percolation the conductivity and permeability exponents for the swiss cheese model of continuum percolation are calculated numerically using the transfer matrix method in two dimensions the conductivity exponent t is shown to equal its universal value tun 1 24 while the permeability exponent e 2 53 is considerably larger the same exponents in three dimensions are also determined and found to be t 2 46 and e 4 1 greatly exceeding the universal value tun 1 95 the results are in fair agreement with earlier theoretical predictions

irreducibility brill noether loci and vojta s inequality this paper deals with generalizations of hilbert s irreducibility theorem the classical hilbert irreducibility theorem states that for any cover f of the projective line defined over a number field k there exist infinitely many k rational points on the projective line such that the fiber of f over p is irreducible over k in this paper we consider similar statements about algebraic points of higher degree on curves of any genus we prove that hilbert s irreducibility theorem admits a natural generalization to rational points on an elliptic curve and thus via a theorem of abramovich and harris to points of degree 3 or less on any curve we also present examples that show that this generalization does not hold for points of degree 4 or more these examples come from an earlier geometric construction of debarre and fahlaoui some additional necessary facts about this construction can be found in the appendix provided by debarre we exhibit a connection between these irreducibility questions and the sharpness of vojta s inequality for algebraic points on curves in particular we show that vojta s inequality is not sharp for the algebraic points arising in our examples

green s conjecture for curves on rational surfaces with an anticanonical pencil green s conjecture is proved for smooth curves c lying on a rational surface s with an anticanonical pencil under some mild hypotheses on the line bundle l mathcal o s c constancy of clifford dimension clifford index and gonality of curves in the linear system vert l vert is also obtained

a superconvergent partial differential equation approach to price variance swaps under regime switching models we present a superconvergent finite difference algorithm to price discretely sampled variance swaps we consider the black scholes model the merton s jump diffusion model stochastic volatility models that use constant elasticity of variance for the instantaneous variance and corresponding regime switching models pde approach provides a universal and efficient framework for pricing under these models to obtain extremely accurate results we solve pdes whose associated terminal conditions can be represented as second order polynomials based on the two popular definitions of realised variance and for which the spatial derivatives greater than second order are all zero we then apply second order finite difference discretisations in space with an exponential time integration we also derive analytical solutions under the merton s model and some regime switching models to validate our superconvergent results

diophantine equations via cluster transformations abstract motivated by fomin zelevinsky s theory of cluster algebras we introduce a variant of the markov equation we show that all natural solutions of the equation arise from an initial solution by cluster transformations

on stability conditions for the quintic threefold we study the clifford type inequality for a particular type of curves c 2 2 5 which are contained in smooth quintic threefolds this allows us to prove some stronger bogomolov gieseker type inequalities for chern characters of stable sheaves and tilt stable objects on smooth quintic threefolds employing the previous framework by bayer bertram macri stellari and toda we construct an open subset of stability conditions on every smooth quintic threefold in mathbf p 4 mathbb c

a transfer matrix approach to random resistor networks introduces a transfer matrix formulation to compute the conductance of random resistor networks the authors apply this method to strips of width up to 40 and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions t 1 28 or 0 03

pricing variance swaps under stochastic volatility and stochastic interest rate in this paper we investigate the effects of imposing stochastic interest rate driven by the cox ingersoll ross process along with the heston stochastic volatility model for pricing variance swaps with discrete sampling times a dimension reduction mechanism based on the framework of little and pant 2001 is applied which later reduces to solving two three dimensional partial differential equations a semi closed form solution to the fair delivery price of a variance swap is obtained via the derivation of characteristic functions practical implementation of this hybrid model is demonstrated through numerical simulations

tilt stability vanishing theorems and bogomolov gieseker type inequalities abstract we investigate the tilt stability of stable sheaves on projective varieties with respect to certain tilt stability conditions depends on two parameters constructed by bridgeland 11 see also 1 6 5 for a stable sheaf we give effective bounds of these parameters such that the stable sheaf is tilt stable these allow us to prove new vanishing theorems for stable sheaves and an effective serre vanishing theorem for torsion free sheaves using these results we also prove bogomolov gieseker type inequalities for the third chern character of a stable sheaf on p 3

exceptional collections for grassmannians of isotropic lines we construct a full exceptional collection of vector bundles in the derived categories of coherent sheaves on the grassmannian of isotropic two dimensional subspaces in a symplectic vector space of dimension 2n and in an orthogonal vector space of dimension 2n 1 for all n

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

t structures on some local calabi yau varieties abstract let z be a smooth fano variety satisfying the condition that the rank of the grothendieck group of z is one more than the dimension of z let z denote the total space of the canonical line bundle of z considered as a non compact calabi yau variety we use the theory of exceptional collections to describe t structures on the derived category of coherent sheaves on z the combinatorics of these t structures is determined by a natural action of an affine braid group closely related to the well known action of the artin braid group on the set of exceptional collections on z

the mercat conjecture for stable rank 2 vector bundles on generic curves we prove the mercat conjecture for rank 2 vector bundles on generic curves of every genus for odd genus we identify the effective divisor in mg where the mercat conjecture fails and compute its slope the clifford index of a smooth curve c defined as the non negative quantity cliff c min deg l 2r l l pic c h c l 2 deg l g 1 stratifies the moduli space mg of curves of genus g with the smallest stratum being that consisting of hyperelliptic curves which have clifford index zero while the definition is obviously inspired by the classical clifford theorem the notion came to real prominence in the works of green lazarsfeld g gl voisin v2 and many others in the context of syzygies of canonical curves for a general curve c of genus g one has cliff c bg 1 2 c the locus of curves c mg with cliff c bg 1 2 c is a subvariety of codimension 1 respectively 2 when g is odd respectively even much studied by harris and mumford hm it has been a long standing problem to find an adequate definition of the clifford index for higher rank vector bundles on curves a higher rank clifford index should not only capture the behavior of the generic curve from the point of view of special higher rank vector bundles but also provide a geometrically meaningful stratification ofmg an interesting definition has been put forward by lange and newstead ln1 for a semistable vector bundle e of rank 2 and slope e on a curve c of genus g one defines its clifford index as cliff e e h c e 2 0 the rank 2 clifford index of c is then defined as the quantity cliff2 c min cliff e e uc 2 d d 2g 2 h c e 4 observe that if l is a line bundle on c with deg l 2h0 c l 2 cliff c that is l computes the clifford index of c then cliff l l cliff c in particular the inequality cliff2 c cliff c holds for every curve c of genus g the main achievement of this paper is the proof of the following result theorem 1 for a general curve c of genus g the following equality holds cliff2 c cliff c g 1 2 equivalently if e is a semistable rank 2 vector bundle on c contributing to cliff2 c then h c e e 2 cliff c mercat me conjectured the equality cliff2 c cliff c for every smooth curve c mg counterexamples to this expectation have been found using noether lefschetz special k3 surfaces in fo1 fo2 and ln2 however these examples are special in moduli and theorem 1 proves mercat s conjecture for generic curves of every genus

bridgeland stability conditions on wild kronecker quivers abstract we prove that for l 3 the space of bridgeland stability conditions on the kronecker quiver with l parallel arrows is biholomorphic to c h as a complex manifold

brill noether loci and the gonality stratification of m g for an irreducible smooth projective complex curve c of genus g the gonality defined as gon c min d z 1 there exists a gd on c is perhaps the second most natural invariant it gives an indication of how far c is from being rational in a way different from what the genus does for g 3 we consider the stratification of the moduli space mg of smooth curves of genus g given by gonality mg 2 m 1 g 3 m 1 g k mg

numerical study of transport properties in continuum percolation the authors present numerical simulations of ac conductance for a random resistor capacitor network the conductance obeys a probability density function p g varies as g alpha 0 alpha 1 they use a highly efficient propagation algorithm to calculate the effective conductance of a long strip of a lattice at low frequencies they find that for the concentration p of conducting bonds less than the percolation threshold pc the imaginary part of conductance is proportional to frequency im geff approximately omega and the real part of conductance shows an anomalous frequency dependence re geff approximately omega 2 alpha the results of simulations in such a continuum system are in agreement with the predictions of the effective medium and the maxwell garnett approximation they also calculate the non universal dc conductivity exponents in continuum percolation the results are consistent with earlier theoretical predictions and numerical calculations

singularities of moduli spaces of sheaves on k3 surfaces and nakajima quiver varieties abstract the aim of this paper is to study the singularities of certain moduli spaces of sheaves on k3 surfaces by means of nakajima quiver varieties the singularities in question arise from the choice of a non generic polarization with respect to which we consider stability and admit natural symplectic resolutions corresponding to choices of general polarizations for sheaves that are pure of dimension one we show that these moduli spaces are locally around a singular point isomorphic to a quiver variety and that via this isomorphism the natural symplectic resolutions correspond to variations of git quotients of the quiver variety

projectivity and birational geometry of bridgeland moduli spaces we construct a family of nef divisor classes on every moduli space of stable complexes in the sense of bridgeland this divisor class varies naturally with the bridgeland stability condition for a generic stability condition on a k3 surface we prove that this class is ample thereby generalizing a result of minamide yanagida and yoshioka our result also gives a systematic explanation of the relation between wall crossing for bridgeland stability and the minimal model program for the moduli space we give three applications of our method for classical moduli spaces of sheaves on a k3 surface 1 we obtain a region in the ample cone in the moduli space of gieseker stable sheaves only depending on the lattice of the k3 2 we determine the nef cone of the hilbert scheme of n points on a k3 surface of picard rank one when n is large compared to the genus 3 we verify the hassett tschinkel huybrechts sawon conjecture on the existence of a birational lagrangian fibration for the hilbert scheme in a new family of cases

a tropical proof of the brill noether theorem we produce brill noether general graphs in every genus confirming a conjecture of baker and giving a new proof of the brill noether theorem due to griffiths and harris over any algebraically closed field

stability conditions on a non compact calabi yau threefold we study the space of stability conditions stab x on the non compact calabi yau threefold x which is the total space of the canonical bundle of mathbb p 2 we give a combinatorial description of an open subset of stab x and state a conjecture relating stab x to the frobenius manifold obtained from the quantum cohomology of mathbb p 2 we give some evidence from mirror symmetry for this conjecture

semiorthogonal decompositions and birational geometry of del pezzo surfaces over arbitrary fields we study the birational properties of geometrically rational surfaces from a derived categorical perspective in particular we give a criterion for the rationality of a del pezzo surface s over an arbitrary field namely that its derived category decomposes into zero dimensional components when s has degree at least 5 we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of s from brauer classes and chern classes associated to these vector bundles