minimal rational curves on moduli spaces of stable bundles let c be a smooth projective curve of genus g 2 and l be a line bundle on c of degree d let m uc r l be the moduli space of stable vector bundles on c of rank r and with the fixed determinant l assume that r d 1 then m is a smooth projective fano variety with picard number 1 for any projective curve on m we can define its degree with respect to the ample anti canonical line bundle km the first result of this paper determines all rational curves of minimal degree passing through a generic point of m which answers a question of jun muk hwang see question 1 in hw

self avoiding polygons on the square l and manhattan lattices transfer matrix techniques are used to extend the self avoiding polygon generating function on the square lattice to terms in x46 corresponding to 46 step polygons these techniques are then extended to apply to directed square lattices such as the l and manhattan lattice and the self avoiding polygon generating function to x48 is found for these lattices series analysis confirms that the specific heat exponent alpha 1 2 for the self avoiding walk problem and gives the following estimates for the connective constants mu sq 2 638155 or 0 000025 mu l 1 5657 or 0 0019 and mu man 1 7328 or 0 0005 some evidence for a correction to scaling exponent delta approximately 0 84 is found from square lattice series

classifying thick subcategories of the stable category of cohen macaulay modules abstract various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics in this paper as a higher dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p group due to benson carlson and rickard we consider classifying thick subcategories of the stable category of cohen macaulay modules over a gorenstein local ring the main result of this paper yields a complete classification of the thick subcategories of the stable category of cohen macaulay modules over a local hypersurface in terms of specialization closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus

algebro geometric analogues of donaldson s polynomials donaldson 5 has introduced polynomials on the second homology of any simply connected closed four manifold m with b odd at least 3 which up to are invariant under orientation preserving diffeomorphisms for every c ibm there is a polynomial 7c m syma h2 s where d c 4c z2 b 1 to define it one considers j c the set which is in a natural way a manifold of gauge equivalence classes of connections on the su 2 bundle on m with c2 c antiself dual with respect to a generic metric on m there is a natural map h2 m h2 j c namely slant product with 88 where p is the universal so 3 principal bundle on m x j c one way of defining 7c m 10 is to construct a compactification of r xc carrying a fundamental class xc one proves that extends to a map h2 m h2 xc then one defines

singular equivalences of functor categories via auslander buchweitz approximations abstract the aim of this paper is to construct singular equivalences between functor categories as a special case we show that there exists a singular equivalence arising from a cotilting module t namely the singularity category of t t and that of mod t are triangle equivalent in particular the canonical module over a commutative noetherian ring r induces a singular equivalence between cm r and mod r which generalizes matsui takahashi s theorem our result is based on a sufficient condition for an additive category a and its subcategory x so that the canonical inclusion x a induces a singular equivalence d sg a d sg x which is a functor category version of xiao wu chen s theorem

models for singularity categories abstract in this article we construct various models for singularity categories of modules over differential graded rings the main technique is the connection between abelian model structures cotorsion pairs and deconstructible classes and our constructions are based on more general results about localization and transfer of abelian model structures we indicate how recollements of triangulated categories can be obtained model categorically discussing in detail krause s recollement k ac inj r k inj r d r in the special case of curved mixed z graded complexes we show that one of our singular models is quillen equivalent to positselski s contraderived model for the homotopy category of matrix factorizations

quiver generalization of a conjecture of king tollu and toumazet abstract stretching the parameters of a littlewood richardson coefficient of value 2 by a factor of n results in a coefficient of value n 1 12 9 19 we give a geometric proof of a generalization for representations of quivers

the stringy e function of the moduli space of rank 2 bundles over a riemann surface of genus 3 we compute the stringy e function or the motivic integral of the moduli space of rank 2 bundles over a riemann surface of genus 3 in doing so we answer a question of batyrev about the stringy e functions of the git quotients of linear representations

counting elements and geodesics in thompson s group f we present two quite different algorithms to compute the number of elements in the sphere of radius n of thompson s group f with standard generating set the first of these requires exponential time and polynomial space but additionally computes the number of geodesics and is generalisable to many other groups the second algorithm requires polynomial time and space and allows us to compute the size of the spheres of radius n with n 1500 using the resulting series data we find that the growth rate of the group is bounded above by 2 62167 this is very close to guba s lower bound of 3 5 2 indeed numerical analysis of the series data strongly suggests that the growth rate of the group is exactly 3 5 2

complete growth functions of hyperbolic groups we study a generalization of the growth functions of finitely generated groups namely the growth functions g g gz g with coefficients in the group ring g rationality and methods of computation of such functions are discussed in particular for hyperbolic groups the complete growth functions of surface groups are explicitly computed the operator and geodesic growth functions are also studied

donaldson invariants of product ruled surfaces and two dimensional gauge theories using the u plane integral of moore and witten we derive a simple expression for the donaldson invariants of product ruled surfaces g s 2 where g is a riemann surface of genus g this expression generalizes a theorem of morgan and szabo for g 1 to any genus g we give two applications of our results 1 we derive thaddeus formulae for the intersection pairings on the moduli space of rank two stable bundles over a riemann surface 2 we derive the eigenvalue spectrum of the fukaya floer cohomology of g s 1

on singular equivalences of morita type abstract stable equivalences of morita type preserve many interesting properties and are proved to be the appropriate concept for studying equivalences between stable categories recently the singularity category attained much attraction and xiao wu chen and long gang sun gave an appropriate definition of singular equivalence of morita type we shall show that under some conditions singular equivalences of morita type have some biadjoint functor properties and preserve positive degree hochschild homology

singular derived categories of factorial terminalizations and maximal modification algebras abstract let x be a gorenstein normal 3 fold satisfying elf with local rings which are at worst isolated hypersurface e g terminal singularities by using the singular derived category d sg x and its idempotent completion d sg x we give necessary and sufficient categorical conditions for x to be q factorial and complete locally q factorial respectively we then relate this information to maximal modification algebras mmas introduced in 20 by showing that if an algebra is derived equivalent to x as above then x is q factorial if and only if is an mma thus all rings derived equivalent to q factorial terminalizations in dimension three are mmas as an application we extend some of the algebraic results in 6 and 14 using geometric arguments

combinatorial properties of thompson s group f we study some combinatorial consequences of blake fordham s theorems on the word metric of thompson s group f in the standard two generator presentation we explore connections between the tree pair diagram representing an element w of f its normal form in the infinite presentation its word length and minimal length representatives of it we estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates in addition we explore some properties of the cayley graph of f with respect to the two generator finite presentation namely we exhibit the form of dead end elements in this cayley graph and show that it has no deep pockets finally we discuss a simple method for constructing minimal length representatives for strictly positive or negative words

donaldson invariants for non simply connected manifolds we study coulomb branch u plane integrals for supersymmetric su 2 so 3 yang mills theory on 4 manifolds x of b 1 x 0 b 2 x 1 using wall crossing arguments we derive expressions for the donaldson invariants for manifolds with b 1 x 0 b 2 x 0 explicit expressions for where f g is a riemann surface of genus g are obtained using kronecker s double series identity the result might be useful in future studies of quantum cohomology

nilpotency in instanton homology and the framed instanton homology of a surface times a circle abstract in the description of the instanton floer homology of a surface times a circle due to munoz we compute the nilpotency degree of the endomorphism u 2 64 we then compute the framed instanton homology of a surface times a circle with non trivial bundle which is closely related to the kernel of u 2 64 we discuss these results in the context of the moduli space of stable rank two holomorphic bundles with fixed odd determinant over a riemann surface

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

automatic structures rational growth and geometrically finite hyperbolic groups we show that the set s mathfrak a g of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic groupg is dense in the product of the sets s mathfrak a p over all maximal parabolic subgroupsp the set bs mathfrak a g of equivalence classes of biautomatic structures ong is isomorphic to the product of the sets bs mathfrak a p over the cusps conjugacy classes of maximal parabolic subgroups ofg each maximal parabolicp is a virtually abelian group so s mathfrak a p and bs mathfrak a p were computed in ns1

self avoiding polygons on the square lattice we have developed an improved algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 90 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 529 27 1 biased and the critical exponent 0 500 0005 10 unbiased the critical point is indistinguishable from a root of the polynomial 581x4 7x2 13 0 an asymptotic expansion for the coefficients is given for all n there is strong evidence for the absence of any non analytic correction to scaling exponent

partial desingularisations of quotients of nonsingular varieties and their betti numbers when a reductive group g acts linearly on a nonsingular complex projective variety x one can define a projective quotient variety x g using mumford s geometric invariant theory if the condition that every semistable point be properly stable is satisfied this quotient is the ordinary topological quotient of an open subset xss of the variety by the group in k a formula is obtained for the rational cohomology of x g under this condition the formula involves the rational cohomology of x and various linear sections of x together with the rational cohomology of the classifying spaces of g and certain reductive subgroups of g in many interesting examples the condition required in k is not satisfied thus the question arises as to what information we can obtain in general the quotient x g can now have serious singularities in contrast to the good case considered in k where the only singularities are those caused by finite isotropy groups it will be shown in this paper that there is a systematic way of blowing up x along a sequence of nonsingular subvarieties to obtain a variety x with a linear action of g such that every semistable point of x is stable the only assumption that we have to make is that there exists at least one stable point of x then x g is almost a resolution of singularities of x1 g in the sense that the most serious singularities have been resolved moreover there is a formula for the rational cohomology of x g again involving the rational cohomology of x and certain linear sections of x together with the rational cohomology of the classifying spaces of g and some reductive subgroups of g see theorem 8 14 for convenience we shall assume throughout that g is connected however the construction of x works in general and it is straightforward to modify the cohomological formulas to apply to the general case the construction of x g can also be modified to apply in some cases when x has no stable points the layout of the paper is as follows section 1 is a review of the basic facts of geometric invariant theory which will be needed and section 2 describes the relationship of geometric invariant theory with symplectic geometry and the moment map in section 3 semistability and stability in a blow up of x along a nonsingular c invariant subvariety are related to semistability and stability in x

walks confined in a quadrant are not always d finite we consider planar lattice walks that start from a prescribed position take their steps in a given finite subset of z2 and always stay in the quadrant x 0 y 0 we first give a criterion which guarantees that the length generating function of these walks is d finite that is satisfies a linear differential equation with polynomial coefficients this criterion applies among others to the ordinary square lattice walks then we prove that walks that start from 1 1 take their steps in 2 1 1 2 and stay in the first quadrant have a non d finite generating function our proof relies on a functional equation satisfied by this generating function and on elementary complex analysis

quantum cohomology of the moduli space of stable bundles over a riemann surface we determine the quantum cohomology of the moduli space m of odd degree rank two stable vector bundles over a riemann surface of genus g 1 this work together with 10 complete the proof of the existence of an isomorphism qh m hf s1

contravariantly finite resolving subcategories over commutative rings contravariantly finite resolving subcategories of the category of finitely generated modules have been playing an important role in the representation theory of algebras in this paper we study contravariantly finite resolving subcategories over commutative rings the main purpose of this paper is to classify contravariantly finite resolving subcategories over a henselian gorenstein local ring in fact there exist only three such subcategories our method to obtain this classification also recovers as a by product the theorem of christensen piepmeyer striuli and takahashi concerning the relationship between the contravariant finiteness of the full subcategory of totally reflexive modules and the gorenstein property of the base ring

on the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface let j be an abelian surface with a generic ample line bundle open image in new window for n 1 the moduli space mj 2 0 2n of open image in new window 1 semistable sheaves f of rank 2 with chern classes c1 f 0 c2 f 2n is a singular projective variety endowed with a holomorphic symplectic structure on the smooth locus in this paper we show that there does not exist a crepant resolution of mj 2 0 2n for n 2 this certainly implies that there is no symplectic desingularization of mj 2 0 2n for n 2

moduli of sheaves fourier mukai transform and partial desingularization we study birational maps among 1 the moduli space of semistable sheaves of hilbert polynomial 4m 2 on a smooth quadric surface 2 the moduli space of semistable sheaves of hilbert polynomial m 2 3m 2 on mathbb p 3 3 kontsevich s moduli space of genus zero stable maps of degree 2 to the grassmannian gr 2 4 a regular birational morphism from 1 to 2 is described in terms of fourier mukai transforms the map from 3 to 2 is kirwan s partial desingularization we also investigate several geometric properties of 1 by using the variation of moduli spaces of stable pairs

singularity categories and singular equivalences for resolving subcategories let mathcal x be a resolving subcategory of an abelian category in this paper we investigate the singularity category mathsf d sg underline mathcal x mathsf d b mathsf mod underline mathcal x mathsf k b mathsf proj mathsf mod underline mathcal x of the stable category underline mathcal x of mathcal x we consider when the singularity category is triangle equivalent to the stable category of gorenstein projective objects and when the stable categories of two resolving subcategories have triangle equivalent singularity categories applying this to the module category of a gorenstein ring we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type mathsf a 1 we also generalize several results of yoshino on totally reflexive modules

the strange duality conjecture for generic curves let sux r be the moduli space of semi stable vector bundles of rank r with trivial determinant over a connected smooth projective algebraic curve x of genus g 1 over c recall that a vector bundle e on x is called semi stable if for any subbundle v deg v rk v deg e rk e points of sux r correspond to isomorphism classes of semi stable rank r vector bundles with trivial determinant up to an equivalence relation for any line bundle l of degree g 1 onx define l e sux r h e l 1 this turns out be a non zero cartier divisor whose associated line bundle l o l does not depend upon l it is known that l generates the picard group of sux r for this and the precise definition of l in terms of determinant of cohomology see dn let u x k be the moduli space of semi stable rank k and degree k g 1 bundles on x recall that on u x k there is a canonical non zero theta cartier divisor k whose underlying set is f u x k h x f 0 put m o k consider the natural map k r sux r u x k u x kr given by tensor product from the theorem of the square it follows that k rm is isomorphic to l m the canonical element kr h0 u x kr m and the kunneth theorem give a map well defined up to scalars

generalized atmospheric rosenbluth methods garm we show that the classical rosenbluth method for sampling self avoiding walks hammersley and morton 1954 j r stat soc b 16 23 rosenbluth and rosenbluth 1955 j chem phys 23 356 can be extended to a general algorithm for sampling many families of objects including self avoiding polygons the implementation relies on an elementary move which is a generalization of kinetic growth rather than only appending edges to the endpoint edges may be inserted at any vertex provided the resulting objects still lie within the same family this gives for the first time a kinetic growth algorithm for sampling self avoiding polygons we implement this method using pruning and enrichment grassberger 1997 phys rev e 56 3682 to sample self avoiding walks and polygons the algorithm can be further extended by mixing it with length preserving moves such as pivots and crank shaft moves

hilbert scheme of rational cubic curves via stable maps the space of smooth rational cubic curves in projective space bbb p r r ge 3 is a smooth quasi projective variety which gives us an open subset of the corresponding hilbert scheme the moduli space of stable maps or the moduli space of stable sheaves by taking its closure we obtain three compactifications bf h bf m and bf s respectively in this paper we compare these compactifications first we prove that bf h is the blow up of bf s along a smooth subvariety parameterizing planar stable sheaves next we prove that bf s is obtained from bf m by three blow ups followed by three blow downs and the centers are described explicitly using this we calculate the cohomology of bf s

descent of coherent sheaves and complexes to geometric invariant theory quotients abstract fix a scheme x over a field of characteristic zero that is equipped with an action of a reductive algebraic group g we give necessary and sufficient conditions for a g equivariant coherent sheaf on x or a bounded above complex of g equivariant coherent sheaves on x to descend to a good quotient x g this gives a description of the coherent derived category of x g as an admissible subcategory of the equivariant derived category of x

desingularizations of the moduli space of rank 2 bundles over a curve let x be a smooth projective curve of genus g 3 and m0 be the moduli space of rank 2 semistable bundles over x with trivial determinant there are three desingularizations of this singular moduli space constructed by narasimhan ramanan nr78 seshadri ses77 and kirwan kir86b respectively the relationship between them has not been understood so far the purpose of this paper is to show that there is a morphism from kirwan s desingularization to seshadri s which turns out to be the composition of two blow downs in doing so we will show that the singularities of m0 are terminal and the plurigenera are all trivial as an application we compute the betti numbers of the cohomology of seshadri s desingularization in all degrees this generalizes the result of bs90 which computes the betti numbers in low degrees another application is the computation of the stringy e function see bat98 for definition of m0 for any genus g 3 which generalizes the result of kie03

syzygies of cohen macaulay modules and grothendieck groups abstract we study the converse of a theorem of butler and auslander reiten we show that a cohen macaulay local ring with an isolated singularity has only finitely many isomorphism classes of indecomposable summands of syzygies of cohen macaulay modules if the auslander reiten sequences generate the relation of the grothendieck group of finitely generated modules this extends a recent result of hiramatsu which gives an affirmative answer in the gorenstein case to a conjecture of auslander

classifying resolving subcategories over a cohen macaulay local ring let r be a cohen macaulay local ring denote by modr the category of finitely generated r modules in this paper we consider the classification problem of resolving subcategories of modr in terms of specialization closed subsets of specr we give a classification of the resolving subcategories closed under tensor products and transposes under restrictive hypotheses we also give better classification results

support varieties and cohomology over complete intersections quillen s geometric approach to the cohomology of finite groups 30 has revolutionized modular representation theory the ideology and techniques involved have been generalized and extended to representations of various hopf algebras culminating in the recent work of friedlander and suslin 17 on finite group schemes in this paper we develop geometric methods for the study of finite modules over a local complete intersection r if r is such a ring and m is its maximal ideal then the m adic completion r has the form q f where f is a regular sequence and q is a regular local ring that can be taken to be a ring of formal power series over a field or a discrete valuation ring the least number of equations needed to cut out r from a regular local ring equals the codimension of r where codim r r m dim r and r m denotes the minimal number of generators of an r module m in 5 a cone that is a homogeneous algebraic set vr m is attached to each finite r module m and used to study its minimal free resolution here we prove

the relative singularity category of a non commutative resolution of singularities abstract in this article we study a triangulated category associated with a non commutative resolution of singularities in particular we give a complete description of this category in the case of a curve with nodal singularities classifying its indecomposable objects and computing its auslander reiten quiver and k group

higher dimensional auslander reiten theory on maximal orthogonal subcategories abstract we introduce the concept of maximal n 1 orthogonal subcategories over artin algebras and orders and develop n 1 dimensional auslander reiten theory on them we give the n auslander reiten translation and the n auslander reiten duality then show the existence of n almost split sequences and n fundamental sequences we give some examples

moduli of vector bundles frobenius splitting and invariant theory let x be an irreducible projective curve of genus g over an algebraically closed field of positive characteristic 2 3 in part i we prove adapting the degeneration arguments of n tr that moduli spaces of rank two parabolic bundles on x are frobenius split m r for generic smooth x a similar result holds for x nodal a consequence is the verlinde formula in positive characteristic in part ii we give a direct proof of the fact that the local rings of the moduli spaces are f split and further that they are cohen macaulay this involves showing that the ring of invariants of g copies of 2 x 2 matrices under the adjoint action of sl 2 is f split and cohen macaulay

on algebras of finite cohen macaulay type abstract we study artin algebras and commutative noetherian complete local rings r in connection with the following decomposition property of gorenstein projective modules any gorenstein projective module is a direct sum of finitely generated modules we show that the class of algebras enjoying coincides with the class of virtually gorenstein algebras of finite cohen macaulay type introduced in beligiannis and reiten 2007 27 beligiannis 2005 24 thus we solve the problem stated in chen 2008 33 this is proved by characterizing when a resolving subcategory is of finite representation type in terms of decomposition properties of its closure under filtered colimits thus generalizing a classical result of auslander 1976 9 and ringel and tachikawa 1974 63 in the commutative case if r admits a non free finitely generated gorenstein projective module then we show that r is of finite cohen macaulay type iff r is gorenstein and satisfies we also generalize a result of yoshino 2005 68 by characterizing when finitely generated modules without extensions with the ring are gorenstein projective finally we study the stable relative auslander algebra of a virtually gorenstein algebra of finite cohen macaulay type and under the presence of a cluster tilting object we give descriptions of the stable category of gorenstein projective modules in terms of the cluster category associated to the quiver of the stable relative auslander algebra in this setting we show that the cluster category is invariant under derived equivalences

conformal field theory and the cohomology of the moduli space of stable bundles let g be a compact riemann surface of genus g 2 and let be a line bundle over g of degree 1 then the moduli space of rank 2 stable bundles v over g such that v was shown by seshadri 19 to be a nonsingular projective variety ng its rational cohomology ring h ng is exceedingly rich and despite years of study has never quite been computed in full the object of this paper is to give an essentially complete characterization of this ring or at least to reduce the problem to a matter of linear algebra in particular we find a proof of newstead s conjecture that pg1 ng 0 we also obtain a formula for the volume of ng which can be regarded as a twisted version of the formula for the degree 0 moduli space recently announced by witten the approach we shall take is not from algebraic geometry but from mathematical physics it relies on the su 2 wess zumino witten model which is a functor zk associating a finite dimensional vector space to each riemann surface with marked points the relationship with the moduli space is that when the level k of the functor is even the vector space associated to g with no marked points can be identified with h ng l where l is a fixed line bundle over ng now the work of verlinde provides us with a means of calculating the dimension of any vector space arising from our functor and in particular dimh ng l which we shall denote d g k on the other hand newstead 14 found explicit generators for h ng and we can also express d g k in terms of them using a riemann roch theorem equating the two formulas enables us to evaluate any monomial in the generators on the fundamental class of ng and by poincare duality this is sufficient at least in principle to determine the ring structure of h ng in the discussion above one important point has been skated over the su 2 wzw model is of course associated to bundles of degree 0 not degree 1 so in order to make use of verlinde s work it is necessary to formulate a twisted version of the field theory this is carried out in 2 but the crucial properties of the twisted theory analogous to those which make the ordinary theory a modular functor are not proved in this paper rather we will confine ourselves to exploring the consequences of these claims and hope to return to justify them in a later paper an outline of the remaining sections goes as follows in 3 we review those parts of verlinde s work we shall need show how they must be modified in the twisted case and work out some explicit formulas for d g k in 4 we study the cohomology of ng which we regard throughout as a space of representations via the theorem of narasimhan and seshadri we define newstead s generators and i of h ng

two theorems about maximal cohen macaulay modules this paper contains two theorems concerning the theory of maximal cohen macaulay modules the first theorem proves that certain ext groups between maximal cohen macaulay modules m and n must have finite length provided only finitely many isomorphism classes of maximal cohen macaulay modules exist having ranks up to the sum of the ranks of m and n this has several corollaries in particular it proves that a cohen macaulay local ring of finite cohen macaulay type has an isolated singularity a well known theorem of auslander gives the same conclusion but requires that the ring be henselian other corollaries of our result include statements concerning when a ring is gorenstein or a complete intersection on the punctured spectrum and the recent theorem of leuschke and wiegand that the completion of an excellent cohen macaulay local ring of finite cohen macaulay type is again of finite cohen macaulay type the second theorem proves that a complete local gorenstein domain of positive characteristic p and dimension d is f rational if and only if the number of copies of r splitting out of r 1 p e divided by p de has a positive limit this result generalizes work of smith and van den bergh we call this limit the f signature of the ring and give some of its properties

hecke correspondence stable maps and the kirwan desingularization we prove that the moduli space m0 0 n 2 of stable maps of degree 2 to the moduli space n of rank 2 stable bundles of flxed odd determinant ox ix over a smooth projective curve x of genus g 3 has two irreducible components which intersect transversely one of them is kirwan s partial desingularization e mx of the moduli space mx of rank 2 semistable bundles with determinant isomorphic to ox yix for some y 2 x the other component is the partial desingularization of the git quotient phom sl 2 w pgl 2 for a vector bundle w r 1 l i2 ix of rank g over the jacobian of x we also show that the hilbert scheme h the chow scheme c of conics in n and m0 0 n 2 are related by explicit contractions