maximal cohen macaulay modules over the cone of an elliptic curve let r k y1 y2 y3 f f y13 y23 y33 where k is an algebraically closed field with chark 3 using atiyah bundle classification over elliptic curves we describe the matrix factorizations of the graded indecomposable reflexive r modules equivalently we describe explicitly the indecomposable bundles over the projective curve v f pk2 using the fact that over the completion r of r every reflexive module is gradable we obtain a description of the maximal cohen macaulay modules over r k y1 y2 y3 f

l szpiro s conjecture on gorenstein algebras in codimension 2 abstract a gorenstein a algebra r of codimension 2 is a perfect finite a algebra such that r ext a 2 r a holds as r modules a being a cohen macaulay local ring with dim a dim a r 2 the aim of this article is to prove a structure theorem for these algebras improving on an old theorem of m grassi koszul modules and gorenstein algebras j algebra 180 1996 918 953 special attention is paid to the question how the ring structure of r is encoded in its hilbert resolution it is shown that r is automatically a ring once one imposes a very weak depth condition on a determinantal ideal derived from a presentation matrix of r over a graded analogues of the aforementioned results are also included questions of applicability to the theory of surfaces of general type namely canonical surfaces in p 4 have served as a guideline in these commutative algebra investigations

existence of a positive solution to kirchhoff type problems without compactness conditions abstract the existence of a positive solution to a kirchhoff type problem on r n is proved by using variational methods and the new result does not require usual compactness conditions a cut off functional is utilized to obtain the bounded palais smale sequences

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

canonical form of symplectic matrix pencils abstract we obtain a real canonical form for real symplectic pencils it extends earlier results which were derived over the complex field so that the canonical form was complex and which were limited to the case where the elementary divisors of eigenvalues on the unit circle have an even degree

obstructions to determinantal representability there has recently been ample interest in the question of which sets can be represented by linear matrix inequalities lmis a necessary condition is that the set is rigidly convex and it has been conjectured that rigid convexity is also sufficient to this end helton and vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices we disprove this conjecture by relating the question of finding lmi representations to the problem of determining whether a polymatroid is representable over the complex numbers we find a real zero polynomial such that no power of it admits a determinantal representation the proof uses recent results of wagner and wei on matroids with the half plane property and the polymatroids associated to hyperbolic polynomials introduced by gurvits

hyperbolic polynomials and interior point methods for convex programming hyperbolic polynomials have their origins in partial differential equations we show in this paper that they have applications in interior point methods for convex programming each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone we give an explicit representation of this cone in terms of polynomial inequalities the function fx log px is a logarithmically homogeneous self concordant barrier function for the hyperbolicity cone with barrier parameter equal to the degree of p the function fx possesses striking additional properties that are useful in designing long step interior point methods for example we show that the long step primal potential reduction methods of nesterov and todd and the surface following methods of nesterov and nemirovskii extend to hyperbolic barrier functions we also show that there exists a hyperbolic barrier function on every homogeneous cone

rank 2 arithmetically cohen macaulay bundles on a nonsingular cubic surface abstract rank 2 indecomposable arithmetically cohen macaulay bundles e on a nonsingular cubic surface x in p 3 are classified by means of the possible forms taken by the minimal graded free resolution of e over p 3 the admissible values of the chern classes of e are listed and the vanishing locus of a general section of e is studied properties of e such as slope semi stability and simplicity are investigated the number of relevant families is computed together with their dimension

determinantal representations of singular hypersurfaces in pn abstract a global determinantal representation of projective hypersurface x p n is a matrix whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface we study the properties of such representations for singular possibly reducible or non reduced hypersurfaces in particular we obtain the decomposability criteria for determinantal representations of globally reducible hypersurfaces further we classify the determinantal representations in terms of the corresponding kernel sheaves on x finally we extend the results to the case of symmetric self adjoint representations with implications to hyperbolic polynomials and the generalized lax conjecture

indefinite hardy spaces on finite bordered riemann surfaces abstract it is well known that one may study hardy spaces with the domain a finite bordered riemann surface rather than the unit disk however for domains with more than one boundary component it is natural to consider besides the usual positive definite inner product on h 2 indefinite inner products obtained by picking up different signs or in the vector valued case different signature matrices when integrating over different components in this paper we obtain a necessary and sufficient condition for such an indefinite inner product to be nondegenerate and show that when this condition is satisfied we actually get a krein space furthermore each holomorphic mapping of the finite bordered riemann surface onto the unit disk which maps boundary to boundary determines an explicit isometric isomorphism between this space and a usual vector valued hardy space on the unit disk with an indefinite inner product defined by an appropriate hermitian matrix as is usual when studying hardy spaces on a multiply connected domain the elements of the space are sections of a vector bundle rather than functions the main point is to construct an appropriate extension of this bundle to the double of the finite bordered riemann surface and to use cauchy kernels for certain vector bundles on a compact riemann surface

asymptotic syzygies of algebraic varieties we study the asymptotic behavior of the syzygies of a smooth projective variety as the positivity of the embedding line bundle grows the main result asserts that the syzygy modules are non zero in almost all degrees allowed by castelnuovo mumford regularity we also give an effective statement for veronese varieties that we conjecture to be optimal

normal forms of symplectic pencils and the discrete time algebraic riccati equation abstract the solution of the discrete time algebraic riccati equation leads to symplectic pencils of matrices normal forms of such pencils under symplectic equivalence are determined special attention is given to characteristic roots of modulus 1 and their corresponding elementary divisors and inertial invariants

the cayley hamilton theorem and inverse problems for multiparameter systems abstract we review some of the current research in multiparameter spectral theory we prove a version of the cayley hamilton theorem for multiparameter systems and list a few inverse problems for such systems some consequences of results on determinantal representations proved by dixon dickson and vinnikov for the inverse problems are discussed

lagrangian subbundles and codimension 3 subcanonical subschemes we show that a gorenstein subcanonical codimension 3 subscheme z x p n 4 can be realized as the locus along which two lagrangian subbundles of a twisted orthogonal bundle meet degenerately and conversely we extend this result to singular z and all quasi projective ambient schemes x under the necessary hypothesis that z is strongly subcanonical in a sense defined below a central point is that a pair of lagrangian subbundles can be transformed locally into an alternating map in the local case our structure theorem reduces to that of d buchsbaum and d eisenbud 6 and says that z is pfaffian we also prove codimension 1 symmetric and skew symmetric analogues of our structure theorems 0 introduction smooth subvarieties of small codimension z x p have been extensively studied in recent years especially in relation to r hartshorne s conjecture that a smooth subvariety of sufficiently small codimension in p is a complete intersection although the conjecture remains open any smooth subvarietyz of small codimension in p is known by a theorem of w barth m larsen and s lefschetz to have the weaker property that it is subcanonical in the sense that its canonical class is a multiple of its hyperplane class more generally a subscheme z of a nonsingular noetherian scheme x is said to be subcanonical if z is gorenstein and its canonical bundle is the restriction of a bundle on x there is a natural generalization to an arbitrary possibly singular scheme x see below in this paper we give a structure theorem for subcanonical subschemes of codimension 3 in p and generalize it to subcanonical subschemes of codimension 3 in an arbitrary quasi projective scheme x satisfying a mild extra cohomological conduke mathematical journal vol 107 no 3 c 2001 received 4 july 1999 revision received 17 july 2000 2000 mathematics subject classification primary 14m07 13d02 secondary 14m12 14f05 authors work partially supported by the national science foundation

quotients of products of curves new surfaces with pg 0 and their fundamental groups we construct many new surfaces of general type with q pg 0 whose canonical model is the quotient of the product of two curves by the action of a finite group g constructing in this way many new interesting fundamental groups which distinguish connected components of the moduli space of surfaces of general type we indeed classify all such surfaces whose canonical model is singular the smooth case was classified in an earlier work as an important tool we prove a structure theorem giving a precise description of the fundamental group of quotients of products of curves by the action of a finite group g

complete description of determinantal representations of smooth irreducible curves abstract determinantal representations of algebraic curves are interesting in themselves and their classification is equivalent to the simultaneous classification of triples of matrices we present a complete description of determinantal representations of smooth irreducible curves over any algebraically closed field we used the notion of the class of divisors of the vector bundle corresponding to a determinantal representation we prove that two determinantal representations of a smooth curve f are equivalent if and only if the classes of divisors of the corresponding vector bundles coincide we give a precise characterization of those classes of divisors that arise from vector bundles corresponding to determinantal representations of f then we obtain a parametrization of determinantal representations of f up to equivalence by the points of the jacobian variety of f not on some exceptional subvariety in particular it follows that any smooth curve of order 3 or greater possesses an infinite number of nonequivalent determinantal representations we also specialize our results to symmetrical and self adjoint representations

existence of positive solutions for the nonlinear kirchhoff type equations in rn abstract in this paper we consider the following nonlinear problem of kirchhoff type a r n u 2 d x u v x u u p 1 u x r n u h 1 r n u 0 where n 3 a are positive constants 1 p max 3 n 2 n 2 and v x is a positive continuous potential satisfying v 0 v x lim inf x v x v using variational methods and a cutoff technique we prove the existence of positive solution to the above equation for all 0 small

livsic type determinantal representations and hyperbolicity abstract hyperbolic homogeneous polynomials with real coefficients i e hyperbolic real projective hypersurfaces and their determinantal representations play a key role in the emerging field of convex algebraic geometry in this paper we consider a natural notion of hyperbolicity for a real subvariety x p d of an arbitrary codimension l with respect to a real l 1 dimensional linear subspace v p d and study its basic properties we also consider a class of determinantal representations that we call livsic type and a nice subclass of these that we call very reasonable much like in the case of hypersurfaces l 1 the existence of a definite hermitian very reasonable livsic type determinantal representation implies hyperbolicity we show that every curve admits a very reasonable livsic type determinantal representation our basic tools are cauchy kernels for line bundles and the notion of the bezoutian for two meromorphic functions on a compact riemann surface that we introduce we then proceed to show that every real curve in p d hyperbolic with respect to some real d 2 dimensional linear subspace admits a definite hermitian or even definite real symmetric very reasonable livsic type determinantal representation

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

infinitely many radial solutions for kirchhoff type problems in rn abstract in this paper we concern with a class of kirchhoff type problems in r n by using the fountain theorem we obtain three existence results of infinitely many radial solutions for the problem

second root vectors for multiparameter eigenvalue problems of fredholm type a class of multiparameter eigenvalue problems involving generally non self adjoint and unbounded operators is studied a basis for the second root subspace at eigenvalues of fredholm type is computed in terms of the underlying multiparameter system a self adjoint version of this result is given under a weak definiteness condition and sturm liouville and finitedimensional examples are considered

real separated algebraic curves quadrature domains ahlfors type functions and operator theory abstract the aim of this paper is to inter relate several algebraic and analytic objects such as real type algebraic curves quadrature domains functions on them and rational matrix functions with special properties and some objects from operator theory such as vector toeplitz operators and subnormal operators our tools come from operator theory but some of our results have purely algebraic formulation we make use of xia s theory of subnormal operators and of the previous results by the author in this direction we also correct in section 5 some inaccuracies in the works of d v yakubovich subnormal operators of finite type i xia s model and real algebraic curves in c 2 rev mat iberoamericana 14 1998 95 115 d v yakubovich subnormal operators of finite type ii structure theorems rev mat iberoamericana 14 1998 623 681 by the author

on linearizations of the quadratic two parameter eigenvalue problems abstract we present several transformations that can be used to solve the quadratic two parameter eigenvalue problem qmep by formulating an associated linear multiparameter eigenvalue problem two of these transformations are generalizations of the well known linearization of the quadratic eigenvalue problem and linearize the qmep as a singular two parameter eigenvalue problem the third replaces all nonlinear terms by new variables and adds new equations for their relations the qmep is thus transformed into a nonsingular five parameter eigenvalue problem the advantage of these transformations is that they enable one to solve the qmep using existing numerical methods for multiparameter eigenvalue problems we also consider several special cases of the qmep where some matrix coefficients are zero

deformation of canonical morphisms and the moduli of surfaces of general type in this article we study the deformation of finite maps and show how to use this deformation theory to construct varieties with given invariants in a projective space among other things we prove a criterion that determines when a finite map can be deformed to a one to one map we use this criterion to construct new surfaces of general type with birational canonical map for different c 1 2 and the canonical map of the surfaces we construct is in fact a finite birational morphism our general results enable us to describe some new components of the moduli of surfaces of general type we also find infinitely many moduli spaces open image in new window having one component whose general point corresponds to a canonically embedded surface and another component whose general point corresponds to a surface whose canonical map is a degree 2 morphism

root vectors for geometrically simple two parameter eigenvalues a class of multiparameter eigenvalue problems involving generally non self adjoint and unbounded operators is studied bases for lower order root subspaces at geometrically simple eigenvalues of fredholm type of arbitrary finite index are computed in terms of the underlying multiparameter system

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

on the structure of commutative matrices ii a finite set a of n n nilpotent commutative matrices that have one dimensional joint kernel is considered the theorem due to suprunenko and tyshkevich that the algebra a generated by a and the identity matrix has dimension equal to n is proved a canonical basis for a is given and related structure constants are discussed

hermitian solutions of the equation x q nx 1n we consider the matrix equation x q nx 1n its hermitian solutions are parametrized in terms of the generalized lagrangian eigenspaces of a certain matrix pencil we show that the equation admits both a largest and a smallest solution the largest solution corresponds to the unique positive definite solution the smallest solution is the unique negative definite solution if and only if n is nonsingular if n is singular no negative definite solution exists an interesting relation between the given equation and a standard algebraic riccati equation of kalman filtering theory is also obtained finally we present an algorithm which converges to the positive definite solution for a wide range of initial conditions

koszul modules and gorenstein algebras abstract i first define koszul modules which are a generalization to arbitrary rank of complete intersections after a study of some of their properties it is proved that gorenstein algebras of codimension one or two over a local or graded cm ring are koszul modules thus generalizing a well known statement for rank one modules the general techniques used to describe koszul modules are then used to obtain a structure theorem for gorenstein algebras in codimension one and two over a local or graded cm ring

elementary transformations of pfaffian representations of plane curves abstract let c be a smooth curve in p 2 given by an equation f 0 of degree d in this paper we consider elementary transformations of linear pfaffian representations of c elementary transformations can be interpreted as actions on a rank 2 vector bundle on c with canonical determinant and no sections which corresponds to the cokernel of a pfaffian representation every two pfaffian representations of c can be bridged by a finite sequence of elementary transformations pfaffian representations and elementary transformations are constructed explicitly for a smooth quartic applications to aronhold bundles and theta characteristics are given

on the geometry of symplectic pencils arising from discrete time matrix equations abstract in this paper general symplectic matrix pencils are considered disregarding the particular matrix equations from which they arise a parameterization of the lagrangian deflating subspaces is given with the only assumption of regularity of the matrix pencil

periodic riccati difference equations approaching equilibria by implicit systems a parameterization of solutions to the periodic riccati difference equation rde is given by using behavioral tools the moving equilibria of the periodic rde are encompassed among the trajectories of the implicit ar model associated to the symplectic pencil of the corresponding hamiltonian equations this allows a parameterization of solutions of periodic rdes even in the case the dynamic matrix is singular

pencils of real symmetric matrices and real algebraic curves abstract let a rational real algebraic curve of degree n in the real projective plane x y z be given by its parametric equations let not containing the point o 0 0 1 have the property that i every line through the point o intersects in n real points counting multiplicities an explicit form is given for real symmetric matrices a and b of order n such that det xa yb zl 0 is an equation for for a diagonal b is a diagonal multiple of a loewner matrix the result partly proves a previous conjecture of the author about general real algebraic curves of degree n having the property i it is also shown that in this general case the conjecture is not true

on the characteristic polynomials of multiparameter pencils abstract in this paper we study the characteristic polynomial of multiparameter pencil z 1 a 1 z 2 a 2 z s a s the main theorem states that a unitary representation of a finitely generated group contains a one dimensional representation if and only if the characteristic polynomial of its generators contains a linear factor it follows that a two or three dimensional unitary representation of a finitely generated group is irreducible if and only if the characteristic polynomial of the pencil of its generators is irreducible the result is of kin to the dedekind and frobenius theorem on finite group determinant

geometry of the algebraic riccati equation part ii we prove that the set of real symmetric solutions of the algebraic riccati equation is isomorphic to the algebraic variety of invariant subspaces of a related n times n matrix by characterizing the structure of this variety we obtain a detailed description of the geometric properties of the solution set of the algebraic riccati equation

on the quadratic two parameter eigenvalue problem and its linearization we introduce the quadratic two parameter eigenvalue problem and linearize it as a singular two parameter eigenvalue problem this together with an example from model updating shows the need for numerical methods for singular two parameter eigenvalue problems and for a better understanding of such problems there are various numerical methods for two parameter eigenvalue problems but only few for nonsingular ones we present a method that can be applied to singular two parameter eigenvalue problems including the linearization of the quadratic two parameter eigenvalue problem it is based on the staircase algorithm for the extraction of the common regular part of two singular matrix pencils

resultants and chow forms via exterior syzygies let w be a vector space of dimension n 1 over a field k the chow divisor of a k dimensional variety x in p p w is the hypersurface in the grassmannian gk 1 of planes of codimension k 1 in p whose points over the algebraic closure of k are the planes that meet x the chow form of x is the defining equation of the chow divisor for example the resultant of k 1 forms of degree e in k 1 variables is the chow form of p embedded by the e th veronese mapping in p with n k e k 1 more generally the chow divisor of a k cycle i ni vi on projective space is defined to be i nidi where di is the chow divisor of vi the chow divisor of a sheaf f with k dimensional support is the chow divisor of the associated k cycle of f in this paper we will give a new expression for the chow divisor and apply it to give explicit formulas in many new cases starting with a sheaf f on p we use exterior algebra methods to define a canonical and effectively computable chow complex of f on each grassmannian of planes in p if f has k dimensional support we show that the chow form of f is the determinant of the chow complex of f on the grassmannian of planes of codimension k 1 the beilinson monad of f beilinson 1978 is the chow complex of f on the grassmannian of 0 planes that is on p itself in particular we are able to give explicit determinantal and pfaffian formulas for resultants in some cases where no polynomial formulas were known for example the horrocks mumford bundle gives rise to polynomial formulas for the resultant of five homogeneous forms of degrees 4 6 or 8 in five variables the easiest of our new formulas to write down is for the resultant of 3 quadratic forms in three variables the chow form of the veronese surface in p using the tangent bundle of p conclude that it can be written in bezout form described below as the pfaffian of the matrix 0 245 345 135 045 035 145 235

general matrix pencil techniques for the solution of algebraic riccati equations a unified approach we present a unified theory of matrix pencil techniques for solving both continuous and discrete time algebraic riccati equations ares under fairly general conditions on the coefficient matrices the theory applies to a large class of ares and riccati like equations arising from the singular h sup spl infin and h sup 2 control problems singular linear quadratic control the 4 block nehari problem or from singular j spectral factorizations the underlying concept is the so called proper deflating subspace of a possibly singular matrix pencil in terms of which necessary and sufficient conditions for the solvability of riccati equations are given it is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm

a compactification of a family of determinantal godeaux surfaces in this paper we present a geometric description of the compactification of the family of determinantal godeaux surfaces via the study of the bicanonical pencil and using classical prym theory in particular we reduce the problem of compactifying the space of bicanonical pencils of determinantal godeaux surfaces to the compactification of the family of twisted cubic curves in p3 with certain given tangent conditions

bloch s conjecture for catanese and barlow surfaces catanese surfaces are regular surfaces of general type with pg 0 they specialize to double covers of barlow surfaces we prove that the ch0 group of a catanese surface is equal to z which implies the same result for the barlow surfaces

an algebraic approach to multiparameter spectral theory root subspaces for multiprameter eigenvalue problems are described using coalgebraic techniques an algorithm is given to construct bases for the root subspaces

geometric aspects of multiparameter spectral theory the paper contains a geometric description of the dimension of the total root subspace of a regular multiparameter system in terms of the intersection multiplicities of its determinantal hypersurfaces the new definition of regularity used is proved to restrict to the familiar definition in the linear case a decomposability problem is also considered it is shown that the joint root subspace of a multiparameter system may be decomposable even when the root subspace of each member is indecomposable

elementary transformations of determinantal representations of algebraic curves abstract we study elementary transformations first introduced by livsic and kravitsky in an operator theoretic context of determinantal representations of algebraic curves we consider determinantal representations of a smooth irreducible curve f over an algebraically closed field when regarded as acting on the corresponding vector bundle the elementary transformations are a matrix generalization of scalar linear fractional transformations they add to the class of divisors of the vector bundle a single zero and a single pole we show that given a determinantal representation of f we can build all the nonequivalent determinantal representations of f by applying to the given representation finite sequences of at most g elementary transformations where g is the genus of f

de branges spaces on compact riemann surfaces and a beurling lax type theorem abstract using the notion of commutative operator vessels this work investigates de branges rovnyak spaces whose elements are sections of a line bundle of multiplicative half order differentials on a compact real riemann surface as a special case we obtain a beurling lax type theorem in the setting of the corresponding hardy space on a finite bordered riemann surface

determinantal representations of invariant hyperbolic plane curves abstract we study hyperbolic polynomials with nice symmetry and express them as the determinant of a hermitian matrix with special structure the goal of this paper is to answer a question posed by chien and nakazato in 2015 3 by properly modifying a determinantal representation construction of dixon 1902 5 we show for every hyperbolic polynomial of degree n invariant under the cyclic group of order n there exists a determinantal representation admitted via some cyclic weighted shift matrix moreover if the polynomial is invariant under the action of the dihedral group of order n the associated cyclic weighted shift matrix is unitarily equivalent to one with real entries

nikulin involutions on k3 surfaces we study the maps induced on cohomology by a nikulin i e a symplectic involution on a k3 surface we parametrize the 11 dimensional irreducible components of the moduli space of algebraic k3 surfaces with a nikulin involution and we give examples of the general k3 surface in various components we conclude with some remarks on morrison nikulin involutions these are nikulin involutions which interchange two copies of e 8 1 in the neron severi group

theta characteristics on algebraic curves the theory of theta characteristics is developed algebraically so that it may be applied to possibly singular and or reducible algebraic curves the configuration of theta characteristics on a curve is described in terms of its singularities with applications to the geometry of plane quartic curves and the problem of appolonius some results on gorenstein local rings are appended 0 introduction the theory of theta characteristics has been studied from different points of view and often inadvertently cf for example the 19th century literature on bitangents to smooth plane quartic curves e g 4 and 10 until recently however virtually all treatments of theta characteristics per se involved the transcendental theory of the theta function associated to a compact riemann surface then in 1971 mumford in 8 gave a completely algebraic treatment of the subject formulating the main results and proving them without reference to the transcendental theory this in turn suggested that theta characteristics could be studied in a more general context for example on possibly singular and or reducible algebraic curves in this paper we undertake such an analysis in 1 following mumford we establish some of the basic facts about theta characteristics algebraically so as to apply to more general curves in fact 1 10 i and 1 13 in their present generality are due to mumford while he assumes throughout 8 that his curve x is smooth his proof seems to apply as given the argument for 1 13 here is somewhat different in order to make the proof more elementary and selfcontained one bonus is the result 1 11 on the subvariety of the moduli space 9thg consisting of curves with large semicanonical linear series this fact seemingly can be proved only via the algebraic set up in 2 we investigate the number and configuration of odd and even theta characteristics on a singular curve the results are expressed in the two main theorems 2 12 and 2 22 giving a more or less complete answer to these questions in terms of the singularities of the curve interestingly a recurrent motif in these arguments is one introduced by mumford in 8 the behavior of the maximal isotropic subspaces for a nondegenerate quadratic form on an even dimensional vector space here the mainspring of theorems 1 13 2 12 and 2 22 is assertion 1 4 below that the sum of the dimensions of the pairwise intersections of any three such subspaces is congruent to n mod 2 received by the editors july 11 1980 and in revised form october 22 1981 1980 mathematics subject classification primary 14h99 14h30 secondary 14h20 14h40 14k25 research partially supported by nsf grant mcs78 04008 aol dl 982 american mathematical society 0002 9947 8 1 0000 0371 10 25

a simply connected numerical campedelli surface with an involution we construct a simply connected minimal complex surface of general type with p g 0 and k 2 2 which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with p g 0 and k 2 1 in order to construct the example we combine a double covering and mathbb q gorenstein deformation especially we develop a method for proving unobstructedness for deformations of a singular surface by generalizing a result of burns and wahl which characterizes the space of first order deformations of a singular surface with only rational double points we describe the stable model in the sense of kollar and shepherd barron of the singular surfaces used for constructing the example we count the dimension of the invariant part of the deformation space of the example under the induced mathbb z 2 mathbb z action

symmetric determinantal representation of polynomials abstract we give an elementary proof only using linear algebra of a result due to helton mccullough and vinnikov which says that any polynomial over the reals can be written as the determinant of a symmetric affine linear pencil we give explicit determinantal representation formulas and extend our results to polynomials with coefficients in a ring of characteristic different from 2