chow groups of projective varieties of very small degree see 12 5 and the references given there these facts together with various conjectures on the cohomology and chow groups of algebraic varieties suggest that the chow groups of x might satisfy chl x q chl pk q q for l 1 compare with remark 5 6 and corollary 5 7 this is explicitly formulated by v srinivas and k paranjape in 16 conjecture 1 8 the chain of reasoning goes roughly as follows suppose x is smooth one expects a good filtration 0 f j 1 f j f 0 ch x x q whose graded pieces f f l 1 are controlled by h2j l x x see 10 according to grothendieck s generalized conjecture 8 the groups h x should be generated by the image under the gysin morphism of the homology of a codimension subset together with the classes coming from p applying this to the diagonal in x x should then force the triviality of the chow groups in the desired range for zero cycles the conjecture follows from roitman s theorem see 17 and 18

on the energy spectrum and parameter spaces of classicalcp n models it is known that a large class of smooth solutions ofcpn models can be constructed starting from holomorphic maps of an algebraic curve into complex projective spaces here we apply results from algebraic geometry to describe the energy spectrum and the parameter spaces for such models

tangent cones to discriminant loci for families of hypersurfaces a deformation of a variety with nonisolated hypersurface singularities such as a projective hypersurface or a theta divisor of an abelian variety determines a rational map of the singular locus to projective space and the resulting projective geometry of the singular locus describes how the singularities propagate in the deformation the basic principle is that the projective model of the singular locus is dual to the tangent cone to the discriminant of the deformation a detailed study of the method which emerged from interpreting andreotti mayer s work on theta divisors in terms of schlessinger s deformation theory of singularities is given along with examples applications and multiplicity formulas introduction this paper is about the geometry associated to deformations of nonisolated hypersurface singularities the basic principle is that a deformation of a hypersurface determines a projective model of the singular locus whose dual variety is the projectivized tangent cone to the discriminant this includes an infinitesimal form of bertini s theorem for a linear system of divisors and a converse to andreotti mayer s condition for double points of theta divisors now we describe the main results more precisely working in the category of complex analytic spaces let x be a space which can locally be written as a hypersurface of a nonsingular space and let x s be a deformation of x ir 1 0 then t1 the sheaf of first order deformations of x is a line bundle on sg x the singular locus of x and ir determines a kodaira spencer linear map to s k h0 t1 i e a linear system on the singular locus if x and s are nonsingular this linear system has no base locus and the corresponding morphism sg x pto s satisfies 0 p ir p tpx for p e sg x hence is called the gauss map of ir for a projective hypersurface of degree d x is the d fold veronese map restricted to the singular locus and for the theta divisor of an abelian variety 0 is the rational map assigning to a double point its quadric tangent cone assuming the restriction of ir to the critical locus c c x is proper consider the discriminant locus d ir c with its reduced analytic structure in addition to the above suppose finally that c is nonsingular of the expected codimension in x duality theorem there is a finite collection of 8ubvarietie8 zz of sg x which is intrinsic to the analytic space sg x and includes the irreducible components received by the editors march 31 1987 the contents of this paper were presented october 18 1986 by the first author in the special session on singularities and algebraic geometry at the charlotte north carolina meeting of the ams 1980 mathematics subject classification 1985 revision primary 14d15 14n05 secondary 32g11 53c57 d1988 american mathematical society 0002 9947 88 1 00 25 per page

the regularity of tor and graded betti numbers let s k x1 xn let a b be finitely generated graded s modules and let m x1 xn s we give bounds for the regularity of the local cohomology of tork a b in terms of the graded betti numbers of a and b under the assumption that dim tor1 a b 1 we apply the results to syzygies grobner bases products and powers of ideals and to the relationship of the rees and symmetric algebras for example we show that any homogeneous linearly presented m primary ideal has some power equal to a power of m and if the first n 1 2 steps of the resolution of i are linear then i 2 is a power of m

a treatment of breakdowns and near breakdowns in a reduction of a matrix to upper j hessenberg form and related topics abstract the reduction of a matrix to an upper j hessenberg form is a crucial step in the s r algorithm which is a q r like algorithm structure preserving for computing eigenvalues and vectors for a class of structured matrices this reduction may be handled via the algorithm jhess or via the recent algorithm jhmsh and its variants the main drawback of jhess or jhmsh is that it may suffer from a fatal breakdown causing a brutal stop of the computations and hence the s r algorithm does not run jhess may also encounter near breakdowns source of serious numerical instability in this paper we focus on these aspects we first bring light on the necessary and sufficient condition for the existence of the s r decomposition which is intimately linked to j hessenberg reduction then we will derive a strategy for curing fatal breakdowns and also for treating near breakdowns hence the j hessenberg form may be obtained numerical experiments are given demonstrating the efficiency of our strategies to cure and treat breakdowns or near breakdowns

nonemptiness of symmetric degeneracy loci let v be a rank n vector bundle on a d dimensional complex projective scheme x assume that v is equipped with a quadratic form with values in a line bundle l and that s 2 v l is ample suppose that the maximum rank of the quadratic form at any point of x is r 0 the main result of this paper is that if d n r the locus of points where the rank of the quadratic form is at most r 1 is nonempty we give some applications to subschemes of matrices and to degeneracy loci associated to embeddings in projective space the paper concludes with an appendix on gysin maps the main result of the appendix which may be of independent interest identifies a gysin map with the natural map from ordinary to relative cohomology

a modified schur method for robust pole assignment in state feedback control recently a schur method was proposed in chu 2007 to solve the robust pole assignment problem in state feedback control it takes the departure from normality of the closed loop system matrix a c as the measure of robustness and intends to minimize it via the real schur form of a c the schur method works well for real poles but when complex conjugate poles are involved it does not produce the real schur form of a c and can be problematic in this paper we propose a modified schur method which improves schur when nonreal poles are to be assigned besides producing the real schur form of a c our approach also leads to a relatively small departure from normality of a c numerical examples show that our modified method produces better or at least comparable results than both place and robpole algorithms with much less computational costs

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

a symplectic qr like algorithm for the solution of the real algebraic riccati equation a method is presented to solve the real algebraic riccati equation xnx xa a t x k 0 where k k t and n n t the solution for the corresponding eigenvalue problem mx lambda x where m is a hamiltonian matrix is computed by an algorithm similar to the qr algorithm special symplectic matrices are used for the transformation of m such that the hamiltonian form is preserved during the computations

a unified method for optimal arbitrary pole placement abstract we consider the classic problem of pole placement by state feedback we offer an eigenstructure assignment algorithm to obtain a novel parametric form for the pole placing feedback matrix that can deliver any set of desired closed loop eigenvalues with any desired multiplicities this parametric formula is then exploited to introduce an unconstrained nonlinear optimisation algorithm to obtain a feedback matrix that delivers the desired pole placement with optimal robustness and minimum gain lastly we compare the performance of our method against several others from the recent literature

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

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

optimal symplectic householder transformations for sr decomposition the sr factorization is a key step for some important structure preserving eigenproblems in this work we introduce symplectic householder transformations and show their main features furthermore we construct a new algorithm for computing the sr factorization based on these transformations the new algorithm is the analogous of the classical qr factorization via householder transformations unlike the latter it involves free parameters we show how to choose these parameters in an optimal way and derive an algorithm implementing this some illustrating numerical experiments comparisons and computational aspects are given

pole assignment via sylvester s equation it is shown that the pole assignment problem can be reduced to solving the linear matrix equations ax xa bg fx g successively for x and then f for almost any choice of g the result is a new pole assignment procedure and proof of the pole assignment theorem that should play an important role in both theoretical and practical applications

ulrich sheaves and higher rank brill noether theory abstract we show that the existence of an ulrich sheaf on a projective variety x p n is equivalent to the solution of a possibly higher rank brill noether problem for a curve on x that is rarely general in moduli in addition we exhibit a large family of curves for which this brill noether problem admits a solution and we show that existence of an ulrich sheaf for a finite morphism of smooth projective varieties of any dimension implies sharp numerical constraints involving the degree of the map and the ramification divisor

matrix factorizations for symplectic qr like methods abstract symplectic qr like methods use symplectic or unitary symplectic similarity transformations instead of the usual unitary similarity transformations in the qr process a fundamental problem for the development of such methods is the choice of a suitable type of decomposition a sr corresponding to the qr decomposition where s is symplectic or unitary symplectic decompositions of this type are studied with regard to their application in a qr like process

linked symplectic forms and limit linear series in rank 2 with special determinant abstract we generalize the prior linked symplectic grassmannian construction applying it to prove smoothing results for rank 2 limit linear series with fixed special determinant on chains of curves we apply this general machinery to prove new results on nonemptiness and dimension of rank 2 brill noether loci in a range of degrees

eigenstructure assignment in linear multivariable systems a parametric solution this paper generalizes a recently reported method 1 of closed loop eigenstructure assignment via state feedback in a linear multivariable system with n states and r control inputs by introducing a lemma on the differentiation of determinants the class of assignable eigenvectors and generalized eigenvectors associated with the assigned eigenvalues is explicitly described by a complete set of n r dimensional free parameter vectors this parametric characterization conveniently organizes the nonuniqueness of the solution of the eigenvalue assignment problem and thereby provides an efficient means of further modifying the system dynamic response a numerical example is worked out to demonstrate the feasibility of the method

linear series with cusps and n fold points a linear series v s on a curve x has an n fold point along a divisor d of degree n if dim vr h x d dim v 1 the linear series has a cuspofordere atapoint p if dim frch0 xs e l p dim z l linear series with cusps and n fold points are shown to exist if certain inequalities are satisfied the dimensions of the families of linear series with cusps are determined for general curves introduction throughout this paper we work over the complex numbers c let x be a smooth projective curve of genus g a gd on x is a linear series of degree d and dimension r on x i e a pair v s consisting of a line bundle s of degree d on x and an r l dimensional subspace vof h x s when p g r d g r 1 g r d o then x has gd s and the family of gd s on x forms a projective scheme gd x of dimension p g r d furthermore if x is general in moduli then gd x is smooth of dimension p g r d acgh definitic n if d is an effective divisor of degree n 2 on x we say that a gd v s has an n fold point along d if dim v n h x 2 d r we say that a gd has a cusp of order e at a point p if it has an e l fold point along the divisor e 1 p the aim of this paper is to prove the following i if g n p g r d n 1 r n o and p g r 1 d n o then there exists a gd on x with an n fold point along some divisor d of degree n ii if p g r d n 1 r o p g r 1 d n 0 and d is any divisor of degree n on x then there exists a gd on x with an n fold point along d iii if p g r d er 1 0 and p g r 1 d e 1 o then there exists a gd on x with a cusp of order e at some point p iv if x is general in moduli then the family of gd s on x with a cusp of order e has dimension p g r d er 1 if it is nonempty assertion iv has been proved independently by marc coppens c it is easy to see that the hypotheses p g r 1 d n 0 in assertions i and ii and p g r 1 d e 1 0 in assertion iii are necessary if x is general in moduli received by the editors january 20 1987 1980 mclthemcltlcs subject clclsslficcltion 1985 revision primary 14h10 1987 american mathematical society 0002 9947 87 1 00 25 per page

globally convergent algorithms for robust pole assignment by state feedback it is observed that an algorithm proposed in 1985 by kautsky nichols and van dooren knv amounts to maximizing at each iteration the determinant of the candidate closed loop eigenvector matrix x with respect to one of its columns with unit length constraint subject to the constraint that it remains an achievable closed loop eigenvector matrix this interpretation is used to prove convergence of the knv algorithm it is then shown that a more efficient algorithm is obtained if det x is concurrently maximized with respect to two columns of x and such a scheme is easily extended to the case where the eigenvalues to be assigned include complex conjugate pairs variations exploiting the availability of multiple processors are suggested convergence properties of the proposed algorithms are established their superiority is demonstrated numerically

vanishing theorems for ample vector bundles the main result of this article is a general vanishing theorem for the cohomology of tensorial representations of an ample vector bundle on a smooth complex projective variety in particular we extend classical theorems of griffiths and le potier to the whole dolbeault cohomology prove a variant of an uncorrect conjecture of sommese and answer a question of demailly as an application we prove conjectures of debarre and kim for branched coverings of grassmannians and extend a well known barth lefschetz type theorem for branched covers of projective spaces due to lazarsfeld we also obtain new restriction theorems for certain degeneracy loci

the surface c c on jacobi varieties and 2nd order theta functions preprint enviat per a la seva publicacio en una revista cientifica acta mathematica volume 157 1986 1 22 https projecteuclid org euclid acta 1485890436

approaches to robust pole assignment abstract robust pole assignment is a non linear optimization problem in many variables we describe numerical methods for determining robust or well conditioned so lutions to the problem of pole assignment by state feedback the solutions are chosen to minimize various objective functions based on the condition number of the eigenvector matrix careful choice of parametrization and objective function avoids singularities and artificial variable constraints explicit formulae for the gradient and hessian permit rigorous stopping criteria and rapid local convergence several computational examples are included

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

codimension and connectedness of degeneracy loci over local rings we deduce results on the dimension and connectedness of degeneracy loci of maps of finite modules f m rightarrow n over a local noetherian ring a mathfrak m we show for instance that the expected determinantal bounds on the dimension of the t th degeneracy locus of f hold if f in mathfrak m mathrm hom m n and that this degeneracy locus is connected in the expected dimension provided hat a is a domain

algebraic cycles on prym varieties this article proposes a generalization of tautological rings introduced by beauville and moonen for jacobians the main result is that under certain hypotheses the special subvarieties of prym varieties are algebraically equivalent and their classes belong to the tautological ring

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

arbitrary pole placement with the extended kautsky nichols van dooren parametric form abstractwe consider the classic problem of pole placement by state feedback the well known eigenstructure assignment algorithm of kautsky nichols and van dooren is extended to obtain a parametric formula for the pole placing feedback matrix that can deliver any desired closed loop eigenvalues with any desired multiplicities

on the existence of certain families of curves introduction 25 w 1 generalities on families of curves in ip 28 w the map 0 d 33 w 3 the number of moduli of a family 40 w 4 plane curves existence of families with the expected number of moduli 43 w 5 smoothing certain reducible curves in ip r r 3 47 w 6 curves in ip r 3 existence of families with the expected number of moduli 52 w curves in ip3 a more general existence theorem 55

a schur method for pole assignment in this paper a new numerical algorithm for pole assignment of linear time invariant systems is presented the proposed algorithm uses reliable numerical techniques based on the schur form of the state matrix and on the use of orthogonal transformations only the bad eigenvalues of the system are modified by a sequential pole shifting procedure and the resulted gains in the feedback matrix are minimized

robust exact pole placement via an lmi based algorithm this technical note deals with the robust exact pole placement problem pole placement algorithms that guarantee a small variation of the assigned poles against possible perturbations the solution to this problem is related to the solvability of a sylvester like equation thus the main issue is to compute a well conditioned solution to this equation also the best candidate solution must possess the minimal condition number to reduce sensitivity to perturbation this problem is cast as a global optimization under linear matrix inequality constraints for which a numerical convergent algorithm is provided and compared with the most attractive methods in the literature

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

divisors on general curves and cuspidal rational curves in t roduc t ion 371 0 pre l iminar ies 377 1 a pl i icker formula 378 2 d imens iona l t ransversa l i ty 380 3 d i m e n s i o n of families of r a t iona l curves with fixed degree and ass igned cusps in a project ive space 383 4 l inear series on curves with universa l s ingular i t ies 386 5 l inear series and project ive embedd ings of general curves 391 6 embedd ings of general cuspidal curves 400 7 degenera t ion of schuber t in tersect ions some special cases 403 8 degene ra t ion of schuber t in tersect ions ma in results 407 9 g c is reduced for general cuspidal r a t iona l c 414

symplectic householder transformations for a qr like decomposition a geometric and algebraic approaches the aim of this paper is to show how geometric and algebraic approaches lead us to a new symplectic elementary transformations the 2 d symplectic householder transformations their features are studied in details their interesting properties allow us to construct a new algorithm for computing a sr factorization this algorithm is based only on these 2 d symplectic householder transformations its new features are highlighted the study shows that in the symplectic case the new algorithm is the corresponding one to the classical qr factorization algorithm via the householder transformations some numerical experiments are given

the severi bound on sections of rank two semistable bundles on a riemann surface let c be a compact riemann surface of genus g 1 for a line bundle l on c of degree d with 0 d 2g 2 the classical theorem of clifford states that the dimension h 0 l of the space of global sections of l satisfies h 0 l d 2 1 it is also quite easy to list the cases which give equality apart from the trivial bundle and the canonical bundle these exist only on hyperelliptic curves this theorem can be extended to a semistable bundle e on c of rank n and degree d with 0 d n 2g 2 when it states that h 0 e d 2 n now let e be a semistable bundle of rank 2 in this case the authors obtain a more refined result depending on the segre invariant s e which is defined as the minimum value of dege l degl for l a line subbundle of e in particular the semistability condition becomes s e 0 the main theorem is that if dege d and s d 4g 4 s then h 0 e d s 2 where 1 or 2 note that d s is always even in a previous paper e arrondo and the second author conjectured this inequality with 1 with a small number of exceptions similar to those for rank 1 however this is not correct and 2 is sometimes necessary the authors call the bundles which attain this bound severi bundles the number is obtained as follows we define polynomials k r n n the krawtchouk polynomials by means of the generating function r k r n n z r 1 z n 1 z n n let r d s 2 1 then we can take 1 if k r g 2g s 0 and 2 otherwise the connection with krawtchouk polynomials is intriguing since these are used in coding theory and recent work of t johnsen has established links between coding theory and semistable bundles of rank 2 on riemann surfaces these links make essential use of the segre invariant s the key to the proof of the inequality is the following lemma let e be a bundle of rank 2 and degree d on a curve of genus g having at least r 1 g independent sections if k r g 2g 2r 2 d 0 then there is a nonzero section of e vanishing at r points this is proved using a computation of chern classes formal properties of k r n n and the porteous formula the authors give examples to show that for any allowable values of g s d the bound 1 can be attained in fact on a hyperelliptic curve a good deal is known about the vanishing of k r n n thus providing a restricted list of candidates for the invariants of severi bundles but these values do not all arise from actual bundles the severi bundles with d s 0 are all known and the authors compute those with d s 2 and d s 4 in particular the bundles obtained by i grzegorczyk ulam quart 3 1996 no 2 41 ff approx 3 pp electronic are severi bundles with g 5 s 4 d 8 the authors also obtain sharp bounds for the numbers of independent sections of non semistable bundles

the brill noether curve and prym tyurin varieties we prove that the jacobian of a general curve c of genus g 2a 1 with a ge 2 can be realized as a prym tyurin variety for the brill noether curve w 1 a 2 c as consequence of this result we are able to compute the class of the sum of secant divisors of the curve c embedded with a complete linear series g a 1 3a 2

limit linear series and ranks of multiplication maps we develop a new technique for studying ranks of multiplication maps for linear series via limit linear series and degenerations to chains of genus 1 curves we use this approach to prove a purely elementary criterion for proving cases of the maximal rank conjecture and then apply the criterion to several ranges of cases giving a new proof of the case of quadrics and also treating several families in the case of cubics our proofs do not require restrictions on direction of approach so we recover new information on the locus in the moduli space of curves on which the maximal rank condition fails

the connectedness of symmetric and skew symmetric degeneracy loci even ranks a degeneracy locus is the set of points where a vector bundle map has rank at most a given integer such a set is symmetric or skew symmetric according as whether the vector bundle map is symmetric or skew symmetric we prove a connectedness result first conjectured by fulton and lazarsfeld for skew symmetric degeneracy loci and for symmetric degeneracy loci of even ranks

an upper j hessenberg reduction of a matrix through symplectic householder transformations abstract in this paper we introduce a reduction of a matrix to a condensed form the upper j hessenberg form via elementary symplectic householder transformations which are rank one modification of the identity features of the reduction are highlighted and a general algorithm is derived then we study different possibilities to specify the general algorithm in order to built better versions we are led to two variants numerically more stables that we compare to jhess algorithm also some numerical experiments for comparing the different algorithms are given