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

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

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

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

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

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

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

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

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

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

on the numerical approximation of an evolution problem in nonlinear viscoelasticity abstract an error analysis is furnished for the spatial approximation via the finite element method of certain nonlinear time dependent problems emanating in the theory of viscoelastic solids numerical examples are given first for a model equation and then for numerical equilibrium solutions in cases of a nonconvex double well potential and a convex potential the latter derived from the nonconvex one via relaxation by an equal area construction

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

a shooting approach to chaos in the lorenz equations abstract we prove that under certain conditions the lorenz equations support a form of chaos the conditions have been verified for a particular set of parameter values showing that there is a natural 1 1 correspondence between a set of solutions and the set of all sequences on countably many symbols the computing required was modest