on schwarz type smoothers for saddle point problems in this paper we consider additive schwarz type iteration methods for saddle point problems as smoothers in a multigrid method each iteration step of the additive schwarz method requires the solutions of several small local saddle point problems this method can be viewed as an additive version of a multiplicative vanka type iteration well known as a smoother for multigrid methods in computational fluid dynamics it is shown that under suitable conditions the iteration can be interpreted as a symmetric inexact uzawa method in the case of symmetric saddle point problems the smoothing property an important part in a multigrid convergence proof is analyzed for symmetric inexact uzawa methods including the special case of the additive schwarz type iterations as an example the theory is applied to the crouzeix raviart mixed finite element for the stokes equations and some numerical experiments are presented

sur les distributions covariantes dans les algebres de lie reductives abstract we study the possibility of factoring a covariant distribution on reductive lie algebras as finite sum of products of an invariant distribution by a covariant polynomial

multigrid methods for saddle point problems stokes and lame systems we develop new multigrid methods for a class of saddle point problems that include the stokes system in fluid flow and the lame system in linear elasticity as special cases the new smoothers in the multigrid methods involve optimal preconditioners for the discrete laplace operator we prove uniform convergence of the w w cycle algorithm in the energy norm and present numerical results for w w cycle and v v cycle algorithms

a bound on the order of jumping lines let e be a stable rank 2 vector bundle on n 3 assume e is normalized i e c e 0 or 1 by grothedieck s theorem the restriction of e on a line l is isomorphic to l bl o gl bl if cl o gl bl 1 t ol b l if c1 1 for some nonnegative integer b l this b l is an upper semicontinuous function of l which in char 0 according to the grauert miilich theorem g m assumes the value 0 on an open set if be 0 then l is said to be a jumping line of order b l for c 1 0 barth b has shown that the set of jumping lines can be given a scheme structure making it a divisor s e on the grassmannian g g 1 3 which via g ip 5 has degree 2c2 e and for any line l in ip 3 b l pl se where s is the multiplicity of s t at l since any point on a projective variety of degree k has multiplicity k it follows that b l 2c 2 however this bound is not sharp in this paper we will give a sharp bound for b l valid for arbitrary c r namely if c 0

a nonconforming mixed multigrid method for the pure displacement problem in planar linear elasticity an optimal order multigrid method is developed for the pure displacement problem in two dimensional linear elasticity it is based on a nonconforming mixed formulation where the displacement is approximated by weakly continuous piecewise linear vector functions and the pressure is approximated by piecewise constants the full multigrid convergence is proved the performance of the multigrid method does not deteriorate as the material becomes nearly incompressible

on the convergence of multi grid methods with transforming smoothers in the present paper we give a convergence theory for multi grid methods with transforming smoothers as introduced in 31 applied to a general system of partial differential equations the theory follows hackbusch s approach for scalar pde and allows a convergence proof for some well known multi grid methods for stokes and navier stokes equations as dgs by brandt dinar 5 tilu from 31 and the simple methods by patankar spalding 23

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

multigrid methods for saddle point problems darcy systems we design and analyze multigrid methods for the saddle point problems resulting from raviart thomas nedelec mixed finite element methods for the darcy system in porous media flow uniform convergence of the w cycle algorithm in a nonstandard energy norm is established extensions to general second order elliptic problems are also addressed

stable rank 2 reflexive sheaves on with small and applications we investigate the moduli spaces of stable rank two reflexive sheaves on p3 with small chern classes as an application to curves of low degree in p3 we prove the curve has maximal rank and that the corresponding hilbert scheme is irreducible and unirational introduction in the past few years the subject of vector bundles on projective spaces and in particular the case of rank 2 on p3 has received much attention several basic theorems have been proved e g the existence of the moduli space of stable vector bundles m though so far very few of these moduli spaces have been studied in detail the interest of vector bundles partly stems from their connection with curves however the class of curves obtained in this way is rather restricted recently hartshorne srs has focused attention on reflexive sheaves of rank 2 on p3 on the one hand most results proved for vector bundles also turn out to be true for reflexive sheaves on the other hand reflexive sheaves have two significant advantages first they correspond to quite general curves second and most importantly one has the reduction step introduced by hartshorne which is an effective tool in studying vector bundles by relating them to simpler reflexive sheaves in this paper we investigate the moduli spaces of stable rank 2 reflexive sheaves on p3 with c2 3 for c2 2 we prove the moduli spaces are irreducible nonsingular and rational we also classify the related unstable planes for c2 3 in most cases we show that the moduli space is irreducible and unirational as one of the applications to curves of low degree in p3 cf srs 4 1 we prove that the curve has maximal rank and that the corresponding hilbert scheme is irreducible and unirational hence as a corollary we conclude that the variety of moduli 1g of curves of genus g is unirational for g 5 6 7 8 in 1 we give facts on multiple lines and deduce a criterion for unstable planes 2 contains our classification of semistable sheaves with c2 2 3 is about sheaves with c2 3 4 gives the applications to curves acknowledgment i would like to thank my thesis advisor robin hartshorne he has been very generous in his advice and time in addition he attracted to received by the editors september 1 1982 and in revised form march 18 1983 1980 mathematics subject classification primary 14f05 secondary 14d22 i1984 american mathematical society 0002 9947 84 1 00 25 per page

convergence of the multigrid v cycle algorithm for second order boundary value problems without full elliptic regularity the multigrid v cycle algorithm using the richardson relaxation scheme as the smoother is studied in this paper for second order elliptic boundary value problems the contraction number of the v cycle algorithm is shown to improve uniformly with the increase of the number of smoothing steps without assuming full elliptic regularity as a consequence the v cycle convergence result of braess and hackbusch is generalized to problems without full elliptic regularity

a multilevel algorithm for mixed problems we describe a multilevel algorithm for the numerical solution of symmetric indefinite problems which arise e g from mixed finite element approximations of the stokes equation the main difficulty besides the indefiniteness is the lack of regularity of the solution of the corresponding continuous problem this is overcome by introducing a scale of mesh dependent norms the convergence rate of the described algorithm is bounded independently of the meshsize for convenience we only discuss jacobi relaxation as smoothing operator in detail in the last section we comment on lanczos type smoothing procedures

multigrid analysis for the time dependent stokes problem certain implicit time stepping procedures for the incompressible stokes or navier stokes equations lead to a singular perturbed stokes type problem at each time step the paper presents a convergence analysis of a geometric multigrid solver for the system of linear algebraic equations resulting from the disretization of the problem using a finite element method several smoothing iterative methods are considered a smoother based on distributive iterations the braess sarazin and inexact uzawa smoother convergence analysis is based on smoothing and approximation properties in special norms a robust independent of time step and mesh parameter estimate is proved for the two grid and multigrid w cycle convergence factors

multigrid methods for saddle point problems oseen system abstract we develop and analyze multigrid methods for the oseen system in fluid flow we show that the w cycle algorithm is a uniform contraction if the number of smoothing steps is sufficiently large numerical results that illustrate the performance of the methods are also presented

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

a nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity a robust optimal order multigrid method for the pure traction problem in two dimensional linear elasticity is studied the finite element discretization is a variant of a mixed method proposed by falk where the displacement is approximated by nonconforming piecewise linear functions and the pressure is approximated by piecewise constant functions on a coarser grid full multigrid convergence is obtained the performance of this multigrid algorithm does not deteriorate as the material becomes nearly incompressible

convergence of nonconforming v cycle and f cycle multigrid algorithms for second order elliptic boundary value problems the convergence of v cycle and f cycle multigrid algorithms with a sufficiently large number of smoothing steps is established for nonconforming finite element methods for second order elliptic boundary value problems

linear finite element methods for planar linear elasticity a linear nonconforming conforming displacement finite element method for the pure displacement pure traction problem in two dimensional linear elasticity for a homogeneous isotropic elastic material is considered in the case of a convex polygonal configuration domain error estimates in the energy l sup 2 norm are obtained the convergence rate does not deteriorate for nearly incompressible material furthermore the convergence analysis does not rely on the theory of saddle point problems 22 refs

the gonality of general smooth curves with a prescribed plane nodal model let c be a smooth curve of genus g it is well known that c possesses a linear system g l if q l g 2n g 2 0 moreover if c is general and q g 2 k l and 6 kd k l 2 3 theorem 2 1 in loc cit and for each d this bound on 6 is sharp examples 4 1 and 4 2 in loc cit in this paper we consider the question for a general choice of f since the space parametrizing integral plane nodal curves with prescribed d and 6 is irreducible famous conjecture of severi proved by harris 9 the meaning of general becomes represented by a point of a particular nonempty zariski open subset of that parameter space note that qn x 3 g 0 if and only if 6 d 2 7d 18 2

multigrid methods for saddle point systems using constrained smoothers the constrained smoother for solving the saddle point system arising from the constrained minimization problem is a relaxation scheme such that the iteration remains in the constrained subspace a multigrid method using constrained smoothers for saddle point systems is analyzed in this paper uniform convergence of two level and w cycle multigrid methods with sufficient many smoothing steps and full regularity assumptions are obtained for some stable finite element discretization of stokes equations for braess sarazin smoother a convergence theory using only partial regularity assumption is also developed

the geometry of the moduli space of curves of genus 23 the problem of describing the birational geometry of the moduli space mg of complex curves of genus g has a long history severi already knew in 1915 thatmg is unirational for g 10 cf sev see also ac1 for a modern proof in the same paper severi conjectured thatmg is unirational for all genera g then for a long period this problem seemed intractable mumford writes in mu p 51 whether more mg s g 11 are unirational or not is a very interesting problem but one which looks very hard too especially if g is quite large the breakthrough came in the eighties when eisenbud harris and mumford proved that mg is of general type as soon as g 24 and that the kodaira dimension of m23 is 1 see hm eh3 we note that mg is rational for g 6 see dol for problems concerning the rationality of various moduli spaces severi s proof of the unirationality of mg for small g was based on representing a general curve of genus g as a plane curve of degree d with nodes this is possible when d 2g 3 2 when the number of nodes is small i e d 1 d 2 6 the dominant map from the variety of plane curves of degree d and genus g to mg yields a rational parametrization of the moduli space the two conditions involving d and can be satisfied only when g 10 so severi s argument cannot be extended for other genera however using much more subtle ideas chang ran and sernesi proved the unirationality of mg for g 11 12 13 see cr1 se1 while for g 15 16 they proved that the kodaira dimension is see cr2 4 the remaining cases g 14 and 17 g 23 are still quite mysterious harris and morrison conjectured in hmo that mg is uniruled precisely when g 23 all these facts indicate that m23 is a very interesting transition case our main result is the following

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

uniform convergence of multigrid v cycle iterations for indefinite and nonsymmetric problems in this paper an analysis of a multigrid method for nonsymmetric and or indefinite elliptic problems is presented in this multigrid method various types of smoothers may be used one type of smoother considered is defined in terms of an associated symmetric problem and includes point and line jacobi and gauss seidel iterations smoothers based entirely on the original operator are also considered one smoother is based on the normal form that is the product of the operator and its transpose other smoothers studied include point and line jacobi and gauss seidel it is shown that the uniform estimates of j h bramble and j e pasciak math comp 60 1993 pp 447 471 for symmetric positive definite problems carry over to these algorithms more precisely the multigrid iteration for the nonsymmetric and or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine but not dependent on the number of multigrid levels

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

multigrid methods for saddle point problems optimality systems abstract we develop multigrid methods for an elliptic distributed optimal control problem on convex domains that are robust with respect to a regularization parameter we prove the uniform convergence of the w cycle algorithm and demonstrate the performance of v cycle and w cycle algorithms through numerical experiments

distributions covariantes sur les espaces hyperboliques resume soient g k p une algebre de lie symetrique reelle de rang 1 k un groupe de lie d algebre de lie k et e une representation complexe irreductible de dimension finie de k l objet de cet article est de montrer que toute distribution k covariante sur p a valeurs dans e est produit d une distribution k invariante et d un polynome k covariant

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

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

on covariant functions and distributions under the action of a compact group abstract let g be a compact subgroup of gl n r acting linearly on a finite dimensional complex vector space e b malgrange has shown that the space c r n e g of c and g covariant functions is a finite module over the ring c r n g of c and g invariant functions first we generalize this result for the schwartz space s r n e g of g covariant functions secondly we prove that any g covariant distribution can be decomposed into a sum of g invariant distributions multiplied with a fixed family of g covariant polynomials this gives a generalization of an oksak result proved in 4

multilevel algorithms for mixed problems ii treatment of the mini element we present a multilevel algorithm for the mixed finite element approximation of the stokes problem by the mini element a local l 2 projection of the bubble functions and a strengthened cauchy inequality are essential tools for the convergence analysis of the multigrid algorithm

fast iterative solvers for discrete stokes equations we consider saddle point problems that result from the finite element discretization of stationary and nonstationary stokes equations three efficient iterative solvers for these problems are treated namely the preconditioned conjugate gradient method introduced by bramble and pasciak the preconditioned minres method and a method due to bank et al we give a detailed overview of algorithmic aspects and theoretical convergence results for the method of bank et al a new convergence analysis is presented a comparative study of the three methods for a three dimensional stokes problem discretized by the hood taylor p2 p1 finite element pair is given

families of nodal curves on projective threefolds and their regularity via postulation of nodes the main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds precisely given x a smooth projective threefold e a rank two vector bundle on x l a very ample line bundle on x and k 0 0 integers and denoted by v e l k the subscheme of p h 0 e l k parametrizing global sections of e l k whose zero loci are irreducible and nodal curves on x we present a new cohomological description of the tangent space t s v e l k at a point s 2 v e l k this description enable us to determine effective and uniform upper bounds for which are linear polynomials in k such that the family v e lk is smooth and of the expected dimension regular for short the almost sharpness of our bounds is shown by some interesting examples furthermore when x is assumed to be a fano or a calaby yau threefold we study in detail the regularity property of a point s 2 v e lk related to the postulation of the nodes of its zero locus c v s x roughly speaking when the nodes of c are assumed to be in general position either on x or on an irreducible divisor of x having at worst log terminal singularities or to lie on a l c i and subcanonical curve in x we find upper bounds onwhich are respectively cubic quadratic and linear polynomials in k ensuring the regularity of v e l k at s finally when x p 3 we also discuss some interesting geometric properties of the curves given by sections parametrized by v e ox k