Showing papers in "Mathematische Zeitschrift in 1998"

Syzygies of codimension 2 lattice ideals

Abstract: The study of semigroup algebras has a long tradition in commutative algebra. Presentation ideals of semigroup algebras are called toric ideals, in reference to their prominent role in geometry. In this paper we consider the more general class of lattice ideals. Fix a polynomial ring S = k[x1, . . . , xn] over a field k and identify monomials x in S with vectors a ∈ N. Let L be any sublattice of Z. Then its associated lattice ideal in S is

Combinatorial vector fields and dynamical systems

Abstract: In this paper we introduce the notion of a combinatorial dynamical system on any CW complex. Earlier in [Fo3] and [Fo4], we presented the idea of a combinatorial vector field (see also [Fo1] for the one-dimensional case), and studied the corresponding Morse Theory. Equivalently, we studied the homological properties of gradient vector fields (these terms were defined precisely in [Fo3], see also Sect. 2 of this paper). In this paper we broaden our investigation and consider general combinatorial vector fields. We first study the homological properties of such vector fields, generalizing the Morse Inequalities of [Fo3]. We then introduce various zeta functions which keep track of the closed orbits of the corresponding flow, and prove that these zeta functions, initially defined only on a half plane, can be analytically continued to meromorphic functions on the entire complex plane. Lastly, we review the notion of Reidemeister Torsion of a CW complex (introduced in [Re], [Fr]) and show that the torsion is equal to the value at $z=0$ of one of the zeta functions introduced earlier. Much of this paper can be viewed as a combinatorial analogue of the work on smooth dynamical systems presented in [P-P], [Fra], [Fri1, 2] and elsewhere.

On vector subdivision

Abstract: In this paper we give a complete characterization of the convergence of stationary vector subdivision schemes and the regularity of the associated limit function. These results extend and complete our earlier work on vector subdivision and its use in the construction of multiwavelets.

Interior gradient estimates for mean curvature equations

Abstract: The interior gradient estimate for the prescribed mean curvature equation has been extensively studied, see [9] and the references therein. For high order mean curvature equations it has also been obtained in [11, 18]. In most articles such estimates were obtained by carrying out analysis on the graphs of solutions and so the arguments depend on the invariance of the equations under rigid motion. From the view point of partial differential equations such estimate should hold for equations with similar structural conditions. In [13, 6] the gradient estimate was obtained for certain fully nonlinear elliptic equations under various conditions. Different proofs for the gradient estimate for mean curvature equations have been given in [1, 3]. In this note we show that for curvature and Hessian equations the interior gradient estimate can be obtained very easily. Our proof, which is based on suitable choice of auxiliary functions, is elementary and avoids geometric computations on the graph of solutions. The technique in this note has actually been widely used in literature, see, e.g., [9].

Chern's conjecture on minimal hypersurfaces

Abstract: In this paper, we study n-dimensional complete minimal hypersurfaces in a unit sphere. We prove that an n-dimensional complete minimal hypersurface with constant scalar curvature in a unit sphere with f3 constant is isometric to the totally geodesic sphere or the Clifford torus if S ≤ 1.8252n−0.712898, where S denotes the squared norm of the second fundamental form of this hypersurface.

On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals

Abstract: In the first part of this paper we show that the Castelnuovo-Mumford regularity of a monomial ideal is bounded above by its arithmetic degree. The second part gives upper bounds for the Castelnuovo-Mumford regularity and the arithmetic degree of a monomial ideal in terms of the degrees of its generators. These bounds can be formulated for an arbitrary homogeneous ideal in terms of any Grobner basis.

On a larger class of stable solutions to the Navier-Stokes equations in exterior domains

Abstract: where w = w(x) = (w1(x), · · · , wn(x)) and π = π(x) denote the unknown velocity vector and the unknown pressure of the fluid at point x ∈ Ω, respectively, while F = F (x) = (F i j (x))i,j=1,··· ,n is the given tensor with div F = ( ∑n j=1 ∂F 1 j ∂xj , · · · ,∑nj=1 ∂F n j ∂xj ) denoting the external force. In our previous paper [23], we established the theory of existence, uniqueness and regularity for solutions to (N − S0) in the class w ∈ Ln,∞(Ω),

Remarks on Seshadri constants

Abstract: which is called the Seshadri constant of L at x. Here the infimum is taken over all irreducible curves C passing through x and mx(C) is the multiplicity of C at x. For example, if L is very ample then e(L, x) ≥ 1. There has been recant interest in trying to give lower bounds for this invariant at a general point. Ein and Lazarsfeld [EL] show that if X is a surface, then e(L, x) ≥ 1 for very general x ∈ X . In higher dimension (n ≥ 3) Ein, Kuchle and Lazarsfeld [EKL] prove that e(L, x) ≥ 1 n for a very general point. We say that a point x ∈ X is very general if x is in the complement X\Z of Z a countably union of proper subvarieties. Examples of Miranda show that e(L, x) can take arbitrarily small values in codimension two (i.e. codim Z = 2), even for an ample line bundle. One may expect that this general bounds are not optimal. An elementary observation (see Remark 1 below for the proof) shows that e(L, x) ≤ n √Ln. A natural question is, are there conditions which guarantee equality? Even in relative simple cases it turns out to be hard to give an answer. Recently, Xu [Xu] improved the surface bound given by Ein and Lazarsfeld. He showed that if L2 ≥ 3(4α − 4α + 5) for a given integer α > 1

On Fatou-Bieberbach domains

Abstract: A Fatou-Bieberbach domain is a domain in C2 which is a biholomorphic image of C2 and is not all of C2. The existence of such domains has been known for a long time. There are several papers dealing with Fatou-Bieberbach domains [BF, BS, DE, E, FS, Gl, K, N1, N2, RR1, RR2, S]. Nevertheless, they remain quite misterious objects and there are many open questions about them. One of them is the following [RR1, Q.11, p.79)]: If Ω is a FatouBieberbach domain and L is a complex line is it possible that (a) L ∩Ω is connected (b) L∩Ω has finitely many components (c) L∩Ω is a circular disc. Let∆ be the open unit disc in C and let P = ∆×∆. Suppose one wants to obtain a Fatou-Bieberbach domain Ω whose intersection with the z-axis {(z, 0) : z ∈ C} is approximately the unit disc. One would try to find Ω such that

Basic hopf algebras and quantum groups

Abstract: This paper investigates the structure of basic finite dimensional Hopf algebras H over an algebraically closed field k. The algebra H is basic provided H modulo its Jacobson radical is a product of the field k. In this case H is isomorphic to a path algebra given by a finite quiver with relations. Necessary conditions on the quiver and on the coalgebra structure are found. In particular, it is shown that only the quivers $\Gamma_G(W)$ given in terms of a finite group G and sequence $W=(w_1,w_2,\ldots,w_r)$ of elements of G in the following way can occur. The quiver $\Gamma_G(W)$ has vertices $\{v_g\}_{g\in G}$ and arrows $\{ (a_i,g)\colon v_{g^{-1}}\to v_{w_ig^{-1}}\mid g\in G, w_i\in W\}$ , where the set $\{ w_1,w_2,\ldots,w_r\}$ is closed under conjugation with elements in G and for each g in G, the sequences W and $(gw_1g^{-1}, gw_2g^{-1},\ldots, gw_rg^{-1})$ are the same up to a permutation. We show how $k\Gamma_G(W)$ is a kG-bimodule and study properties of the left and right actions of G on the path algebra. Furthermore, it is shown that the conditions we find can be used to give the path algebras $k\Gamma_G(W)$ themselves a Hopf algebra structure (for an arbitrary field k). The results are also translated into the language of coverings. Finally, a new class of finite dimensional basic Hopf algebras are constructed over a not necessarily algebraically closed field, most of which are quantum groups. The construction is not characteristic free. All the quivers $\Gamma_G(W)$ , where the elements of W generates an abelian subgroup of G, are shown to occur for finite dimensional Hopf algebras. The existence of such algebras is shown by explicit construction. For closely related results of Cibils and Rosso see [Ci-R].

Correction and linearization of resonant vector fields and diffeomorphisms

Abstract: We extend the classical Siegel-Brjuno-Russmann linearization theorem to the resonant case by showing that under A. D. Brjuno's diophantine condition, any resonant local analytic vector field (resp. diffeomorphism) possesses a well-defined correction which (1) depends on the chart but, in any given chart, is unique (2) consists solely of resonant terms and (3) has the property that, when substracted from the vector field (resp. when factored out of the diffeomorphism), the vector field or diffeomorphism thus “corrected” becomes analytically linearizable (with a privileged or “canonical” linearizing map). Moreover, in spite of the small denominators and contrary to a hitherto prevalent opinion, the correction's analyticity can be established by pure combinatorics, without any analysis.

On Holomorphic Curves in Semi-Abelian Varieties

Abstract: The algebraic degeneracy of holomorphic curves in a semi-Abelian variety omitting a divisor is proved (Lang's conjecture generalized to semi-Abelian varieties) by making use of the {\it jet-projection method} and the logarithmic Wronskian jet differential after Siu-Yeung. We also prove a structure theorem for the locus which contains all possible image of non-constant entire holomorphic curves in a semi-Abelian variety omitting a divisor.

On the genus and hartshorne-rao module of projective curves

Abstract: In this paper optimal upper bounds for the genus and the dimension of the graded components of the Hartshorne-Rao module of curves in projective n-space are established. This generalizes earlier work by Hartshorne [H] and Martin-Deschamps and Perrin [MDP]. Special emphasis is put on curves in ${\bf P}^4$ . The first main result is a so-called Restriction Theorem. It says that a non-degenerate curve of degree $d \geq 4$ in ${\bf P}^4$ over a field of characteristic zero has a non-degenerate general hyperplane section if and only if it does not contain a planar curve of degree $d-1$ (see Th. 1.3). Then, using methods of Brodmann and Nagel, bounds for the genus and Hartshorne-Rao module of curves in ${\bf P}^n$ with non-degenerate general hyperplane section are derived. It is shown that these bounds are best possible in a very strict sense. Coupling these bounds with the Restriction Theorem gives the second main result for curves in ${\bf P}^4$ . Then curves of maximal genus are investigated. The Betti numbers of their minimal free resolutions are computed and a description of all reduced curves of maximal genus in ${\bf P}^n$ of degree $\geq n+2$ is given. Finally, all pairs (d,g) of integers which really occur as the degree d and genus g of a non-degenerate curve in ${\bf P}^4$ are described.

Courbes spectrales et compactifications de jacobiennes

Abstract: We generalize here a result of A. Beauville, M.S. Narasimhan and S. Ramanan about spectral curves and use it to realize a compactiication of the generalized jacobian of a singular spectral curve as a ber of the morphism of Hitchin. Introduction. pour une courbe spectrale : X ! Y , lorsque X est int egre, il y a une correspondance biunivoque entre les classes d'isomorphismes de faisceaux sans torsion de rang un, M, sur X et les classes d'isomorphismes de paires (E; u) o u E est un faisceau localement libre sur Y de rang egal au degr e de , L le faisceau inversible sur Y associ e a la courbe spectrale et u : E ! E L un homomorphisme dont le polyn^ ome caract eristique d eenit X.

Modular diagonal quotient surfaces

Abstract: The main objective of this paper is to analyze the geometry of the modular diagonal quotient surface ZN,e = ∆e\(X(N) × X(N)) which classifies pairs of elliptic curves E1 and E2 together with an isomorphism of “determinant e” between their associated modular representations mod N . In particular, we calculate some of the numerical invariants of its minimal desingularization ZN,e such as its Betti and Chern numbers and determine its place in the Enriques-Kodaira classification table.