Showing papers in "Collectanea Mathematica in 2003"

Castelnuovo-Mumford regularity of products of ideals

Abstract: The Castelnuovo-Mumford regularity reg$(M)$ is one of the most important invariants of a finitely generated graded module $M$ over a polynomial ring $R$. For instance, it measures the amount of computational resources that working with $M$ requires. In general one knows that the regularity of a module can be doubly exponential in the degrees of the minimal generators and in the number of the variables. On the other hand, in many situations one has or one conjectures a much better behavior. One may ask, for instance, whether the Castelnuovo-Mumford regularity reg($IM$) of the product of an ideal $I$ with a module $M$ is bounded by the sum reg($I$) + reg($M$). In general this is not the case. But we show that it is indeed the case if either dim $R/I\leq 1$ or $I$ is generic (in a very precise sense). Further we show that products of ideals of linear forms have always a linear resolution and that the same is true for products of determinantal ideals of a generic Hankel matrix.

Fully absolutely summing and Hilbert-Schmidt multilinear mappings

Abstract: The space of the fully absolutely (r;r1,...,rn)-summing n-linear mappings between Banach spaces is introduced along with a natural (quasi-)norm on it. If r,rk C [1,+infinite], k=1,...,n, this space is characterized as the topological dual of a space of virtually nuclear mappings. Other examples and properties are considered and a relationship with a topological tensor product is stablished. For Hilbert spaces and r = r1 = ... = rn C [2,+infinite[ this space is isomorphic to the space of the Hilbert-Schmidt multilinear mappings.

Transferring monotonicity in weighted norm inequalities

Abstract: Certain weighted norm inequalities for integral operators with non-negative, monotone kernels are shown to remain valid when the weight is replaced by a monotone majorant or minorant of the original weight. A similar result holds for operators with quasi-concave kernels. To prove these results a careful investigation of the functional properties of the monotone envelopes of a non-negative function is carried-out. Applications are made to function space embeddings of the cones of monotone functions and quasi-concave functions. Under weaker partial orders on non-negative functions, monotone envelopes are re-examined and the level function is recognized as a monotone envelope in two ways. Using the level function, monotonicity can be transferred from the kernel to the weight in inequalities restricted to a cone of monotone functions.

Hurwitz spaces of genus 2 covers of an elliptic curve

Abstract: Let $E$ be an elliptic curve over a field $K$ of characteristic $ ot =$ 2 and let $N > 1$ be an integer prime to char ($K$). The purpose of this paper is to construct the (twodimensional) Hurwitz moduli space $H (E / K, N, 2)$ which "classifies" genus 2 covers of $E$ of degree $N$ and to show that it is closely related to the modular curve $X$ ($N$) which parametrizes elliptic curves with level -$N$- structure. ewline More precisely, we introduce the notion of a normalized genus 2 cover of $E / K$ and show that the corresponding moduli space is $H_{E / K, N}$ is an open subset of (a twist of) $X$ ($N$), and that the connected components of the Hurwitz space $H (E/K, N, 2)$ are of the form $E\times H_{E'/K,M}$ for suitable elliptic curves $E'\sim E$ and divisors $M\vert N$.

On the existence of wavelets for non-expansive dilation matrices

Abstract: Sets which simultaneously tile R n by applying powers of an invertible matrix and translations by a lattice are studied. Diagonal matrices A for which there exist sets that tile by powers of A and by integer translations are characterized. A sufficient condition and a necessary condition on the dilations and translations for the existence of such sets are also given. These conditions depend in an essential way on the interplay between the eigenvectors of the dilation matrix and the translation lattice rather than the usual dependence on the eigenvalues. For example,it is shown that for any values |a| > 1 > |b|,there is a (2 ×2) matrix A with eigenvalues a and b for which such a set exists,and a matrix A with eigenvalues a and b for which no such set exists. Finally,these results are related to the existence of wavelets for non-expansive dilations.

Integral transforms of the Kontorovich-Lebedev convolution type

Abstract: We deal with a class of integral transformations of the form \begin{flushleft}$f(x) \rightarrow\frac{1}{2x}\prod_{n=1}^\infty(1+\frac{x(x-\frac{d}{dx}-\frac{d^2} {dx^2})}{(2n-1)^2})\int_{\mathbb{R}_+^2}e^{-{\frac{1}{2} (x\frac{u^2+y^2} {uy}+ \frac{yu}{x})}_{f(u)h(y)dudy,x\in\mathbb{R}+}}$ \end{flushleft}in $L_2(\mathbb {R}_+;xdx)$, which is associated with the Kontorovich-Lebedev operator \begin{center} $K_{i\tau}[f]=\int_0^\infty K_{i\tau}(x)f(x)dx,\tau\in\mathbb{R}+$. \end{center} Necessary and sufficient conditions on h to establish that the transformation is unitary in $L_2(\mathbb{R}+;xdx)$ are obtained. A reciprocal inversion formula and an example of the unitary convolution transformation are given. given.

Interpolation properties of a scale of spaces

Abstract: A scale of function spaces is considered which proved to be of considerable importance in analysis. Interpolation properties of these spaces are studied by means of the real interpolation method. The main result consists in demonstrating that this scale is interpolated in a way different from that for $L^{ p }$ spaces, namely, the interpolation space is not from this scale.

Ample vector bundles with zero loci having a bielliptic curve section

Abstract: Let X be a smooth complex projective variety and let $Z \subset X$ be a smooth submanifold of dimension $\geq 2$, which is the zero locus of a section of an ample vector bundle $\mathcal{E}$ of rank dim $X$ - dim $Z \geq 2$ on $X$. Let $H$ be an ample line bundle on $X$ whose restriction $H_Z$ to $Z$ is very ample. Triplets $(X, \mathcal{E}, H)$ as above are studied and classified under the assumption that $Z$ is a projective manifold of high degree with respect to $H_Z$, dmitting a curve section which is a double cover of an elliptic curve.

Curves on a smooth quadric

Abstract: We associate to every curve on a smooth quadric a polynomial equation that defines it as a divisor; this polynomial is defined through a matrix. In this way we can study several properties of these curves; in particular we can give a geometrical meaning to the rank of the matrix which defines the curve.

Orbifold principal bundles on an elliptic fibration and parabolic principal bundles on a Riemann surface

Abstract: Let X be a compact Riemann surface and associated to each point p-i of a finite subset S of X is a positive integer m-i. Fix an elliptic curve C. To this data we associate a smooth elliptic surface Z fibered over X. The group C acts on Z with X as the quotient. It is shown that the space of all vector bundles over Z equipped with a lift of the action of C is in bijective correspondence with the space of all parabolic bundles over X with parabolic structure over S and the parabolic weights at any p-i being integral multiples of 1 / m-i. A vector bundle V over Z equipped with an action of C is semistable (respectively, polystable) if and only if the parabolic bundle on X corresponding to V is semistable (respectively, polystable). This bijective correspondence is extended to the context of principal bundles.

Frames associated with expansive matrix dilations

Abstract: We construct wavelet-type frames associated with the expansive matrix dilation on the Anisotropic Triebel-Lizorkin spaces. We also show the a.e. convergence of the frame expansion which includes multi-wavelet expansion as a special case.

Properties of extensions of algebraically maximal fields

Abstract: We prove some properties similar to the theorem Ax-Kochen-Ershov, in some cases of pairs of algebraically maximal fields of residue characteristic p > 0. This properties hold in particular for pairs of Kaplansky fields of equal characteristic, formally p-adic fields and finitely ramified fields. From that we derive results about decidability of such extensions.

A note about the isotropy groups of 2-plane bundles over closed surfaces

Abstract: Let $\xi$ be a $2$-plane bundle over a closed surface $S$. The line bundles $\lambda$ over $S$ such that $\xi\otimes\lambda\cong\xi$ form a group $\mathcal{J}(\xi)$ (the isotropy group of $\xi$); the scope of this paper is to describe $\mathcal{J}(\xi)$.

Curves on a Double Surface

Abstract: Let F be a smooth projective surface contained in a smooth threefold T, and let X be the scheme corresponding to the divisor 2F on T. A locally Cohen-Macaulay curve C included in X gives rise to two effective divisors on F, namely the largest divisor P contained in C intersection F and the curve R residual to C intersection F in C. We show that under suitable hypotheses a general deformation of R and P lifts to a deformation of C on X, and give applications to the study of Hilbert schemes of locally Cohen-Macaulay space curves.

Curves on a ruled cubic surface

Abstract: For the general ruled cubic surface S (with a double line) in P3 = P3 sub k, k any algebraically closed field, we find necessary conditions for which curves on S can be the specialization of a flat family of curves on smooth cubics. In particular, no smooth curve of degree > 10 on S is such a specialization.

The support theorem for the complex Radon transform of distributions

Abstract: The properties of the complex Radon transform of compactly supported distributions are considered. For such distributions, we prove a support theorem allowing us to describe the support of the distribution in terms of the support of its Radon transform.

On some properties of partial intersection schemes

Abstract: Partial intersection subschemes of Pr of codimension c were used to furnish various graded Betti numbers which agree with a fixed Hilbert function. Here we study some further properties of such schemes; in particular, we show that they are not in general licci and we give a large class of them which are licci. Moreover, we show that all partial intersections are glicci. We also show that for partial intersections the first and the last Betti numbers, say m and p respectively, give bounds each other; in particular, in codimension 3 case we see that [(p+5)/2] = m = 2p+1 and each m and p satisfying the above inequality can be realized.

Extreme positive definite double sequences which are not moment sequences

Abstract: From the fact that the two-dimensional moment problem is not always solvable, we can deduce that there must be extreme ray generators of the cone of positive definite double sequences which are nor moment sequences. Such an argument does not lead to specific examples. In this paper it is shown how specific examples can be constructed if one is given an example of an N-extremal indeterminate measure in the one-dimensional moment problem (such examples exist in the literature). Konrad Schmudgen had an example similar to ours.