It has been proved by various authors that a normalized,
l-Buchsbaum rank 2 vector bundle on P3 is a nullcorrelation bundle,
while a normalized, 2-Buchsbaum rank 2 vector bundle on P3 is an
instanton bundle of charge 2. We find that the same is not true for
3-Buchsbaum rank 2 vector bundles on P3, and propose a conjecture
regarding the classification of such objects.
We use Fink’s identity to obtain new identities related to
generalizations of Steffensen’s inequality. Ostrowski-type inequalities
related to these generalizations are also given. Using inequalities for
the Cebysev functional we obtain bounds for these identities. Further,
we use these identities to obtain new generalizations of Steffensen’s inequality for n-convex functions. Finally, we use these generalizations to
construct a linear functional that aenerates exvonentiallv convex functions.
Università degli Studi di Trieste, Dipartimento di Matematica e Informatica
Rendiconti dell’Istituto di Matematica dell’Università di Trieste was founded in 1969 by Arno Predonzan, with the aim of publishing original research articles in all fields of mathematics and has been the first Italian mathematical journal to be published also on-line. The access to the electronic version of the journal is free. All published articles are available on-line. The journal can be obtained by subscription, or by reciprocity with other similar journals. Currently more than 100 exchange agreements with mathematics departments and institutes around the world have been entered in.
In 1991 Campana and Petemell proposed, as a natural
algebro-geometric extension of Mori’s characterization of the projective
space, the problem of classifying the complex projective Fano manifolds
whose tangent bundle is nef, conjecturing that the only varieties satisfy
ing these properties are rational homogeneous. In this paper we review
some background material related to this problem, with special attention
to the partial results recently obtained by the authors.
For a generic set M of 3x3 matrices over C we find necessary and sufficient conditions when A4 is simultaneously self-adjoint.
Moreover, for a set of complex hermitian matrices we can tell if there
exists a linear combination of matrices which is positive definite. Every
M can be identified with a determinantal representation of a cubic hypersurface. This allows us to use the tools of algebraic geometry. The question of definiteness can be solved by using semidefinite programming.
We consider the parametrization (f0, f1, f2) of plane
rational curve c, and we want to relate the splitting type of C (i.e.
the second Betti numbers of the ideal (f0, f1, f2) with the
singularities of the associated Poncelet surface in p3. We are able of doing this for Ascenzi curves, thus generalizing a result in  in the
case of plane curves. Moreover we prove that if the Poncelet surface
s C p3 is singular then it is associated with a curve C which possesses
at least a point of multiplicity >_ 3.
This is an introductory survey, from a geometric perspective, on the Singular Value Decomposition (SVD) for real matrices,
focusing on the role of the Terracini Lemma. We extend this point of
view to tensors, we define the singular space of a tensor as the space
spanned by singular vector tuples and we study some of its basic properties.
Given P nk with k algebraically closed field of characteristic p > 0, and X C Pnk integral variety of codimension 2 and degree d,
let Y = X n H be the general hyperplane section of X. In this paper
we study the problem of lifting, i.e. extending, a hypersurface in H of
degree s containing Y to a hypersurface of same degree s in Pn containing X. For n = 3 and n = 4, in the case in which this extension
does not exist we get upper bounds for d depending on s and p.
Several families of rank-two vector bundles on Hirzebruch
surfaces are shown to consist of all very ample, uniform bundles. Under
suitable numerical assumptions, the projectivization of these bundles,
embedded by their tautological line bundles as linear scrolls, are shown
to correspond to smooth points of components of their Hilbert scheme,
the latter having the expected dimension. If e = 0,1 the scrolls fill up
the entire component of the Hilbert scheme, while for e = 2 the scrolls
exhaust a subvariety of codimension 1.
Let A = (aij) be a symmetric non-negative integer 2k X 2k
matrix. A is homogeneous if aij + ail = an + akj for any choice of the
four indexes. Let A be a homogeneous matrix and let F be a general
form in C[xi,....xn] with 2deg(F) = trace(A). We look for the least
integer s(A), so that F = pfaff(M1) + ••• + pfaff(Ms(A)) where the
Mi = (Fim) are 2k X 2k skew-symmetric matrices of forms with degree
matrix A. We consider this problem for n = 4 and we prove that
s (A)_< k for all A.
We consider congruences of multisecant lines to a non
linearly or non quadratically normal variety of codimension two or three
in a projective space. We give a uniform way to compute the degree of
the dual variety of their focal locus. Then we focus on the geometry
of the non quadratically normal variety of codimension three in Pg. In
particular we construct a component of the double locus of its dual from
the Hyper-Kahler 4-fold of Debarre-Voisin.
In this note we present a notion of fundamental scheme for
Cohen-Macaulay, order I, irreducible congruences of lines. We show
that such a congruence is formed by the k-secant lines to its fundamental
scheme for a number k that we call the secant, index of the congruence.
if the fundamental scheme X is a smooth connected variety in FN, then
k = (N — l)/(c — 1) (where c is the codimension of X) and X comes
equipped with a special tangency divisor cut out by a virtual hypersurface of degree k — 2 (to be precise, linearly equivalent to a section by
an hypersurface of degree (k — 2) without being cut by one). This is
explained in the main theorem of this paper. This theorem is followed
by a complete classification of known locally Cohen-Macaulay order 1
congruences of lines with smooth fundamental scheme. To conclude we
remark that according to Zak’s classification of Severi Varieties and
Hartshome conjecture for low codimension varieties, this classification
We present some existence and multiplicity results for positive solutions to the Dirichlet problem associated with; under suitable conditions on the nonlinearity g(u)and thew eight function a(x): The assumptions considered are related to classical theorems about positive solutions to a sublinear elliptic equation due to Brezis-Oswald and Brown-Hess.