Related papers: Principal parts on the projective line over arbitr…
Matrices over the ring of formal power series are considered. Normal forms with respect to various sub-groups of the two-sided transformations are constructed. The construction is based on the special property of the action: it induces a…
We study the rigid analytic geometry of cyclic coverings of the projective line. We determine the defining equation of a cyclic covering of degree $p$ of the projective line by a Mumford curve over a complete discrete valuation field of…
In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singular versions of them) over smooth projective varieties over algebraically closed ground fields of positive characteristic.
A standard procedure in classical projective geometry, using pencils of lines to extend an incidence plane to a projective plane, is examined from a constructive viewpoint. Brouwerian counterexamples reveal the limitations of traditional…
Following our previous work, we develop an algorithm to compute a presentation of the fundamental group of certain partial compactifications of the complement of a complex arrangement of lines in the projective plane. It applies, in…
We study zero divisors and minimal prime ideals in semirings of characteristic one. Thereafter we find a counterexample to the most obvious version of primary decomposition, but are able to establish a weaker version. Lastly, we study…
We study complex analytic (possibly singular) projective connections on the plane. We characterize some of them in terms of their families of integral curves. We also give a beginning of classification of second order odes polynomial in the…
We develop a constructive process which determines all extreme points of the unit ball of the space of $m$--linear forms, $m\geq1.$ Our method provides a full characterization of the geometry of that space through finitely many elementary…
We discuss a general procedure for using characteristic classes to study the components of the gauge group for a principal G-bundle. To illustrate this, we work out the case where G is the projective unitary group.
In these notes we aim at bringing together design theory and projective geometry over a ring. Both disciplines are well established, but the results on the interaction between them seem to be rare and scattered over the literature. Thus our…
We consider a subset of projective space over a finite field and give bounds on the minimal degree of a non-vanishing form with respect to this subset.
The projective line over a field carries structure of a groupoid with a certain correspondence between objects and arrows. We discuss to what extent the field can be reconstructed from the groupoid.
We generalize Horrocks' criterion for the splitting of vector bundles on projective space. We establish an analogous splitting criterion for vector bundles on a class of smooth complex projective varieties of dimension at least four, over…
We discuss to what extent the local techniques of resolution of singularities over fields of characteristic zero can be applied to improve singularities in general. For certain interesting classes of singularities, this leads to an embedded…
Vector fields with components which are generalized zero-forms are constructed. Inner products with generalized forms, Lie derivatives and Lie brackets are computed. The results are shown to generalize previously reported results for…
We study linear series on curves inducing injective morphisms to projective space, using zero-dimensional schemes and cohomological vanishings. Albeit projections of curves and their singularities are of central importance in algebraic…
The paper gives a succinct appraisal of the properties of the projective line defined over the direct product ring $R\_{\triangle} \equiv$ GF(2)$\otimesGF(2)\otimes$GF(2). The ring is remarkable in that except for unity, all the remaining…
In this paper we collect the main properties of free curves in the complex projective plane and a lot of conjectures and open problems, both old and new. In the quest to understand the mystery of free curves, many tools were developed and…
Any set of $\sigma$-Hermitian matrices of size $n \times n$ over a field with involution $\sigma$ gives rise to a projective line in the sense of ring geometry and a projective space in the sense of matrix geometry. It is shown that the two…
In this article, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either a…