A characterization of the projective line

Let be a set (with at least three different points) and let be a group of bijections of . If the action of on satisfies three natural conditions, then admits a canonical structure of a projective line over a commutative field, such that is the group of all projective transformations of .

     

Orbits on the projective line

Let PGL(2, q) act in the natural way on the four-sets and the five-sets in PG(1, q). We determine the number of orbits of any given size and use this to construct some 3-designs on q + 2 points.     

Affine Kac-Moody Lie algebras as torsors over the punctured line

We interpret and develop a theory of loop algebras as torsors (principal homogeneous spaces) over Spec (k[t, t⁻¹]). As an application, we recover Kac's realization of affine Kac-Moody Lie algebras.     

Regular sets on the projective line

We show that if G is the group PL(2,q)(for q a prime-power) acting on the points of the projective line in the usual way, then for q>27 there is a set of 5 points such that no non-trivial element of G fixes .     

Principal bundles on the projective line

We classify principalG-bundles on the projective line over an arbitrary fieldk of characteristic ≠ 2 or 3, whereG is a reductive group. If such a bundle is trivial at ak-rational point, then the structure group can be reduced to a maximal torus.     

Modules of principal parts on the projective line

The modules of principal partsP ^k (E) of a locally free sheaf ε on a smooth schemeX is a sheaf ofO _X -bimodules which is locally free as left and rightO _X -module. We explicitly split the modules of principal partsP ^k (O(n)) on the projective line in arbitrary characteristic, as left and rightO _p1-modules. We get examples when the splitting-type as left module differs from the splitting-type as right module. We also give examples showing that the splitting-type of the principal parts changes with the characteristic of the base field. Research supported by the EAGER foundation, the Emmy Noether research institute for mathematics, the Minerva foundation of Germany and the excellency centerGroup theoretic methods in the study of algebraic varieties.     

Mumford coverings of the projective line

A Mumford covering of the projective line over a complete non-archimedean valued field K is a Galois covering X? P¹_K X\rightarrow {\bf P}^1_K such that X is a Mumford curve over K. The question which finite groups do occur as Galois group is answered in this paper. This result is extended to the case where P¹_K {\bf P}^1_K is replaced by any Mumford curve over K.     

Weierstrass points on cyclic covers of the projective line

We are interested in cyclic covers of the projective line which are totally ramified at all of their branch points. We begin with curves given by an equation of the form , where is a polynomial of degree . Under a mild hypothesis, it is easy to see that all of the branch points must be Weierstrass points. Our main problem is to find the total Weierstrass weight of these points, . We obtain a lower bound for , which we show is exact if and are relatively prime. As a fraction of the total Weierstrass weight of all points on the curve, we get the following particularly nice asymptotic formula (as well as an interesting exact formula):

where is the genus of the curve. In the case that (cyclic trigonal curves), we are able to show in most cases that for sufficiently large primes , the branch points and the non-branch Weierstrass points remain distinct modulo .

     

On the classification of fuzzy projective planes of fuzzy 3-dimensional projective space

In this work, the classifications of fuzzy 3-dimensional vector spaces of fuzzy 4-dimensional vector space and fuzzy projective planes of fuzzy 3-dimensional projective space from fuzzy 4-dimensional vector space are given.     

On cyclic covers of the projective line

We construct configuration spaces for cyclic covers of the projective line that admit extra automorphisms and we describe the locus of curves with given automorphism group. As an application we provide examples of arbitrary high genus that are defined over their field of moduli and are not hyperelliptic.     

On linear sets on a projective line

Linear sets generalise the concept of subgeometries in a projective space. They have many applications in finite geometry. In this paper we address two problems for linear sets: the equivalence problem and the intersection problem. We consider linear sets as quotient geometries and determine the exact conditions for two linear sets to be equivalent. This is then used to determine in which cases all linear sets of rank 3 of the same size on a projective line are (projectively) equivalent. In (Donati and Durante, Des Codes Cryptogr, 46:261–267), the intersection problem for subgeometries of PG(n, q) is solved. The intersection of linear sets is much more difficult. We determine the intersection of a subline PG(1, q) with a linear set in PG(1, q ^h ) and investigate the existence of irregular sublines, contained in a linear set. We also derive an upper bound, which is sharp for odd q, on the size of the intersection of two different linear sets of rank 3 in PG(1, q ^h ).     

Linear sets on the projective line with complementary weights

Linear sets on the projective line have attracted a lot of attention because of their link with blocking sets, KM-arcs and rank-metric codes. In this paper, we study linear sets having two points of complementary weight, that is, with two points for which the sum of their weights equals the rank of the linear set. As a special case, we study those linear sets having exactly two points of weight greater than one, by showing new examples and studying their equivalence issue. Also, we determine some linearized polynomials defining the linear sets recently introduced by Jena and Van de Voorde [30].