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 seven elements are zero-divisors, the non-trivial ones forming two distinct sets of three; elementary ('slim') and composite ('fat'). Due to this fact, the line in question is endowed with a very intricate structure. It contains twenty-seven points, every point has eighteen neighbour points, the neighbourhoods of two distant points share twelve points and those of three pairwise distant points have six points in common -- namely those with coordinates having both the entries `fat' zero-divisors. Algebraically, the points of the line can be partitioned into three groups: a) the group comprising three distinguished points of the ordinary projective line of order two (the 'nucleus'), b) the group composed of twelve points whose coordinates feature both the unit(y) and a zero-divisor (the 'inner shell') and c) the group of twelve points whose coordinates have both the entries zero-divisors (the 'outer shell'). The points of the last two groups can further be split into two subgroups of six points each; while in the former case there is a perfect symmetry between the two subsets, in the latter case the subgroups have a different footing, reflecting the existence of the two kinds of a zero-divisor. The structure of the two shells, the way how they are interconnected and their link with the nucleus are all fully revealed and illustrated in terms of the neighbour/distant relation. Possible applications of this finite ring geometry are also mentioned.
We compute the equivariant fundamental class of the orbit closure of a linear series on the projective line. We also describe the boundary of the orbit closure and how the orbits specialise in one parameter families.
We develop techniques to split the principal parts on the projective line over an arbitrary ring and apply these techniques to give a complete classification of the principal parts on the projective line over any field of characteristic zero.
Our goal is to study the syzygies of the projective embeddings defined by ample line bundles on a fake projective plane S. The syzygies are studied in terms of the property $N_p$. For various kinds of ample line bundles, we give explicit lower bounds for their powers above which the property $N_p$ is satisfied.
We prove a bound on the number of lines on a smooth degree-d surface in three-dimensional projective space for $d \geq 3$. This bound improves a bound due to Segre and renders some of his arguments rigorous. It is the best known bound for $d \geq 6$.
It is well known that there are no stable bundles of rank greater than 1 on the projective line. In this paper, our main purpose is to study the existence problem for stable coherent systems on the projective line when the number of sections is larger than the rank. We include a review of known results, mostly for a small number of sections.
In this paper, we establish equivariant mirror symmetry for the weighted projective line. This extends the results by B. Fang, C.C. Liu and Z. Zong, where the projective line was considered [{\it Geometry \& Topology} 24:2049-2092, 2017]. More precisely, we prove the equivalence of the $R$-matrices for A-model and B-model at large radius limit, and establish isomorphism for $R$-matrices for general radius. We further demonstrate that the graph sum of higher genus cases are the same for both models, hence establish equivariant mirror symmetry for the weighted projective line.
