Volume 1, Issue 3

We classify, up to four open cases, all near hexagons with four points on a line and two points at distance 2.

First, the paper contains new necessary and sufficient conditions for a set of subspaces of PG(n,q) to be an SPG regulus. Then a new construction method for semipartial geometries is given. As particular cases we find known classes of partial and semipartial geometries, but also new classes of semipartial geometries. The new semipartial geometries have parameters s=q n - 1 , t = q n+1 , α = 2q n-1 , μ = 2q n (q n - 1) , with either q any prime power and n=2 , or q=2 h , h ≥ 1 , and n ≥ 3 .

We study smooth projective varieties X ⊆ ℙ N of dimension 3, such that there are two very ample invertible sheaves ℒ , ℳ on X , and there exist two sections of ℒ , ℳ which intersect along a bielliptic curve C . We give a classification of such threefolds X under some hypotheses on the degree of C with respect to the two embeddings given by ℒ , ℳ .

The moduli space of (1,3) polarized abelian surfaces with full level-2 structure is birational to a double cover of the Barth-Nieto quintic. Barth and Nieto have shown that these varieties have Calabi-Yau models Z and Y , respectively. In this paper we apply the Weil conjectures to show that Y and Z are rigid and we prove that the L function of their common third étale cohomology group is modular, as predicted by a conjecture of Fontaine and Mazur. The corresponding modular form is the unique normalized cusp form of weight 4 for the group~ Γ 1 (6) . By Tate's conjecture, this should imply that Y , the fibred square of the universal elliptic curve~ S 1 (6) , and Verrill's rigid Calabi-Yau~ 𝒵 A 3 , which all have the same L function, are in correspondence over ℚ . We show that this is indeed the case by giving explicit maps.

We describe an algorithm to compute finite presentations for the mapping class group of a connected, compact, orientable surface, possibly with boundary and punctures. By an inductive process, such an algorithm, starting from a presentation well known for the mapping class group of the sphere and the torus with ``few'' boundary components and/or punctures, produces a presentation for the mapping class group of any other surface.