Some deformations of codimension two discrete groups

Deformations of Codimension 2 Toric Varieties

We prove Sturmfels’ conjecture that toric varieties of codimension two have no other flat deformations than those obtained by Gröbner basis theory.

Some Finite and Discrete Groups

We have have encountered several finite groups in the previous chapter: the cyclic group Zn, the symmetric group Sn, and its subgroup the alternating group An. The properties of these groups will be elaborated in this chapter. We will also discuss three other groups obtained by incorporating ‘reflections’ into Zn, the dihedral group Dn, the generalized quaterion group Qn, and the dicyclic group...

Some numerical results on two classes of finite groups

In this paper, we consider the finitely presented groups $G_{m}$ and $K(s,l)$ as follows;$$G_{m}=langle a,b| a^m=b^m=1,~[a,b]^a=[a,b],~[a,b]^b=[a,b]rangle $$$$K(s,l)=langle a,b|ab^s=b^la,~ba^s=a^lbrangle;$$and find the $n^{th}$-commutativity degree for each of them. Also we study the concept of $n$-abelianity on these groups, where $m,n,s$ and $l$ are positive integers, $m,ngeq 2$ and $g.c.d(s,...

Structural Properties of Inner Amenable Discrete Groups

Towards Connectivity for Codimension 2 Cycles: Infinitesimal Deformations

Let X be a smooth projective variety over an algebraically closed field k ⊂ C of characteristic zero, and Y ⊂ X a smooth complete intersection. The Weak Lefschetz theorem states that the natural restriction map H(X(C), Q) → H(Y (C), Q) on singular cohomology is an isomorphism for all i < dim(Y ). The Bloch-Beilinson conjectures on the existence of certain filtrations on Chow groups combined wit...