On Zariski problem

An Algebraic Approach to Zariski Problem

Zariski Problem (Cancellation of indeterminates) is settled affirmatively, that is, it is proved that : Let k be an algebraically closed field of characteristic zero and let n, m ∈ N. If R[Y1, . . . , Ym] ∼=k k[X1, . . . , Xn+m] as k-algebras, where Y1, . . . , Ym, X1, . . . , Xn+m are indetermoinates, then R ∼=k k[X1, . . . , Xn]. Zariski Problem is the following : Zariski Problem. Let k be an...

Zariski Pairs on Sextics Ii

We continue to study Zariski pairs in sextics. In this paper, we study Zariski pairs of sextics which are not irreducible. The idea of the construction of Zariski partner sextic for reducible cases is quit different from the irreducible case. It is crucial to take the geometry of the components and their mutual intersection data into account. When there is a line component, flex geometry (i.e.,...

A Note on Zariski Pairs

Definition. A couple of complex reduced projective plane curves C1 and C2 of a same degree is said to make a Zariski pair, if there exist tubular neighborhoods T (Ci) ⊂ P of Ci for i = 1, 2 such that (T (C1), C1) and (T (C2), C2) are diffeomorphic, while the pairs (P, C1) and (P , C2) are not homeomorphic; that is, the singularities of C1 and C2 are topologically equivalent, but the embeddings ...

Zariski Pairs on Sextics I

We study Zariski pairs of sextics which are distinguished by the Alexander polynomials. For this purpose, we present two constructive methods to produce explicit sextics of non-torus type with given configuration of simple singularities.

Zariski Dense Random Walks on Homogeneous Spaces