Mutsuo Oka

Alexander Polynomial of Sextics

Alexander polynomials of sextics are computed in the case of sextics with only simple singularities or sextics of torus type with arbitrary singularities. We will show that for irreducible sextics, there are only 4 possible Alexander polynomials: (t2-t+1)j, j=0,1,2,3. For the computation, we use the method of Libgober and Loeser-Vaquie [5, 7] and the classification result in our previous papers [12, 11].

Flex Curves and their Applications

In this paper, we define the notion of the flex curve F(ℙ)(f; P) at a nonsingular point P of a plane curve Ca. We construct interesting plane curves using a cyclic covering transform, branched along F(ℙ)(f; P). As an application, we show the moduli space of projective curves of degree 12 with 27 cusps has at least three irreducible components. Simultaneously, we give an example of Alexander-equivalent Zariski pair of irreducible curves.

FUNDAMENTAL GROUP OF SEXTICS OF TORUS TYPE

We show that the fundamental group of the complement of any irreducible tame torus sextics in ℙ2 is isomorphic to ℤ2 * ℤ3 except one class. The exceptional class has the configuration of the singularities {C 3,9, 3A2} and the fundamental group is bigger than ℤ2 * ℤ3. In fact, the Alexander polynomial is given by (t 2 −t+1)2. For the proof, we first reduce the assertion to maximal curves and then we compute the fundamental groups for maximal tame torus curves.

Examples of Algebraic Surfaces with q = 0 and p

Publisher Summary This chapter presents the examples of algebraic surfaces with q = 0 and pg ≤ 1, which are locally hypersurfaces. It discusses a canonical way of the compactification M of Ma through the toroidal embedding theory and three algebraic surfaces M1, M2, M3 with q = pq = 0. M1 and M3 are known as an Enriques surfaces and a Godeaux surface, respectively. The chapter also presents a theorem that states that the exceptional divisor E(P) is a smooth compactification of M.