The Lefshetz vanishing cycles on a projective nonsingular plane curve

Abstract

We deal in this paper with an explicit construction of Lefschetz vanishing cycles, which arise from a pencil of plane algebraic curves. Our main goal is Theorem 1, which helps to prove the Topological Decomposit ion Theorem for algebraic surfaces in CP 3 (Mandelbaum and Moishezon [7]). We consider a polynomial equation F(x,y)=O, which defines an affine plane algebraic curve C. We assume that the closure of C in CP 2 is nonsingular, and not tangent to the line at infinity. We consider a pencil of curves V~ :F(x, y) =s. There is a finite set Z of critical values of s, such that V~ has singularities. The noncritical values of s are called regular values. By our assumptions s = 0 is a regular value. Let c be a critical value, such that ~ possesses only one nondegenerate singular point P. Then e is called a simple critical value. We can choose analytic local coordinates u(x, y), v(x,y) in a neighbourhood U of P in the plane, such that F(x, y) = c + u(x, y)2 + v(x, y)2 in U and (u, v) (P) = (0, 0). We can also choose a neighbourhood D of c in the parameter plane, such that Vsc~ U has a 1-sphere as a deformation retract for each seD. This sphere represents the Lefschetz vanishing cycle on Vs corresponding to c. We will choose a special representation of the Lefschetz cycle. Let ),(t), 0 < t < 1, be a simple path in the parameter plane, such that 7(0)=0, ?(1)=c and ?(t) is not a critical value for t # 1. Then there exists a continuous deformation I,: Vo~ Vm),