A GENERALIZATION OF LEFSCHETZ-ZARISKI THEOREM ON FUNDAMENTAL-GROUPS OF ALGEBRAIC-VARIETIES

Let X and Y be the complements of divisors on non-singular irreducible closed subvarieties (X) over bar and (Y) over bar in P-n, respectively. Suppose that dim X + dim Y greater than or equal to n + 2. Then, for a general g is an element of PGL (n + 1), the natural homomorphism pi(1)(g(X) boolean AND Y) --> pi(1)(Y) induces a surjection from Ker (pi(1)(g(X) boolean AND Y) --> pi(1)(g(X))) onto pi(1)(Y), and there is a surjection to its kernel from the cokernel of pi(2)(X) --> pi(2)(P-n). In particular, if E subset of P-n is a hypersurface and 2 . dim (X) over bar > n, then pi(1)(g(X)\E) is isomorphic to pi(1)(P-n\E) for a general g is an element of PGL (n + 1).