A deterministic polynomial-time algorithm for the first Bertini theorem. II

Consider a projective algebraic variety $W$ which is an irreducible component of a set of all common zeroes of a family of homogeneous polynomials of degrees less than $d$ in $n+1$ variables in zero-characteristic. Consider a linear system on $W$ given by homogeneous polynomials of degree $d'$. Under the conditions of the first Bertini theorem for $W$ and this linear system we show how to construct an irreducible divisor in general position from the statement of this theorem. This algorithm is deterministic and polynomial in $(dd')^n$ and the size of input. This paper is the second in the tree-part series.