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

Consider a projective algebraic variety $W$ that is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degree 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. The algorithm is deterministic and polynomial in $(dd')^n$ and the size of the input.