Effective estimates for unirationality (Q807705)

Let k be an algebraically closed field. All varieties are defined over k. The paper deals with unirational varieties V that are complete intersections of a fixed number of hypersurfaces of a projective space \({\mathbb{P}}_ k^ r\) of degrees \(d_ 1,...,d_ m\). It is known that there exists a number \(r_ 0=r_ 0(d_ 1,...,d_ m)\) such that if \(r>r_ 0\) then the generic complete intersection \(V\subseteq {\mathbb{P}}_ k^ r\) is unirational over k. - The main concern is to find an explicit formula for \(r_ 0\). The author gives a formula that sharpens the known value for \(r_ 0\). The determination of the lowest possible \(r_ 0\) remains an open problem.