Rational curves with prescribed branches (Q1358601)

The only currently available means for describing groups of birational automorphisms (and, in a more general situation, birational correspondences) of multidimensional Fano varieties is the technique of greatest singularities, developed for three-dimensional varieties by \textit{V. A. Iskovskikh} and \textit{Yu. I. Manin} [Math. USSR, Sb. 15(1971), 141-166 (1972); translation from Mat. Sb., Nov. Ser. 86(128), 140-146 (1971; Zbl 0222.14009)] based on some ideas dating back to \textit{M. Noether} and \textit{G. Fano}. The subsequent development and generalization of this method demonstrated the deficiencies of the employed trial class procedure, which works worse the higher the variety degree. In order to avoid these difficulties, a new version of the technique was proposed by \textit{A. V. Pukhlikov} [in: Proc. Steklov Inst. Math. 208, 244-254 (1995); translation from Tr. Mat. Inst. Steklova 208, 278-289 (1995; Zbl 0880.14020)], based on the trial surface procedure. However, the key statement on the existence of a family of limited-degree rational curves with the prescribed branches was left unproved in Pukhlikov's paper cited above. The present paper is aimed at filling this gap.