Galois skew lines of smooth surfaces in P 3

Abstract

It is known that the degree of irrationality irr ( X ) of a smooth surface X of degree d ≥ 5 in P 3 is at least d – 2. In this paper, we study a Galois rational map f : X − − → P 2 of degree d – 2, and we show that the Galois group of f is a cyclic group, and f is defined by the family of lines intersecting two fixed skew lines. In addition, we give a necessary and sufficient condition that X has skew lines which give a Galois rational map of degree d – 2 by using automorphisms of X.