Galois skew lines of smooth surfaces in P3

Abstract

It is known that the degree of irrationality $\text{irr}(X)$ of a smooth surface X of degree $d\ge 5$ in ${\mathbb{P}}^3$ is at least d – 2. In this paper, we study a Galois rational map $f:X--\to {\mathbb{P}}^2$ 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.

KEYWORDS:

• Automorphism
• Galois rational cover
• smooth hypersurface

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 14J50
• Secondary: 12F10
• 14E20