Sextic variety as Galois closure variety of smooth cubic

TL;DR

This paper characterizes certain algebraic varieties as Galois closure varieties of smooth cubics, linking geometric properties with Galois group actions in algebraic geometry.

Contribution

It provides a criterion for when a variety with specific divisor properties is a Galois closure of a smooth cubic, establishing a new connection between geometry and Galois theory.

Findings

Characterization of Galois closure varieties as smooth cubic projections

Conditions for a variety to admit a D_6-Galois embedding

Link between divisor properties and Galois group actions

Abstract

Let V be a nonsingular projective algebraic variety of dimension n. Suppose there exists a very ample divisor D such that D^n=6 and dim H^0(V, O(D))=n+3. Then, (V, D) defines a D_6-Galois embedding if and only if it is a Galois closure variety of a smooth cubic in P^{n+1} with respect to a suitable projection center such that the pull back of hyperplane of P^n is linearly equivalent to D.