Classification of sextic curves in the Fano 3-fold $\mathcal{V}_5$ with rational Galois covers in ${\mathbb P}^3$
Quo-Shin Chi, Zhenxiao Xie, Yan Xu
TL;DR
The paper addresses the problem of classifying sextic curves in the Fano 3-fold ${oldsymbol{ u}}_5$ that admit rational Galois covers in ${oldsymbol{P}}^3$, showing a complex complex-dimension-2 moduli with a 1-dimensional exceptional transversal locus. It develops a construction framework based on binary polyhedral groups, invariants, and the decomposition of $V_noxtimes{f C}^2$ to produce 2-planes corresponding to independent 1-dimensional irreducible representations, then realizing Galois lifts via a Galois diagram involving a subgroup $G^* o S_4^*$. The paper systematically treats cyclic, dihedral, and exceptional groups $A_4$ and $S_4$, giving explicit parametrizations and divisor-theoretic criteria that separate generally ramified from exceptional transversal families, and shows that every sextic with a rational Galois lift arises from these constructions. This clarifies the moduli of sextics in ${oldsymbol{ u}}_5$ through their Galois covers in ${oldsymbol{P}}^3$, linking invariant theory, representation theory, and the geometry of the ${oldsymbol{ u}}_5$ orbit structure. The results have potential implications for understanding holomorphic curves in $G(2,5)$ and the algebraic structure of Galois lifts in projective spaces.
Abstract
In this paper, we classify sextic curves in the Fano $3$-fold $\bf \mathcal{V}_5$ (the smooth quintic del Pezzo $3$-fold) that admit rational Galois covers in the complex ${\mathbb P}^3$. We show that the moduli space of such sextic curves is of complex dimension $2$ through the invariants of the engaged Galois groups for the explicit constructions. This raises the intriguing question of understanding the moduli space of sextic curves in ${\mathcal V}_5$ through their Galois covers in ${\mathbb P}^3$.