Fano 3-folds and classification of constantly curved holomorphic $2$-spheres of degree $6$ in the complex Grassmannian $G(2,5)$

TL;DR

This paper classifies and analyzes the rarity of constantly curved holomorphic 2-spheres of degree 6 in the complex Grassmannian G(2,5), revealing that such curves are mostly nonhomogeneous and establishing their moduli space structure.

Contribution

It provides a detailed classification of constantly curved sextic curves in G(2,5), showing their rarity and describing the structure of their moduli space, including explicit examples.

Findings

Constantly curved sextic curves are rare and mostly nonhomogeneous.

The moduli space of these curves is 2-dimensional and semialgebraic.

Most such curves are unramified and not equivalent to the Veronese curve.

Abstract

Up to now the only known constantly curved sextic curve, i.e., holomorphic 2-sphere of degree 6, in the complex $G(2,5)$ has been the first associated curve of the Veronese curve of degree 4, which indicates that such curves are rare to find. Exploring the rich interplay between the ramification of harmonic sequences in differential geometry and algebro-geometric properties of projectively equivalent Fano 3-folds of index 2 and degree 5, we invoke the moduli space structure of sextic curves in the Fano 3-fold often referred to as $V_5$ to confirm the rarity of constancy of curvature, by establishing that the harmonic sequence of a generic sextic curve in $G(2, 5)$ is totally unramified. This paper proposes to investigate from the Galois viewpoint the way ramification can appear in relation to the constancy of curvature among nongeneric sextic curves in $G(2, 5)$. We prove through elaborate $PSL_2$-transvectant and engaged unitary analyses that, up to the ambient unitary equivalence, the moduli space of constantly curved sextic curves in $G(2,5)$ that are $GL(5,{\mathbb C})$-equivalent to those in $V_5$ ramified at the $PSL_2$-invariant 1-dimensional singular locus somewhere, is semialgebraic of dimension 2 all members of which barring the above Veronese curve are nonhomogeneous. Many explicit examples can be constructed.