2022/08/17 by Chi, Quo-Shin, Xie, Zhenxiao, Xu, Yan · 1 citation
#14H45 #14J45 #53C55 #Differential Geometry (math.DG) #FOS: Mathematics #Primary: 53C42 #Secondary: 14M15
paper · doi:10.48550/arxiv.2208.08525
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 V5 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 PSL2-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 V5 ramified at the PSL2-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.