On the rank index of projective curves of almost minimal degree

2024/11/26 by Jung, Jaewoo, Moon, Hyunsuk, Park, Euisung
#14A25 #14H45 #14N05 #15A63 #16E45 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2411.17494

Abstract

In this article, we investigate the rank index of projective curves \mathscrC ⊂ ℙ^r of degree r+1 when \mathscrC = π_p (\mathscrC) for the standard rational normal curve \mathscrC ⊂ ℙ^r+1 and a point p ∈ ℙ^r+1 ∖ \mathscrC³. Here, the rank index of a closed subscheme X ⊂ ℙ^r is defined to be the least integer k such that its homogeneous ideal can be generated by quadratic polynomials of rank ≤ k. Our results show that the rank index of \mathscrC is at most 4, and it is exactly equal to 3 when the projection center p is a coordinate point of ℙ^r+1. We also investigate the case where p ∈ \mathscrC³ ∖ \mathscrC².

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