On the rank index of some quadratic varieties

TL;DR

This paper introduces methods to compute the rank index of quadratic varieties and determines its values for classical varieties like scrolls, del Pezzo, Segre, and Grassmannians, expanding understanding of their ideal generation.

Contribution

It provides new techniques for calculating the rank index and applies them to various classical projective varieties, broadening the scope of known results.

Findings

Veronese embeddings have rank index 3 in characteristic not 2 or 3

Calculated rank index for rational normal scrolls, del Pezzo, Segre, and Grassmannians

Enhanced understanding of quadratic ideal generation in classical varieties

Abstract

Regarding the generating structure of the homogeneous ideal of a projective variety $X \subset \mathbb{P}^r$, we define the rank index of $X$ to be the smallest integer $k$ such that $I(X)$ can be generated by quadratic polynomials of rank at most $k$. Recently it is shown that every Veronese embedding has rank index $3$ if the base field has characteristic $\ne 2, 3$. In this paper, we introduce some basic ways of how to calculate the rank index and find its values when $X$ is some other classical projective varieties such as rational normal scrolls, del Pezzo varieties, Segre varieties and the Pl\"{u}cker embedding of the Grassmannian of lines.