Non-Secant Defectivity via Osculating Projections

TL;DR

This paper develops a new method to bound the non secant defectivity of projective varieties using osculating spaces and projections, with applications to Grassmannians showing they are not h-defective within certain bounds.

Contribution

Introduces a novel approach linking osculating space behavior to defectivity bounds, improving previous results for Grassmannians when r ≥ 4.

Findings

Established bounds for non secant defectivity of Grassmannians.

Demonstrated asymptotic non-defectivity for certain parameters.

Enhanced previous defectivity bounds by a logarithmic factor.

Abstract

We introduce a method to produce bounds for the non secant defectivity of an arbitrary irreducible projective variety, once we know how its osculating spaces behave in families and when the linear projections from them are generically finite. Then we analyze the relative dimension of osculating projections of Grassmannians, and as an application of our techniques we prove that asymptotically the Grassmannian $\mathbb{G}(r,n)$, parametrizing $r$-planes in $\mathbb{P}^n$, is not $h$-defective for $h\leq (\frac{n+1}{r+1})^{\lfloor\log_2(r)\rfloor}$. This bound improves the previous one $h\leq \frac{n-r}{3}+1$, due to H. Abo, G. Ottaviani and C. Peterson, for any $r\geq 4$.