Determinantal characterization of higher secant varieties of minimal degree

TL;DR

This paper characterizes higher secant varieties of minimal degree using determinantal presentations, extending classical classifications and revealing structural decompositions of their embedding line bundles.

Contribution

It provides a determinantal characterization of higher secant varieties of minimal degree, generalizing the del Pezzo-Bertini classification and identifying special line bundle decompositions.

Findings

Higher secant varieties of minimal degree have determinantal presentations.

The classification extends the del Pezzo-Bertini results.

Embedding line bundles decompose into two line bundles in these cases.

Abstract

A variety of minimal degree is one of the basic objects in projective algebraic geometry and has been classified and characterized in many aspects. On the other hand, there are also minimal objects in the category of higher secant varieties, and their algebraic and geometric structures seem to share many similarities with those of varieties of minimal degree. We prove in this paper that higher secant varieties of minimal degree have determinantal presentation of two types, i.e. scroll type and Veronese type. Our result generalizes the del Pezzo-Bertini classification for varieties of minimal degree. Also, as a consequence, we show that for any smooth projective variety having higher secant variety of minimal degree, the embedding line bundle admits a special decomposition into two line bundles as so do those of the well-known examples: varieties of minimal degree, smooth del Pezzo varieties, Segre varieties and 2-Veronese varieties.