Tangential varieties of Segre-Veronese surfaces are never defective

TL;DR

This paper determines the dimensions of all secant varieties to the tangential varieties of Segre-Veronese surfaces, using a novel degeneration technique to compute Hilbert functions of special schemes.

Contribution

It introduces a new degeneration method to compute Hilbert functions, enabling the complete classification of secant varieties of tangential Segre-Veronese surfaces.

Findings

All secant varieties have the expected dimension, showing no defectiveness.

The new degeneration technique simplifies Hilbert function computations.

The results provide a comprehensive understanding of the secant varieties in this context.

Abstract

We compute the dimensions of all the secant varieties to the tangential varieties of all Segre-Veronese surfaces. We exploit the typical approach of computing the Hilbert function of special 0-dimensional schemes on projective plane by using a new degeneration technique.