Generalizing M. Dale's results from secants to joins

TL;DR

This paper extends Magnar Dale's results on secant varieties of curves to the broader context of embedded joins of varieties, providing methods to compute related degrees and generalize existing lemmas.

Contribution

It generalizes Dale's secant variety results to embedded joins, offering detailed proofs and a new approach to calculating degrees of canonical maps.

Findings

Extended Dale's lemmas to embedded joins

Provided formulas for degree calculations of canonical maps

Enhanced understanding of secant and join relationships

Abstract

Magnar Dale's paper ``Terracini's lemma and the secant variety of a curve" contains various facts about secant varieties, nearly all of whose proofs can immediately be extended to the situation of embedded joins of varieties. This note provides the necessary details on how to do so, and as an application shows how to use this information to calculate the degree of the canonical map from the ruled join down to the embedded join.