Secant degeneracy index of the standard strata in the space of binary forms

TL;DR

This paper investigates the secant degeneracy index of standard strata in the space of binary forms, revealing minimal projectively dependent point configurations within these algebraic varieties.

Contribution

It introduces the secant degeneracy index for each stratum and its closure, providing new insights into the geometric structure of binary forms.

Findings

Defined the secant degeneracy index for strata in binary forms

Analyzed minimal configurations of projectively dependent points

Explored differences between strata and their closures

Abstract

The space $Pol_d\simeq \bC P^d$ of all complex-valued binary forms of degree $d$ (considered up to a constant factor) has a standard stratification, each stratum of which contains all forms whose set of multiplicities of their distinct roots is given by a fixed partition $\mu \vdash d$. For each such stratum $S_\mu,$ we introduce its secant degeneracy index $\ell_\mu$ which is the minimal number of projectively dependent pairwise distinct points on $S_\mu$, i.e., points whose projective span has dimension smaller than $\ell_\mu-1$. In what follows, we discuss the secant degeneracy index $\ell_\mu$ and the secant degeneracy index $\ell_{\bar \mu}$ of the closure $\bar S_\mu$.