On the variety parametrizing completely decomposable polynomials

TL;DR

This paper explores the geometric structure of the variety of completely decomposable polynomials, relating it to Grassmannians and Veronese varieties, and computes dimensions of certain secant varieties, challenging existing conjectures.

Contribution

It establishes new relationships between the variety of decomposable polynomials and classical algebraic varieties, and provides explicit dimension calculations and counterexamples to conjectures.

Findings

Computed dimensions of secant varieties to the decomposable polynomial variety.

Found a counterexample to a conjecture relating dimensions of secant varieties.

Analyzed intersections with Grassmannians and Veronese varieties.

Abstract

The purpose of this paper is to relate the variety parameterizing completely decomposable homogeneous polynomials of degree $d$ in $n+1$ variables on an algebraically closed field, called $\Split_{d}(\PP n)$, with the Grassmannian of $n-1$ dimensional projective subspaces of $\PP {n+d-1}$. We compute the dimension of some secant varieties to $\Split_{d}(\PP n)$ and find a counterexample to a conjecture that wanted its dimension related to the one of the secant variety to $\GG (n-1, n+d-1)$. Moreover by using an invariant embedding of the Veronse variety into the Pl\"ucker space, then we are able to compute the intersection of $\GG (n-1, n+d-1)$ with $\Split_{d}(\PP n)$, some of its secant variety, the tangential variety and the second osculating space to the Veronese variety.