Spaces of non-resultant systems of real bounded multiplicity determined by a toric variety

Abstract

For each field 𝔽 and positive integers 𝑚, 𝑛, 𝑑 with (𝑚, 𝑛) ≠ (1, 1), Farb and Wolfson defined the certain affine variety Poly^𝑑,𝑚_𝑛(𝔽) as generalizations of spaces first studied by Arnol'd, Vassiliev, Segal and others. As a natural generalization, for each fan Σ and 𝑟-tuple 𝐷 = (𝑑₁, …, 𝑑_𝑟) of positive integers, the authors also defined and considered a more general space Poly^𝐷,Σ_𝑛(𝔽), where 𝑟 is the number of one dimensional cones in Σ. This space can also be regarded as a generalization of the space Hol^*_𝐷(𝑆², 𝑋_Σ) of based rational curves from the Riemann sphere 𝑆² to the toric variety 𝑋_Σ of degree 𝐷, where 𝑋_Σ denotes the toric variety (over ℂ) corresponding to the fan Σ.

In this paper, we define a space Q^𝐷,Σ_𝑛(𝔽) (𝔽 = ℝ or ℂ) which is its real analogue and can be viewed as a generalization of spaces considered by Arnol'd, Vassiliev and others in the context of real singularity theory. We prove that homotopy stability holds for this space and compute the stability dimension explicitly.