The problem of spaces of resultants related to rational curves on a toric variety

2017/12/25 月 16:00 - 17:30

For positive integers $d, m, n$ and a field $F$, let $\mathrm{Poly}(d;m,n)(F)$ denote the space of all $m$-tuples of $F$-coefficients monic polynomials of the same degree $d$ such that these polynomials have no common root in the algebraic closure of $F$ of multiplicity greater or equal to $n$. These spaces were defined and studied by B. Farb and J. Wolfson as a generalization of the spaces first studied by V. Arnold, V. Vassiliev and G. Segal in different contexts.

As the first step of this talk, the author studies the homotopy type of these spaces when the field $F$ is the complex field $\mathbb{C}$. Note that this space is homeomorphic to the space of based rational curves on the $(m-1)$-dimensional complex projective space if $F=\mathbb{C}$ and $n=1$ with $m>1$. Moreover, by the classical theory of resultants we see that the space $\mathrm{Poly}(d;m,n)(F)$ is an affine variety over $F$. So it may be interesting to consider the generalization of these spaces defined by using the resultants induced from the rational curves on a general compact toric variety $X$.

The second purpose of this talk is to consider this problem from the point of view of toric topology.

This talk is based on the joint work with A. Kozlowski (University of Warsaw).