The Configuration Space X(2,5)

Abstract

The configuration space X(4) = X(2,4) was studied in Chapter I; it is a simple space. Indeed it is the unique configuration space which is 1-dimensional. In the next chapter we study modular interpretations of the configuration spaces X(2,n)(n ≥ 5). Recall that the configuration space of n distinct points on the line is given as

$$X\left( {2,n} \right) = GL\left( 2 \right)\backslash {M^ * }\left( {2,n} \right)/{H_n} $$

, where M*(2,n) is the space of 2 × n complex matrices such that any 2×2-minor does not vanish. Since every point x ∈ X(2,n) can be represented by

$$\left( {\begin{array}{*{20}{c}}1 & 0 & 1 & 1 & {...} & 1 \\ 0 & 1 & 1 & {{x_1}} & {...} & {{x_{n - 3}}} \\ \end{array} } \right) $$

, the space can be seen as an open set in ℂ^n-3:

$$X\left( {2,n} \right) \cong \left\{ {\left( {{x_1},...,{x_{n - 3}}} \right) \in {\mathbb{C}^{n - 3}}|{x_j} \ne 0,1,{x_k}\left({j \ne k} \right)} \right\} $$

.