# Deligne-Mumford moduli spaces and compactification of symmetric matrices

The real Deligne-Mumford moduli space $\bar M_{0,n+1}(\mathbb R)$ of stable genus zero curves with $n+1$ marked points is a compactification of the space of configurations of $n$ distinct ordered points on the real line modulo affine transformations. This space is a smooth projective variety and carries a natural action of the permutation group $S_n$. We can interpret the configurations of distinct points on the real line as diagonal $n\times n$ matrices with distinct eigenvalues, $S_n$ acting via conjugation with signed permutation matrices.

Question: What is the right compactification of the space of arbitrary symmetric $n\times n$ matrices with distinct eigenvalues modulo affine transformations? I am looking for a projective variety carrying a natural $SO(n)$-action, such that if we restrict to diagonal matrices, we recover $\bar M_{0,n+1}(\mathbb R)$ with the above $S_n$-action. Is this variety smooth? Does it have an interpretation in terms of curves with marked points?

• Can you please remind me how the group of affine transformations of $\mathbb{C}$ acts on the space of symmetric $n\times n$ matrices? – Jason Starr Jan 27 '14 at 18:47
• Everything is over $\mathbb R$. The group of affine transformations $x\mapsto ax+b$ of $\mathbb R$ acts on symmetric matrices by $S\mapsto aS+bI$, where $I$ is the identity matrix. – Konrad Schöbel Jan 27 '14 at 20:12