Compactifications of spaces of symmetric matrices and pointed Kontsevich spaces of isotropic Grassmannians

We study two closely related families of varieties arising from genus $0$ stable maps to the Lagrangian Grassmannian $\operatorname{LG}(n,2n)$. First, we construct the Kausz--type compactification $\mathcal {TL}_n$ of the space of symmetric matrices and give an explicit description of its birational geometry. Second, we realize $\mathcal {TL}_n$ as a general evaluation fiber in a Kontsevich space, and then exploit this modular interpretation to derive consequences for the birational geometry of the space of pointed conics $\overline{M}_{0,1}(\operatorname{LG}(n,2n),2)$. Analogous compactifications related to orthogonal Grassmannians are also presented.