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Complete Homogeneous Varieties via Representation Theory

Given an algebraic variety $X\subset\mathbb{P}^N$ with stabilizer $H$, the quotient $PGL_{N+1}/H$ can be interpreted a parameter space for all $PGL_{N+1}$-translates of $X$. We define $X$ to be a $\textit{homogeneous variety}$ if $H$ acts on it transitively, and satisfies a few other properties, such as $H$ being semisimple. Some examples of homogeneous varieties are quadric hypersurfaces, rational normal curves, and Veronese and Segre embeddings. In this case, we construct new compactifications of the parameter spaces $PGL_{N+1}/H$, obtained compactifying $PGL_{N+1}$ to the classically known space of \textit{complete collineations}, and taking the G.I.T. quotient by $H$, and we will call the result space of \textit{complete homogeneous varieties}; this extends the same construction for quadric hypersurfaces in [36]. We establish a few properties of these spaces: in particular, we find a formula for the volume of divisors that depends only on the dimension of $H$-invariants in irreducible representations of $SL_{N+1}$. We then develop some tools in invariant theory, combinatorics and spline approximation to calculate such invariants, and carry out the entire calculations for the case of $SL_2$-invariants in irreducible representations of $SL_4$, that gives us explicit values for the volume function in the case of $X$ being a twisted cubic. Afterwards, we focus our attention on the case of twisted cubics, giving a more explicit description of these compactifications, including the relation with the previously known moduli spaces. In the end, we make some conjectures about how the volume function might be used in solving some enumerative problems.