Linear orbits of smooth quadric surfaces

The linear orbit of a degree d hypersurface in $\mathbb{P}^n$ is its orbit under the natural action of PGL(n+1), in the projective space of dimension $N =\binom{n+d}{d} - 1$ parameterizing such hypersurfaces. This action restricted to a specific hypersurface $X$ extends to a rational map from the projectivization of the space of matrices to $\mathbb{P}^N$. The class of the graph of this map is the predegree polynomial of its corresponding hypersurface. The objective of this paper is threefold. First, we formally define the predegree polynomial of a hypersurface in $\mathbb{P}^n$, introduced in the case of plane curves by Aluffi and Faber, and prove some results in the general case. A key result in the general setting is that a partial resolution of said rational map can contain enough information to compute the predegree polynomial of a hypersurface. Second, we compute the leading term of the predegree polynomial of a smooth quadric in $\mathbb{P}^n$ over an algebraically closed field with characteristic 0, and compute the other coefficients in the specific case n = 3. In analogy to Aluffi and Faber's work, the tool for computing this invariant is producing a (partial) resolution of the previously mentioned rational map which contains enough information to obtain the invariant. Third, we provide a complete resolution of the rational map in the case $n = 3$, which in principle could be used to compute more refined invariants.