Linear orbits of smooth quadric surfaces
Franquiz Caraballo Alba
TL;DR
The paper studies the action of $\mathbb{P}\mathrm{GL}(n+1)$ on degree $d$ hypersurfaces in $\mathbb{P}^n$ and introduces the predegree polynomial $\mathcal{P}_X(t)$, encoding refined orbit-incidence data via the graph of the induced rational map. It formalizes the predegree via multidegrees and Segre classes, showing that partial resolutions of the indeterminacy suffice to compute the coefficients, and interprets them combinatorially as counts of translates containing general points. Focusing on quadrics, it proves that the leading coefficient equals the degree of the stabilizer $\mathrm{PO}(n+1)$, and computes the full predegree polynomial for smooth quadrics in $\mathbb{P}^3$, while also giving a general leading-term formula in $\mathbb{P}^n$. Finally, it constructs a complete resolution of the rational map in the $n=3$ case via a sequence of blow-ups along $Z_1\cap Z_2$ and related centers, enabling potential computation of further invariants beyond the predegree. The results yield new enumerative data for the action of projective linear transformations on quadrics and illustrate a concrete resolution framework for higher-dimensional orbit problems.
Abstract
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.