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# Linear orbits of smooth quadric surfaces

Mar 2023

The linear orbit of a degree d hypersurface in \$\mathbb{P}^n\$ is its orbitunder the natural action of PGL(n+1), in the projective space of dimension \$N=\binom{n+d}{d} - 1\$ parameterizing such hypersurfaces. This action restrictedto a specific hypersurface \$X\$ extends to a rational map from theprojectivization of the space of matrices to \$\mathbb{P}^N\$. The class of thegraph of this map is the predegree polynomial of its correspondinghypersurface. The objective of this paper is threefold. First, we formallydefine the predegree polynomial of a hypersurface in \$\mathbb{P}^n\$, introducedin the case of plane curves by Aluffi and Faber, and prove some results in thegeneral case. A key result in the general setting is that a partial resolutionof said rational map can contain enough information to compute the predegreepolynomial of a hypersurface. Second, we compute the leading term of thepredegree polynomial of a smooth quadric in \$\mathbb{P}^n\$ over analgebraically closed field with characteristic 0, and compute the othercoefficients 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 ofthe previously mentioned rational map which contains enough information toobtain the invariant. Third, we provide a complete resolution of the rationalmap, which in principle could be used to compute more refined invariants.

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