Moduli of Conics on General Plucker Linear Sections of Grassmannians

Abstract

Let $G=\operatorname{Gr}(k,V)$ be a complex Grassmannian in its Pl$\"u$cker embedding, and let $Y_E$ be a general codimension-$r$ linear section. We study the open Hilbert scheme $R_2(Y_E)$ of smooth conics using the kernel and span of a conic. These define a flag variety $B=\operatorname{Fl}(k-2,k+2;V)$ and a relative Grassmannian of three-planes in a rank-six bundle. For $0\le r\le3$, we prove that $R_2(Y_E)$ is nonempty, smooth, irreducible and rational of dimension $e=2\dim V+k(\dim V-k)-3-3r$, with birational model $B\times\operatorname{Gr}(3,6-r)$. For $r\ge4$, a general-position argument gives a componentwise birational correspondence with a rank-three degeneracy locus and two smooth projective incidence resolutions. Every component meets the locus of smooth minimal-envelope conics; no additional density assumption is needed for general $E$. If $0\le e\le r$, the degeneracy locus itself is smooth. In dimension zero we obtain a top Chern class formula, recovering the known counts $1225$ and $1176$ by exact localization.