Moduli of Conics on General Plucker Linear Sections of Grassmannians

Abstract: Let $V$ be a complex vector space of dimension $n$, let $G=\operatorname{Gr}(k,V)$ be Pl$ü$cker-embedded, and let $Y_E=G\cap\mathbf{P}(E<sup>\perp)$ be a general codimension-$r$ linear section. We study the open Hilbert scheme $R_2(Y_E)$ of smooth conics. A conic of minimal flag-envelope type determines a flag in $\operatorname{Fl}(k-2,k+2;V)$ and a plane in a naturally associated rank-six bundle, yielding a uniform relative-Grassmannian model. Using a relative Schubert general-position argument, we prove that for $0\le r\le3$, $R_2(Y_E)$ is nonempty, smooth, irreducible, and rational of dimension $2n+k(n-k)-3-3r$, and is birational to $$\operatorname{Fl}(k-2,k+2;V)\times\operatorname{Gr}(3,6-r)$$. For $r\ge4$, the minimal-envelope locus is related, over the rank-three stratum, to a rank-three degeneracy locus with a natural incidence model. In expected dimension zero, equivariant localization gives $1225$ and $1176$ conics on general linear sections of $\operatorname{Gr}(2,7)$ and $\operatorname{Gr}(3,6)$, respectively.