I am looking for references which might study the following problem in non-trivial cases:


Let $X\subset\mathbb{P}^n\_{\mathbb{C}}$ be a general hypersurface of degree $d$ and consider $G:=G(k+1,n+1)$ the Grassmannian of $k$-planes $\mathbb{P}^k\subset\mathbb{P}^n$. We get an induced family $\pi:Y\to G$ with fibre $Y\_g=\mathbb{P}^k\_g\cap X$ for $g\in G$.
Give a description, as a degeneracy locus of some morphism of vector bundles or otherwise (at least of the classes in cohomology), of the loci in $G$ where $\pi$ has fibres of a prescribed singularity.


In particular I am interested in the case where $k=2$ and the intersection $\mathbb{P}^2\cap X$ is as degenerate as possible, i.e., $d$ times a line.



Loci of singular plane sections of a generic hypersurface
Consider the flag variety $F = Fl(k,k+1;V)$ and let
$$
U\_k \subset U\_{k+1} \subset V \otimes \mathcal{O}
$$
be the tautological flag of subbundles. We have an exact sequence
$$
0 \to L \to U\_{k+1}^\vee \to U\_k^\vee \to 0,
$$
where $L$ is a line bundle (in fact, $L \cong \det(U\_{k+1}^\vee) \otimes \det(U\_k)$). It induces a natural embedding
$$
L^d \to S^dU\_{k+1}^\vee
$$
and the quotient bundle
$$
E := S^dU\_{k+1}^\vee / L^d
$$
has a filtration with factors $L^{d-i} \otimes S^{i}U\_k^\vee$, $1 \le i \le d$). The equation of the hypersurface gives a section of $S^dU\_{k+1}^\vee$, hence a section of its quotient bundle $E$. Let
$$
M \subset F
$$
be the zero locus of this section. Then the locus you are interested in is the image of $M$ under the natural projection $F \to G$. Note that the structure sheaf of $M$ has a Koszul resolution (with terms $\wedge^p E^\vee$), and pushing it forward and using the filtration of $E$ and the Borel-Bott-Weil theorem one can construct a resolution for the structure sheaf of the image of $M$.


