# Universal nondegenrate map over the space of complete linear maps

Let $E$ and $F$ be vector spaces and assume for simplicity that $\dim E = \dim F = n$.

Recall that the moduli space of complete linear maps from $E$ to $F$ is the space $M(E,F)$ parameterizing all collections $(f_1,\dots,f_r)$, where

$f_1$ is a nonzero linear map $E_1 := E \to F =: F_1$ defined up to a constant;

$f_2$ is a nonzero linear map $E_2 := Ker f_1 \to Coker f_2 =: F_2$ defined up to a constant;

$\dots$

$f_r$ is a nondegenerate map $E_r := Ker f_{r-1} \to Coker f_{r-1} =: F_r$.

This space can be obtained by taking ${\mathbb P}(Hom(E,F))$ and then blowing up consecutively degeneration loci of the universal map starting from the smallest one.

Denote $E'_i = E_i/E_{i+1}$, $F'_i = Ker(F_i \to F_{i+1})$. Each $f_i$ factors through $f'_i:E'_i \to F'_i$ and that $f'_i$ is nondegenerate. Note that over each point of $M(E,F)$ with given $r$ we have a nondegenerate map of $n$-dimensional vector spaces. $$\oplus f'_i: \oplus E'_i \to \oplus F'_i$$

## Question.

Is it possible to define vector bundles ${\mathcal E}$ and ${\mathcal F}$ over $M(E,F)$ and a morphism $\varphi:{\mathcal E} \to {\mathcal F}$ which coincides with the above map at each point of $M(E,F)$?

The natural candidates for ${\mathcal E}$ and ${\mathcal F}$ are appropriate Hecke transformations of $E$ and $F$ along the exceptional divisors of the blowup $M(E,F) \to {\mathbb P}(Hom(E,F))$. But the construction of $\phi$ is not clear for me.

Also I am interested in the analogous space of complete quadrics for which I have the same question --- is it possible to define a universal nondegenerate quadric in an appropriate vector bundle over this space?

-