I am currently looking at *stratified Mukai flop*. Roughly speaking, this is a construction that, starting with a grassmannian $G$ inside a hyperkahler $X$ produces a birational manifold $X^\*$ (with a dual grassmannian $G^\*$ inside). The easiest example is the standard Mukai flop, namely the case in which the grassmannian $G$ is a projective space $\mathbb{P}^n$. 


I was looking for a concrete example of this situation.


In the case $G=\mathbb{P}^n$, I can construct such an example starting from a rational curve $C\cong\mathbb{P}^1$ inside a $K3$ surface $S$. The Hilbert scheme of points then provides $C^{[n]}\cong\mathbb{P}^n\subset S^{[n]}$. 


What about a generic grassmannian $G=G(k,n)$ with $k\neq1,n-1$? 


Thank you very much for any comments, remarks and answers.



Grassmannian inside a hyperkahler manifold
This is just the general case of the example from my comment above. Let $S\subset \mathbb{P}^g\_{\mathbb{C}}$ be a K3 surface of degree $2g-2$ that contains no curve that spans a $\mathbb{P}^r$ with $r<g-1$. Any such curve would be contained in (many) hyperplane sections of $S$, hence would have degree $< 2g-2$. So if $\text{Pic}(S)$ is generated by $[\mathcal{O}\_{\mathbb{P}^g}(1)|\_S]$, then $S$ contains no such curve.


Now let $G$ be the Grassmannian of dimension $2g-2$ parameterizing linear subspace $\Lambda\subset \mathbb{P}^g$ of dimension $g-2$. There is a morphism $G \to \text{Hilb}^{2g-2}\_{S/\mathbb{C}}$ by $[\Lambda] \mapsto [\Lambda \cap S]$. Since $\text{Hilb}^{2g-2}\_{S/\mathbb{C}}$ has dimension $4g-4$, $G$ is a maximal isotropic subvariety of $\text{Hilb}^{2g-2}\_{S/\mathbb{C}}$. So it does not factor through any projective spaces embedded in $\text{Hilb}^{2g-2}\_{S/\mathbb{C}}$. 


**Edit.** I originally had an incorrect comment here speculating on a modular interpretation of the Mukai flop.


