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.

  • $\begingroup$ For a quartic surface $S$ in $\mathbb{P}_{\mathbb{C}}^3$ that contains no lines, the Grassmannian $G$ of lines $L$ in $\mathbb{P}_{\mathbb{C}}^3$ embeds in $\text{Hilb}^4_{S/\mathbb{C}}$ via the rule $[L] \mapsto [L\cap S]$. $\endgroup$ – Jason Starr Mar 17 '16 at 19:06
  • $\begingroup$ By the way, in the example above, since $\text{Hilb}^4_{S/\mathbb{C}}$ is an $8$-dimensional hyperKaehler manifold, and since $G$ is $4$-dimensional, it is a maximal isotropic subvariety. So the embedding of $G$ does not factor through some projective space inside of $\text{Hilb}^4_{S/\mathbb{C}}$ of dimension strictly larger than $4$ (isotropic varieties have dimension less than or equal to $4$). $\endgroup$ – Jason Starr Mar 17 '16 at 19:08

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.

  • $\begingroup$ Thank you very much. This is exactly what I was looking for! $\endgroup$ – User3773 Mar 18 '16 at 11:48

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