I'm looking for an example of the following:

$X$ is a hyperkahler fourfold (deformation equivalent to $K3^{[2]}$); $D$ is a movable divisor on $X$ with $D^4=0$; and the base locus of $D$ is a smooth rational surface $\not \simeq \mathbb P^2$.

Note that it is relatively easy to construct such divisors with base locus equal to $\mathbb P^2$: $X=S^{[2]}$ for a degree 2 K3 surface works (Here $X$ contains a $\mathbb P^2$ which can be flopped, and the resulting birational model has a lagrangian fibration). But finding such examples with e.g. blow-ups of $\mathbb P^2$ or $\mathbb P^1\times \mathbb P^1$ seems a bit more difficult.

Any ideas on how to construct such examples would be greatly appreciated!

  • $\begingroup$ This is not quite what you are asking, but you can modify that construction to make an example where the base locus is a union of rational surfaces, some components of which equal $\mathbb{P}^1\times \mathbb{P}^1$. The construction is too long for a comment, so I will write is as an "answer". $\endgroup$ – Jason Starr Jun 28 '18 at 19:05

You can make more examples using the natural $K$-equivalence between the Hilbert scheme of a $K3$ Deligne-Mumford stack to the corresponding Hilbert scheme of the crepant resolution of its coarse moduli space. The example in your post arises this way when the stack has a single stacky point whose inertia group is cyclic of order $2$, so that the crepant resolution has a $(-2)$-curve whose $\text{Sym}^2$ is the flopped $\mathbb{P}^2$ (I assume that your example arises by flopping $\text{Sym}^2$ of one of the preimages of a bitangent line to the plane sextic, which preimage is the $(-2)$-curve in my interpretation). The following example is the next case, where the stack has a single stacky point whose inertia group is cyclic of order $3$, so that the crepant resolution is a chain of two $(-2)$-curves.

Let $S$ be a smooth quartic K3 surface in $\mathbb{P}^3$ that is a general member of the $5$-dimensional projective linear system of quartics that contain a specified line $L$ as well as a specified disjoint curve $C=M\cup N$ that is a union of two lines intersection transversally at one point $p$. The plane cubics residual to $L$ form an elliptic fibration on $S$, $$\pi:S\to \mathbb{P}^1.$$ This induces an Abelian fibration on $S^{[2]}$. Let $D'$ denote the pullback to $S^{[2]}$ of the hyperplane class on $\text{Sym}^2(\mathbb{P}^1) = \mathbb{P}^2$.

The curve $M\cup N$ can be contracted to form a normal projective surface $S'$ with a single $A_2$-singularity $q$. Explicitly, the $2$-plane spanned by $M$ and $N$ intersects $S$ in the union of $M$, of $N$, and of a plane conic $C$. The linear system of sections of $\mathcal{O}_{\mathbb{P}^3}(2)$ that contain $C$ induces a closed immersion of $S'$ as a sextic surface in $\mathbb{P}^4$ (probably there is a direct construction of this sextic suface).

There is a smooth, proper Deligne-Mumford stack $\mathcal{S}$ whose coarse moduli space is a $1$-morphism, $$\nu:\mathcal{S}\to S',$$ that is an isomorphism over the open subset $U:=S'\setminus\{q\}$, and whose inertia group over $q$ is cyclic of order $3$. The Hilbert scheme $X$ of length $2$ closed substacks of $\mathcal{S}$ has a unique connected component that contains $U^{[2]}$. This is a projective hyperkähler $4$-fold that is K-equivalent, hence deformation equivalent (by Huybrechts), to $S^{[2]}$.

The surface in $X$ that is the fundamental locus of the K-equivalence with $S^{[2]}$ equals the punctual Hilbert scheme of $\mathcal{S}$ for the point $q$. Because $K$-equivalences preserve classes in the Grothendieck ring of varieties, etc., we know that this fundamental locus has the same class as the fiber over $2[q]$ for the Hilbert-Chow morphism, $$FC:S^{[2]}\to \text{Sym}^2(S').$$ In particular, the fundamental locus has $3$ irreducible components, each of which is a rational surface.


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