Birational geometry of the moduli space of rational curves of degree 4

Ch. Lehn, M. Lehn, Ch. Sorger and D. van Straten constructed a family of irreducible holomorphic symplectic manifolds via a contraction of the compactified moduli space $M_3(Y)$ of rational curves of degree 3 on a smooth cubic fourfold Y.

This suggests that also the moduli space $M_4(Y)$ could be connected to a family of holomorphic symplectic manifolds.

In order to understand $M_4(Y)$, we study the Hilbert scheme of curves on $P^4$ with Hilbert polynomial 4n+1, a moduli space of Kronecker modules, a moduli space of semi-stable sheaves and their relations.