Hyperk\"ahler Manifolds of Curves in Twistor Spaces

We discuss hypercomplex and hyperk\"ahler structures obtained from higher degree curves in complex spaces fibring over ${\mathbb{P}}^1$.Comment: v4: added the missing assumption on dimension in Theorem 3.1

Birational automorphisms of a three-dimensional double quadric with an elementary singularity

It is proved that the group of birational automorphisms of a three-dimensional double quadric with a singular point arising from a double point on the branch divisor is a semidirect product of the free group generated by birational involutions of a special form and the group of regular automorphisms. The proof is based on the method of `untwisting' maximal singularities of linear systems.Comment: 18 page

A short proof of the G\"ottsche conjecture

We prove that for a sufficiently ample line bundle $L$ on a surface $S$, the number of $\delta$-nodal curves in a general $\delta$-dimensional linear system is given by a universal polynomial of degree $\delta$ in the four numbers $L^2,\,L.K_S,\,K_S^2$ and $c_2(S)$. The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of [PT3] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, G\"ottsche and Lehn. We are also able to weaken the ampleness required, from G\"ottsche's $(5\delta-1)$-very ample to $\delta$-very ample.Comment: 8 pages. Published versio