## Online, hosted by the ICMS Edinburgh, 28th May 2020

**Arend Bayer.** Hyperkähler varieties from Fano varieties via stability conditions.

I will explain how to obtain (families of) Hyperkaehler varieties from (families of) some types of Fano varieties. The construction currently applies to cubic fourfolds, and Gushel-Mukai fourfolds. It goes via moduli spaces of stable objects for stability condition on their “Kuznetsov categories”, a certain component of the derived category of these Fano varieties. This, for example, gives two infinite collections of locally complete unirational families of polarised Hyperkaehler varieties. Conversely, I will explain some applications to the geometry of cubic fourfolds.

**Margherita Lelli-Chiesa.** Genus two curves on abelian surfaces.

Let (S,L) be a general (d_1,d_2)-polarized abelian surfaces. The minimal geometric genus of any curve in the linear system |L| is two and there are finitely many curves of such genus. In analogy with Chen’s results concerning rational curves on K3 surfaces, it is natural to ask whether all such curves are nodal. In the seminar I will prove that this holds true if and only if d_2 is not divisible by 4. In the cases where d_2 is a multiple of 4, I will construct curves in |L| having a triple, 4-tuple or 6-tuple point, and show that these are the only types of unnodal singularities a genus 2 curve in |L| may acquire. This is joint work with A. L. Knutsen.