Hodge numbers of a Fano eightfold of K3 type

TL;DR

This paper constructs a specific degeneration of a Fano eightfold of K3 type, determines its Hodge numbers, and explores its geometric properties, including Picard rank and connections to Hilbert schemes of K3 surfaces.

Contribution

It provides an explicit semistable degeneration of a Fano eightfold of K3 type and computes its Hodge numbers, revealing new geometric insights.

Findings

Fano eightfold has Picard rank one

Explicit degeneration constructed

Connection to Hilbert schemes of K3 surfaces

Abstract

We construct an explicit semistable degeneration of a Fano eightfold of index three and deduce its Hodge numbers, in particular we show that it has Picard rank one. The Fano variety is of K3 type and it is defined as a connected component of the fixed locus of a suitable antisymplectic involution on a projective variety that is deformation equivalent to the Hilbert scheme of eight points on a K3 surface. We also obtain a description of a projective model of the Hilbert square of a K3 surface of genus eight in terms of secant lines to the surface.