Some degenerations of $G_2$ and Calabi-Yau varieties

TL;DR

The paper introduces a degeneration of the $G_2$ variety, explores its geometric properties, and constructs transitions between Calabi-Yau threefolds, linking these degenerations to K3 surfaces with specific properties.

Contribution

It defines a new degenerate variety $ ilde{G}_2$, analyzes its geometric structure, and connects it to Calabi-Yau transitions and K3 surfaces with particular Picard number and genus.

Findings

$ ilde{G}_2$ is a degeneration of the $G_2$ variety.

Degenerations lead to toric Gorenstein Fano fivefolds.

Every Picard number 2, genus 10 K3 surface with a $g^1_5$ appears as a linear section of $ ilde{G}_2$.

Abstract

We introduce a variety $\hat{G}_2$ parameterizing isotropic five-spaces of a general degenerate four-form in a seven dimensional vector space. It is in a natural way a degeneration of the variety $G_2$, the adjoint variety of the simple Lie group $\mathbb{G}_2$. It occurs that it is also the image of $\mathbb{P}^5$ by a system of quadrics containing a twisted cubic. Degenerations of this twisted cubic to three lines give rise to degenerations of $G_2$ which are toric Gorenstein Fano fivefolds. We use these two degenerations to construct geometric transitions between Calabi--Yau threefolds. We prove moreover that every polarized K3 surface of Picard number 2, genus 10, and admitting a $g^1_5$ appears as linear sections of the variety $\hat{G}_2$.