Fano fourfolds of type A beta

About this database

A preliminary catalogue of 823 families, with 919 zero-locus presentations. These are smooth Fano fourfolds (complex dimension 4) presented on products of type A partial flag varieties, including ordinary Grassmannians. Read more about what we are doing on the blog.

We enumerate globally generated, completely reducible equivariant bundles $\mathcal{E}$ whose general sections have four-dimensional zero loci $X\subset Y$. The current search uses bounds on codimension and on the number of ambient factors.

A presentation is strongly Fano when the line bundle $L=\omega_Y^\vee\otimes(\det\mathcal{E})^\vee$ is ample on the ambient variety $Y$. Adjunction gives $\omega_X^\vee\cong L|_X$, so the zero locus is Fano: its anticanonical bundle is ample.

Presentations are grouped into deformation families, each with a numerical label and a representative construction. Explore the families or read about the labels. This is work in progress: more examples and data will follow.

Available data includes:

• The ambient variety and defining bundle
• Hodge and Chern numbers
• Numerical invariants: $\mathrm{h}^0(X, \omega_X^\vee)$, $\operatorname{vol}(X) = \mathrm{c}_1^4$, $\chi(X, \mathrm{T}_X)$
• Hochschild cohomology and the polyvector parallelogram
• Primitive Hilbert series and Fano indices
• Regularized quantum periods and flag-ambient quantum spectra

Please contact us at [email protected] (or email one of us) if you have any comments, remarks, or questions.