On smooth Fano threefolds with coregularity zero

TL;DR

This paper demonstrates that every family of smooth Fano threefolds contains at least one example with coregularity zero, expanding understanding of their geometric properties.

Contribution

It provides explicit examples of smooth Fano threefolds with coregularity zero and shows this property holds across all families of such threefolds.

Findings

Explicit examples of smooth Fano threefolds with coregularity zero.

All families of smooth Fano threefolds contain an element with coregularity zero.

Abstract

We provide examples of smooth three-dimensional Fano complete intersections of dergee 2, 4, 6, and 8 that have coregularity 0. Considering the main theorem of arXiv:2309.16784 on the remaining 101 families of smooth Fano threefolds, our result implies that each family of smooth Fano threefolds has an element of coregularity zero.