Coregularity of smooth Fano threefolds

TL;DR

This paper investigates the coregularity of smooth Fano threefolds, showing that most families have general members with coregularity 0, but some families have members with positive coregularity, revealing new structural insights.

Contribution

It provides a comprehensive classification of coregularity for most families of smooth Fano threefolds, including partial results for the remaining families.

Findings

100 out of 105 families have general members with coregularity 0

92 families have all members with coregularity 0

Existence of families with members having positive coregularity

Abstract

We study the coregularity of smooth Fano threefolds. We prove that for 100 out of 105 families of smooth Fano threefolds, a general member in the family has coregularity 0; moreover, for 92 families out of these 100, any member in the family has coregularity 0; for the remaining 5 families, we obtain some partial results. In particular, we show that there exist families of smooth Fano threefolds whose general elements have positive coregularity.