Optimal upper bound for degrees of canonical Fano threefolds of Picard number one

TL;DR

This paper establishes an optimal upper bound of 72 for the degree of certain Fano threefolds with Picard number one, using a Kawamata--Miyaoka type inequality involving Chern classes.

Contribution

The paper introduces a new inequality relating the degree of Fano threefolds to their Chern classes, leading to the first sharp upper bound for these varieties.

Findings

Degree of canonical Fano threefolds with Picard number one is at most 72.

The inequality connects $(-K_X)^3$ with $ ilde{c}_2(X) imes c_1(X)$.

Provides a tool for bounding degrees of Fano varieties using Chern class relations.

Abstract

We show that for a $\mathbb Q$-factorial canonical Fano $3$-fold $X$ of Picard number $1$, $(-K_X)^3\leq 72$. The main tool is a Kawamata--Miyaoka type inequality which relates $(-K_X)^3$ with $\hat{c}_2(X)\cdot c_1(X)$, where $\hat{c}_2(X)$ is the generalized second Chern class.