# Remarks on defective Fano manifolds

## TL;DR

This paper discusses the properties of prime Fano manifolds, completing previous research on secant defective and dual defective manifolds, and provides additional insights into their geometric structure.

## Contribution

It offers new remarks on prime Fano manifolds, advancing the understanding of their secant and dual defectiveness, and completes earlier classifications.

## Key findings

- Prime Fano manifolds are now better understood in the context of secant and dual defectiveness.
- The paper completes previous classifications of special defective manifolds.
- Additional geometric properties of Fano manifolds are discussed.

## Abstract

This note continues our previous work on special secant defective (specifically, conic connected and local quadratic entry locus) and dual defective manifolds. These are now well understood, except for the prime Fano ones. Here we add a few remarks on this case, completing the results in our papers \cite{LQEL I}, \cite{LQEL II}, \cite{CC}, \cite{HC} and \cite{DD}; see also the recent book \cite{Russo}