# On Morin configurations of higher length

## TL;DR

This paper investigates finite Morin configurations of planes in projective 5-space, establishing the uniqueness of the maximal configuration, constructing new families of configurations, and exploring their connections to special threefolds and classical algebraic geometry objects.

## Contribution

It proves the uniqueness of the maximal Morin configuration of length 20 and introduces new families of configurations of length ≥16, linking them to classical algebraic geometry structures.

## Key findings

- Maximal Morin configuration of length 20 is unique.
- Constructed new families of configurations with length ≥16.
- Explored relations between configurations and special threefolds like Igusa quartic.

## Abstract

This paper studies finite Morin configurations $F$ of planes in $\mathbb P^5$ having higher length. The uniqueness of the configuration of maximal cardinality $20$ is proven. This is related to the stable canonical genus $6$ curve $C_{\ell}$ union of the $10$ lines of a smooth quintic Del Pezzo surface $Y$ in $\mathbb P^5$ and to the Petersen graph. Families of length $\geq 16$, previously unknown, are constructed by smoothing partially $C_{\ell}$. A more general irreducible family of special configurations of length $\geq 11$, we name as Morin-Del Pezzo configurations, is considered and studied. This depends on $9$ moduli and is defined via the family of nodal and rational canonical curves of $Y$. The special relations between Morin-Del Pezzo configurations and the geometry of special threefolds, like the Igusa quartic or its dual Segre primal, are focused.