On Morin Configurations of Higher Length. (9th July 2020)

• Record Type: Journal Article
• Authors: Kapustka, Grzegorz; Verra, Alessandro
• Is Part Of: International mathematics research notices. Volume 2022:Number 1(2022)
• Page Start: 727
• Page End: 772
• Publication Date: 2020-07-09
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa170
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: This paper studies finite Morin configurations $F$ of planes in $\mathbb P^5$ having higher length—a question naturally related to the theory of Gushel–Mukai varieties. The uniqueness of the configuration of maximal cardinality $20$ is proven. This is related to the canonical genus $6$ curve $C_{\ell }$ union of the $10$ lines in a smooth quintic Del Pezzo surface $Y$ in $\mathbb P^5$ and to the Petersen graph. More in general an irreducible family of special configurations of length $\geq 11$, we name as Morin–Del Pezzo configurations, is considered and studied. This includes the configuration of maximal cardinality and families of configurations of lenght $\geq 16$, previously unknown. It 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.