Abstract
AbstractThis 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.
Highlights
Let G be the Grassmannian of n-spaces of P2n+c, c ≥ 1
For any u ∈ G we denote by Pu its corresponding n-space and by σu the codimension c Schubert variety σu := {e ∈ G | Pu ∩ Pe = ∅}
Relying on the geometry of singular genus 6 canonical curves, we describe these configurations of length k ∈ [11, 20]
Summary
Let G be the Grassmannian of n-spaces of P2n+c, c ≥ 1. This family depends on 9 moduli and defines a divisor in the moduli space of finite Morin configurations. We prove that any smooth configuration F of length k ≥ 16 is Morin-Del Pezzo and that: Theorem 1.3. There exists a natural biregular map between F − {u} and Sing VA This relates the study of Morin configurations F of higher length to the study of hypersurfaces V of bidegree (2, 2) in P2 × P2 such that Sing V is finite. Since a finite Morin configuration spans a 9-space, it follows that FC has length k ≥ 11 and, Sing C necessarily spans P5C. Further notations X linear span of X. [X] vector space generated by X
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