On Morin Configurations of Higher Length

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.

Wahl maps and extensions of canonical curves and K⁢3K3 surfaces

Let C be a smooth projective curve (resp. {(S,L)} a polarized {K3} surface) of genus {g\geqslant 11}, with Clifford index at least 3, considered in its canonical embedding in {\mathbb{P}^{g-1}} (resp. in its embedding in {|L|^{\vee}\cong\mathbb{P}^{g}}). We prove that C (resp. S) is a linear section of an arithmetically Gorenstein normal variety Y in {\mathbb{P}^{g+r}}, not a cone, with {\dim(Y)=r+2} and {\omega_{Y}=\mathcal{O}_{Y}(-r)}, if the cokernel of the Gauss–Wahl map of C (resp. {\operatorname{H}^{1}(T_{S}\otimes L^{\vee})}) has dimension larger than or equal to {r+1} (resp. r). This relies on previous work of Wahl and Arbarello–Bruno–Sernesi. We provide various applications.

Syzygies of Prym and paracanonical curves of genus 8

By analogy with Green's Conjecture on syzygies of canonical curves, the Prym-Green conjecture predicts that the resolution of a general level p paracanonical curve of genus g is natural. The Prym-Green Conjecture is known to hold in odd genus for almost all levels. Probabilistic arguments strongly suggested that the conjecture might fail for level 2 and genus 8 or 16. In this paper, we present three geometric proofs of the surprising failure of the Prym-Green Conjecture in genus 8, hoping that the methods introduced here will shed light on all the exceptions to the Prym-Green Conjecture for genera with high divisibility by 2. Comment: 23 pages; to appear in EPIGA Vol 1, paper no. 7