Rational curves on O'Grady's tenfolds

Abstract

We study the existence of ample uniruled divisors on irreducible holomorphic symplectic manifolds that are deformation of the ten dimensional example introduced by O'Grady in [O'G99]. In particular, we show that for any polarized OG 10 manifold lying in four specific connected components of the moduli space of polarized OG 10 manifolds there exists a multiple of the polarization that is the class of a uniruled divisor.

Deformations of rational curves on primitive symplectic varieties and applications

“…Ample uniruled divisors on irreducible symplectic manifolds of K3 [n] , Kum n or OG10 deformation types are investigated respectively in [CMP19, MP18,Ber21]. The OG6 deformation type is the object of an ongoing project by Bertini, Grossi, and Onorati.…”

Rational curves on primitive symplectic varieties of OG6 singular type

“…This problem has been firstly investigated in the smooth setting, where the existence of ample uniruled divisors on an ihs manifold allows to build a canonical subgroup of the 0-Chow group of the ihs manifold [CMP19, Theorem 1.5], giving a first evidence of Voisin's geometrical realization of the conjectural Bloch-Beilinson filtration of the 0-Chow group [Voi16]. Uniruled divisors are known to exist in almost any ample linear system on ihs manifolds of K3 [n] -type [CMP19] and of K n (A)-type [MP17] [MP19], and in any ample linear system with some fixed numerical invariants for ihs manifolds of OG 10 -type [Ber21]. The key ingredient in all these cases is a result by Charles-Mongardi-Pacienza [CMP19] about deformations of rational curves ruling a divisor on an ihs manifold.…”