We study the existence of ample uniruled divisors on irreducible holomorphic symplectic manifolds that are deformations of the 10-dimensional example introduced by O’Grady. In particular, we show that for any polarized O’Grady’s 10-dimensional ($OG10$) manifold lying in four specific connected components of the moduli space of polarized $OG10$ manifolds, there exists a multiple of the polarization that is the class of a uniruled divisor.

In this paper, we prove an explicit matching theorem for some Hecke elements in the case of (possibly ramified) cyclic base change for general linear groups over local fields of characteristic zero with odd residual characteristic under a mild assumption. A key observation, based on the works of Waldspurger and Ganapathy–Varma, is to regard the base change lifts with twisted endoscopic lifts and replace the condition for the matching orbital integrals with one for semisimple descent in the twisted space according to Waldspurger’s fundamental work, “L’endoscopie tordue n’est pas si tordue.”

Questions that seek to determine whether a hyperplane arrangement property, be it geometric, arithmetic, or topological, is of a combinatorial nature (i.e., determined by the intersection lattice) are abundant in the literature. To tackle such questions and provide a negative answer, one of the most effective methods is to produce a counterexample. To this end, it is essential to know how to construct arrangements that are lattice equivalent. The more different they are, the more efficient it will be. In this paper, we present a method to construct arrangements of complex projective lines that are lattice equivalent but lie in distinct connected components of their moduli space. To illustrate the efficiency of the method, we apply it to reconstruct all the classical examples of arrangements with disconnected moduli spaces: MacLane, Falk–Sturmfels, Nasir–Yoshinaga, and Rybnikov. Moreover, we employ this method to produce novel examples of arrangements of eleven lines whose moduli spaces are formed by four connected components.