Monodromy arguments and deformation-and-specialization are used to prove existence of a pencil of Enriques surfaces with no section and index 1. The same technique “completes” the strategy from Graber et al. (2005) proving that the family of witness curves for dimension depends on the integer .
"A pencil of Enriques surfaces of index one with no section." Algebra Number Theory 3 (6) 637 - 652, 2009. https://doi.org/10.2140/ant.2009.3.637