Preheat oven to 350°.
Place hazelnuts on a baking sheet. Bake at 350° for 10 minutes, stirring once. Cool 2 minutes; turn nuts out onto a towel. Roll up towel; rub off skins. Chop nuts; set aside.
Combine vinegar, olive oil, salt, and pepper in a small bowl, stirring well with a whisk.
Combine fennel, persimmon, and arugula in a large bowl. Drizzle with vinegar mixture; toss well to coat. Place 1 1/2 cups fennel mixture on each of 6 plates. Top each serving with about 1 teaspoon hazelnuts. Serve immediately.