To prepare the arugula pesto, in a food processor or blender, combine the arugula, cheese, pecans and garlic and process until finely chopped. With the machine running, slowly add the olive oil and lemon juice and process until smooth. Add salt to taste. Set aside.

To prepare the vegetables and pasta, bring a large pot of salted water to a boil over high heat. Add the potatoes and cook until tender but not too soft. Using a slotted spoon, transfer the potatoes to a large bowl. Add the green beans to the boiling water and cook for 2 minutes. Drain and transfer to the bowl with the potatoes.

Meanwhile, bring another pot of water to a boil. Add the pasta and cook until al dente. Drain the pasta and transfer to the bowl with the potatoes and green beans. 

Add the arugula pesto to the bowl and toss to coat the pasta and vegetables, adding a little olive oil if it seems too dry. Top with grated cheese and serve.