Most references to Italian pesto call it pesto alla Genovese. But this is not a way of making pesto. The olive oil used for pesto, like the basil, should be delicate in quality. Ligurian oil is fruity and light, a far cry from the throat-catching Tuscan oils. Because pesto is not a cooked sauce, the quality of the salt is important.
Table salt can be too harsh.
Coarse sea salt is preferred, but kosher salt will do in a pinch. Speaking of harsh, go easy on the garlic. Italian garlic tends to be smaller and less pungent than the garlic most Americans use. Though pine nuts are preferred for their oiliness and sweetness, it is not uncommon to see walnuts, which give a drier, more tannic bite to the sauce. The classic pesto usually has two cheeses, Parmigiano-Reggiano, which is rich and creamy, and the sharper Pecorino Romano.