Meanwhile, bring a large pot of water to a boil. Add the pasta to the boiling water and cook until just al dente'. Drain the pasta, and then add it to the skillet with the chorizo. Toss well.
Add the arugula to the skillet, stirring as it wilts down into the pasta and chorizo. (Cover the skillet to aid the wilting process.) After the arugula has wilted, add the Parmesan cheese. Toss again until the cheese has melted and all ingredients are well combined.