When talking to students in mathematics at the beginner's level, one often gets the questions: How is it possible to do research in mathematics? Isn't all mathematics already explored? It is rather hard to be able to explain that the converse is true, that the research and knowledge increases at greater speed every decade. One reason for the difficulty to discuss this is of course that the level of abstraction of mathematics has constantly increased. This is true for most parts of mathematics, in particular for commutative algebra and (especially for) algebraic geometry. Since the 1960s there is however also a trend in the opposite direction in these fields. Interest in constructiveness and concrete calculations is also now increasing. One important reason for this is the growth in the capability of computers. It has become possible to carry out calculations that one could only dream of earlier. It is also possible to perform experiments on a large scale.