Geometric modelers study parameterized curves and surfaces in the plane and 3-space. Their method of "moving curves" and "moving surfaces" led them to discover the polynomial relations defining the Rees algebra of the ideal generated by the polynomials that give the parametrization. At the time they did this, they had no idea what a Rees algebra was. Meanwhile, the commutative algebra community had studied for many years the Rees algebras of various types of ideals, though not those that arise from geometric modeling. Recently, the interests of these groups have converged, leading to new results and new hard problems to think about. My lecture will describe joint work with Kustin, Polini and Ulrich on the test case of rational plane sextics and the special role played by double and triple points. K3 surfaces will make a brief appearance toward the end of the talk in connection with my fantasy that they might help distinguish between certain types of Rees algebras.

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