Interactive Visualizations of Blowups of the Plane

Blowups are an important technique in algebraic geometry that permit the smoothing of singular algebraic varieties. It is a challenge to visualize this process even in the case of blowups of points $(X)$ in the affine plane $({\hbox{\rlap{A}\kern 2.0pt{\hbox{A}}}}^2_{{\rlap{\rm I}\kern 2.0pt{{\rm R}}}})$. First results were obtained by Brodmann with the aid of the so-called toroidal blowup, a compact embedding of the blowup into affine 3-space. In fact, Brodmann provides a rational parametrization of the toroidal blowup, but its visualization fails in the neighborhood of $(X)$ because the parametrization tends to indefinite terms of the form $({0\over 0})$. Our approach is based on implicitization of the parametric form. By methods from commutative algebra we are able to reduce the implicitization to the computation of a single, fairly simple resultant. This provides an algebraic equation of the implicit surface of the toroidal blowup including the so-called exceptional fiber associated with $(X)$. Surprisingly, the degree of the equation grows only linearly with the degree of the parametrization. By applying additional clipping techniques to the implicit surface we are able to visualize the toroidal blowup as well as its deformations by several parameters interactively in real time using GPU-based ray casting techniques. The methods of the paper provide insights in the structure of blowups of points, even if the points are interactively moved or tend to degenerations.