Hyperplane sections, Gröbner bases, and Hough transforms

The purpose of this paper is twofold. In the first part we concentrate on hyperplane sections of algebraic schemes, and present results for determining when Gröbner bases pass to the quotient and when they can be lifted. The main difficulty to overcome is the fact that we deal with non-homogeneous ideals. As a by-product we hint at a promising technique for computing implicitization efficiently.In the second part of the paper we deal with families of algebraic schemes and the Hough transforms. We compute the dimension of the Hough transforms, and show that in some interesting cases it is zero. Then we concentrate on their hyperplane sections; some results and examples hint at the possibility of reconstructing external and internal surfaces of human organs from the parallel cross-sections obtained by tomography.