On the Coordinate Ring of a Projection of a Degree Two Veronese Variety

The goal of this thesis is to study Rees algebra R(I) and the special fiber ring F(I) for a family of ideals. Given a map between projective spaces parameterizing a variety, the Rees algebra is the coordinate ring of the graph and the special fiber ring is the coordinate ring of the image. We will compute the defining ideal of these algebras. Let R=k[x_1, …, x_d] for d greater than or equal to 4 be a polynomial ring with homogeneous maximal ideal m. We study the R-ideals I which are m-primary, Gorenstein, generated in degree 2, and have a Gorenstein linear resolution. The defining ideal of the Rees algebra will be of fiber type. That is, the defining ideal of the Rees algebra is generated by the defining ideals of the special fiber ring and of the symmetric algebra. The defining ideal of the symmetric algebra is well understood, so we concentrate on computing the defining ideal of the special fiber ring. In Chapter 4, the defining ideal of the special fiber ring F(I) will be given as a sub-ideal of the 2x2 minors of a symmetric matrix of variables modeled after the defining ideal of F(m^2). In Chapter 5, the defining ideal of the special fiber ring of I will be described as a saturation of the maximal minors of the Jacobian dual. We include both descriptions of the defining ideal in this manuscript because while the methods in Chapter 4 give explicit polynomial generators of the defining ideal, the methods in Chapter 5 are more likely to generalize to the larger class of m-primary Gorenstein ideals having a Gorenstein linear resolution.