Various Special Fiber Ring Problems

In this thesis we explore three projects, all related to the special fiber ring somehow. Consider a rational map from projective d-1 space to projective m-1 space given by m many degree δ forms, g_1,...,g_m, in d variables. Let I be the ideal generated by these forms. The special fiber ring is the ring corresponding to the closure of the image of this map, denoted F(I).

In the first project we restrict to the case d = 2 and m = 3. In this case Proj(F(I)) is a rational plane curve, and we analyze its singularities using the syzygy matrix of I. We are specifically interested in cusps. Motivated by this and the classical Plücker relations, we exhibit a curious categorization of the dual curve using the syzygy matrix. We then use this to give previously unknown bounds on the number of cusps for fixed splitting types.

In the second project we ask for fixed d, m, and δ, as the g_i vary what is the minimal multiplicity of F(I)? This study leads us to investigate the case when I is a strongly stable ideal. We give classification results for minimal multiplicity strongly stable elements and also study when an equigenerated 2-Borel ideal is Cohen-Macaulay.

Finally, in the third project, we investigate the case where I defines a Gorenstein-linear ideal for d ≥ 4. In this case, we are most interested in the defining equations of the special fiber ring. We describe the symmetric Gorenstein-linear ideals when δ = 3 and use SAGBI basis techniques to exhibit some evidence for a conjecture.