Leah Gold

Abstract: We discuss a bound on the multiplicity of ideals in the first chapter. Herzog and Srinivasan have conjectured that for any homogeneous k-algebra of codimension d, the multiplicity is bounded by
$$
\prod_{i=1}^d\frac{M_i}{d!},
$$
where $M_i$ is the maximal degree of an ith syzygy. Using the minimal free resolutions as constructed by Peeva and Sturmfels, we show that this bound holds for codimension 2 lattice ideals.

The topic of chapter two is ideals having three generators. It is known due to work by Burch and by Kohn that any projective dimension may be realized by a 3-generated ideal. Buchsbaum and Eisenbud conjectured that the tail of any resolution may be realized by a 3-generated ideal. This result and a generalization was shown by Bruns. We introduce a family of ideals and prove that for each n the resolution of the ideal in n variables from the family has the same tail as the Koszul resolution on those n variables.

In chapter three we attempt to extend the result of chapter two to ideals generated by binomials. We display binomial examples in 4, 5 and 6 variables. A binomial example having the same tail as the Koszul complex for n = 7 is not found, but we do display 3-generated binomial examples having projective dimensions seven and eight.