(The) degree complexity via Generic initial ideals and applications

Abstract

In this thesis, we study the degree complexity of an integral curve in $\mathbb{p^{3}}$ and a smooth surface in $\mathbb{p^{4}}$ with respect to the lexicographic order. Let $\It{I}$ be the defining ideal of an integral curve in $\mathbb{p^{3}}$ of degree d and arithmetic genus $\rho_{a}}. If we fix a term order as the lexicographic order. Then the degree complexity of $\It{I}$ in generic coordinates is $1+(\binom{d-1}{2})-\rho_{a}}$ with exception of two cases. Additionally if $S \subset\p^{4}$ is smooth surface cut out by quadric and $\It{I_S}$ is the defining ideal of $\It{S}$ then the degree complexity of $I_S$ in generic coordinates $\It{M(I_{S})}$ is $1+\binom{d_{1}-1}{2}-\rho_{a}(Y_{1})$, where $d_{1}=\deg(Y_{1}(S))=\binom{d-1}{2}-\rho_{a}(S \cap H)$ with exception of three cases. If $\It{S}$ is a rational normal scroll then $\It{M(I_{S})=3}$, if $\It{S}$ is a complete intersection of (2,2)-type then $\It{M(I_{S})=4}$, and if $\It{S}$ is a Castelnuovo surface then $\It{M(I_{S})=5}$.