Intersection multiplicty and scheme-theoretic length of algebraic varieties

Abstract

In this thesis, we first see the intersection multiplicity and the scheme-theoretic length of the intersection. And then we look around Cohen-Macaulayness and its meaning in the geometric case. Next we will show the boundedness of the length when $X$ is locally Cohen-Macaulay. More precisely, $length (X ∩ L) ≤ d - e + β if $X^n ⊂ \textbf{P}^{n+e}$ is locally Cohen-Macaulay and $L = \textbf {P}^{β} (1≤ β ≤ e)$ is a linear secant subspace of dimension β to X. Finally, we find an example which shows that the bound is false in case of non-locally Cohen-Macaulay variety.