On projective varieties of almost minimal degree

Abstract

In this thesis, we study varieties of almost minimal degree (i.e. $\mbox{deg} (X) = \mbox{codim}(X, \P^r)+2$). To understand these varieties, many people have investigated their structure theories which are cohomological property, local property and homological property. First we study the problem to determine all occurring Betti diagrams of varieties $X \subset \P^r$ of almost minimal degree and describe a realistic picture of how many different kind of Betti diagrams exist at all. Next we study the problem to classify non-normal varieties of almost minimal degree with low codimension in the viewpoint of projective equivalence. If $X \subset \P^r$ is an irreducible non-normal variety of almost minimal degree which is not a cone. Then for $\mbox{codim}(X, \P^r)=1$, we prove that $r \leq 4$ and there are precisely five (resp. six) irreducible non-normal cubic equations when $\mbox{char}~K \neq 2,3$ (resp. when $\mbox{char}~K = 2,3$), up to projective equivalence. Finally for $\mbox{codim}(X, \P^r)=2$, we prove that $r \leq 5$ and there are six (resp. nine) irreducible non-normal complete intersection of two quadrics when char $\mbox{K} \neq 2$ (resp. whenchar $\mbox{K} = 2$), up to projective equivalence. Also we describe the normalization of $X$ in detail.